MBRIDGE TRACTS IN MATHEMATICS
125
THE
HARDY-LITTLEWOOD
METHOD
SECOND EDITION
R. C. VAUGHAN

CAMBRIDGE TRACTS IN MATHEMATICS
General Editors
B. BOLLOBAS, F. KIRWAN, C.T.C. WALL &
P. SARNAK
125 The Havdy-Littlewood method

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R.C. VAUGHAN
Professor of Pure Mathematics and EPSRC Senior Fellow
Imperial College, University of London
The Hardy-Littlewood
method
Second Edition
V
i i
Cambridge
UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY
OF CAMBRIDGE
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CAMBRIDGE UNIVERSITY PRESS
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© Cambridge University Press 1982, 1997
This book is in copyright. Subject to statutory exception
and to the provisions of relevent collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1981
Second edition 1997
Printed in the United Kingdom at the University Press, Cambridge
Typeset in Times 10/12pt
A catalogue record of this book is available from the British Library
Library of Congress cataloguing in publication data
Vaughan, R. C.
The Hardy-Littlewood method/R. C. Vaughan. - 2nd ed.
p. cm - (Cambridge tracts in mathematics; 125)
Includes bibliographical references (p. - ) and index.
ISBN 0 521 57347 5
1. Hardy-Littlewood method. I. Title. II. Series.
QA241.V34 1997
512'.74-dc20 96-19434 CIP
ISBN 0 521 57347 5 hardback
VN

Contents
Preface
Preface to second edition
Notation
1 Introduction and historical background
1.1 Waring's problem
1.2 The Hardy-Littlewood method
1.3 Goldbach's problem
1.4 Other problems
1.5 Exercises
2 The simplest upper bound for G(k)
2.1 The definition of major and minor arcs
2.2 Auxiliary lemmas
2.3 The treatment of the minor arcs
2.4 The major arcs
2.5 The singular integral
2.6 The singular series
2.7 Summary
2.8 Exercises
3 Goldbach's problems
3.1 The ternary Goldbach problem
3.2 The binary Goldbach problem
3.3 Exercises
4 The major arcs in Waring's problem
4.1 The generating function
4.2 The exponential sum S(q, a)
4.3 The singular series
4.4 The contribution from the major arcs
ix
xi
xiii
1
1
3
6
7
7
8
8
9
14
14
18
20
24
25
27
27
33
36
38
38
45
48
51

vi Contents
4.5 The congruence condition 53
4.6 Exercises 55
5 Vinogradov's methods 57
5.1 Vinogradov's mean value theorem 57
5.2 The transition from the mean 63
5.3 The minor arcs in Waring's problem 69
5.4 An upper bound for G(k) 70
5.5 Wooley's refinement of Vinogradov's mean
value theorem 75
5.6 Exercises 92
6 Davenport's methods 94
6.1 Sets of sums of /cth powers 94
6.2 G(4)= 16 105
6.3 Davenport's bounds for G(5) and G(6) 108
6.4 Exercises 109
7 Vinogradov's upper bound for G(k) 111
7.1 Some remarks on Vinogradov's mean
value theorem 111
7.2 Preliminary estimates 112
7.3 An asymptotic formula for JS(X) 119
7.4 Vinogradov's upper bound for G(k) 122
7.5 Exercises 125
8 A ternary additive problem 127
8.1 A general conjecture 127
8.2 Statement of the theorem 128
8.3 Definition of major and minor arcs 128
8.4 The treatment of u 130
8.5 The major arcs y\(q, a) 135
8.6 The singular series 136
8.7 Completion of the proof of Theorem 8.1 144
8.8 Exercises 146

Contents vii
9 Homogeneous equations and Birch's theorem 147
9.1 Introduction 147
9.2 Additive homogeneous equations 147
9.3 Birch's theorem 151
9.4 Exercises 154
10 A theorem of Roth 155
10.1 Introduction 155
10.2 Roth's theorem 156
10.3 A theorem of Furstenburg and Sarkozy 161
10.4 The definition of major and minor arcs 162
10.5 The contribution from the minor arcs 164
10.6 The contribution from the major arcs 164
10.7 Completion of the proof of Theorem 10.2 165
10.8 Exercises 166
11 Diophantine inequalities 167
11.1 A theorem of Davenport and Heilbronn 167
11.2 The definition of major and minor arcs 168
11.3 The treatment of the minor arcs 169
11.4 The major arc 172
11.5 Exercises 174
12 Wooley's upper bound for G(k) 175
12.1 Smooth numbers 175
12.2 The fundamental lemma 177
12.3 Successive efficient differences 186
12.4 A mean value theorem 187
12.5 Wooley's upper bound for G(k) 191
12.6 Exercises 193
Bibliography 195
Index 229

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Preface
There have been two earlier Cambridge Tracts that have touched
upon the Hardy-Littlewood method, namely those of Landau, 1937,
and Estermann, 1952. However there has been no general account of
the method published in the United Kingdom despite the not
inconsiderable contribution of English scholars in inventing and
developing the method and the numerous monographs that have
appeared abroad.
The purpose of this tract is to give an account of the classical forms
of the method together with an outline of some of the more recent
developments. It has been deemed more desirable to have this
particular emphasis as many of the later applications make important
use of the classical material.
It would have been useful to devote some space to the work of
Davenport on cubic forms, to the joint work of Davenport and Lewis
on simultaneous equations, to the work of Rademacher and Siegel
that extends the method to algebraic numbers, and to the work of
various authors, culminating in the recent work of Schmidt, on
bounds for solutions of homogeneous equations and inequalities.
However this would have made the tract unwieldy. The interested
reader is referred to the Bibliography.
It is assumed that the reader has a familiarity with the elements of
number theory, such as is contained in the treatise of Hardy and
Wright. Also, in dealing with one or two subjects it is expected that the
reader has a working acquaintance with more advanced topics in
number theory. Where necessary, reference is given to a standard text
on the subject.
The contents of Chapters 2, 3, 4, 5, 9, 10 and 11 have been made
the basis of advanced courses offered at Imperial College over a
number of years, and could be used as part of any normal
postgraduate training in analytic number theory.

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Preface to second edition
At the time that the first edition was written, there had been relatively
little recent work on the central theory of the Hardy-Littlewood
method, namely that surrounding Waring's problem and associated
questions. Indeed, the work of Davenport and Vinogradov had taken
on the aspect of being written on tablets of stone. This is in complete
contrast to the current situation. In the last decade or so there has
been a series of important developments in the area. The tract is,
therefore, ripe for revision, and the opportunity has been taken to
give an introduction to this new material, and especially to the
important work of Wooley. Chapter 5 has been extensively rewritten
to take account of our new understanding of Vinogradov's mean
value theorem, and a completely new chapter has been added to
describe the new work on Waring's problem. Fortunately the large
bulk of the material has not been superseded and the underlying ideas
still play an important role in many of the new developments.

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Notation
The letter k denotes a natural number, usually with k ^ 2, and the
statements in which e appear are true for every positive real number s.
The letter p is reserved for prime numbers.
The Vinogradov symbols <^ , > have their usual meaning, namely
that for functions / and g with g taking non-negative real values / <^ g
means \f\ ^ Cg where C is a constant, and if moreover f is also non-
negative, then f 5> g means g <^ f.
Implicit constants in the O, <^ and > notations usually depend on
/c, s and e. Additional dependence will be mentioned explicitly.
As usual in number theory, the functions e(oc) and || a || denote e2m*
and min |a — h\ respectively. Occasionally the expression
heZ
min(X, 1/0) occurs, and is taken to be X.
The notation pr \\ n is used to mean that pr is the highest power of p
dividing n.

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1
Introduction and historical
background
1.1 Waring's problem
In 1770 E. Waring asserted without proof in his Meditationes
Algebraicae that every natural number is a sum of at most nine
positive integral cubes, also a sum of at most 19 biquadrates, and so
on. By this it is usually assumed that he believed that for every natural
number k > 2 there exists a number s such that every natural number
is a sum of at most s /cth powers of natural numbers, and that the least
such s, say g(k\ satisfies g(3) = 9, g(4) = 19.
It was probably known to Diophantus, albeit in a different form,
that every natural number is the sum of at most four squares. The four
square theorem was first stated explicitly by Bachet in 1621, and a
proof was claimed by Fermat but he died before disclosing it. It was
not until 1770 that one was given, by Lagrange, who built on earlier
work of Euler. For an account of this theorem see Chapter 20 of
Hardy & Wright (1979).
In the 19th century the existence of g(k) was established for many
values of /c, but it was not until the present century that substantial
progress was made. First of all Hilbert (1909a, b) demonstrated the
existence of g(k) for every k by a difficult combinatorial argument
based on algebraic identities (see Rieger, 1953a, b, c; Ellison, 1971).
His method gives a very poor bound for g(k).
In the early 1920s Hardy and Littlewood introduced an analytic
method which has been the basis for numerical work by Dickson,
Pillai and others, and has led to an almost complete evaluation of
g(k). Since the integer
is smaller than 3fc it can only be a sum of/cth powers of 1 and 2. Clearly
the most economical representation is by [(f)fc] — 1 fcth powers of 2

Introduction and historical backround
and 2fc — 1 /eth powers of 1. Thus
g(k) >2k +
-2.
(1.1)
It is very plausible that this always holds with equality, and the
current state of knowledge is as follows.
It has been shown that when
^2'
one has
g(k) = 2k +
-2
but when
>2'
one has either
g(k) = 2k +
+
-2
or
g{k) = 2k +
+
-3
according as
(1.2)
(1.3)
+
+
is equal to 2k or is larger than 2k. For the various contributions to the
proof of this, see the Bibliography.
Stemmler (1964) has verified on a computer that (1.2) (and so (1.3))
holds whenever k ^200000, and this has been extended to
471 600000 by Kubina and Wunderlich (to appear). Mahler (1957)
has shown that if there are any values of k for which (1.2) is false, then
there can only be a finite number of such values. No exceptions are
known, and unfortunately the method will not give a bound beyond
which there are no exceptions.

The Hardy-Littlewood method 3
1.2 The Hardy-Littlewood method
Nearly all the above conclusions have been obtained in the following
way. A theoretical argument based on the analytic method of Hardy
and Littlewood produces a number Ck such that every natural
number larger than Ck is the sum of at most sk /cth powers of natural
numbers where sk does not exceed the expected value of g(k). Then a
rather tedious, but often very ingenious, calculation enables a check
to be made on all the natural numbers not exceeding Ck.
One of the features of the Hardy-Littlewood method is that it can
be adapted to attack many other problems of an additive nature. The
method has its genesis in a paper of Hardy & Ramanujan (1918)
concerned mainly with the partition function, but also dealing with
the representation of numbers as sums of squares.
Let s& = (am) denote a strictly increasing sequence of non-negative
integers and consider
00
F(z)= ^ zflm (M<1)
and its sth power
m= 1
00 00 00
F(zY= I ... I **»■ + ■••+«»..= £ Rs(n)z",
mi = 1 ms= 1 n = 0
where Rs(n) is the number of representations of n as the sum of s
members of stf. The objective is an estimate for Rs(n\ at least when n is
large. By Cauchy's integral formula
R,(n) =
2ni
F(z)sz-"_1dz
where ^ is a circle centre 0 of radius p, 0 < p < 1.
Hardy and Ramanujan discovered an alternative way of evaluating
the integral when am = m2. Suppose that p = 1 — £ and that n is large,
and write e(a) = elnm. Then the function F has 'peaks' when z = pe(a)
is 'close' to the point e(a/q) with q 'not too large'. In fact, F has an
asymptotic expansion in the neighbourhood of such points, roughly
speaking valid when |a —a/q\ ^ l/(q^/n) and q ^y/n. By Dirichlet's
theorem on diophantine approximation every z under consideration
is in some such neighbourhood.

Introduction and historical backround
The asymptotic expansion takes the form
C
S(q9a)(l-pe(p)y112 (1.4)
where
S(q, fl) = X e(#w2/g).
m=l
This can be seen by dealing first with the case /? = 0 by partitioning the
squares into residue classes modulo q and then applying partial
summation. Thus, for s ^ 5 one can obtain
Ra(n)~<5s(n)Js(n) (1.5)
where
oo q
©,(«)= I I <TsS(<?,a)se(-anA?)
q = 1 a = 1
(a, q) = 1
and
Js(n) = Cs I (l-M/?)rs/VM-/?")ci/?.
J- 1/2
The integral in Js(n) is quite easy to estimate, and the series Ss(h)
reflects certain interesting number theoretic properties of the
sequence of squares.
The expansion (1.4) corresponds to a singularity of the series F at
e(a/q) on its circle of convergence, and in view of this Hardy and
Littlewood coined the terms singular series and singular integral for
®s(rc) and Js(n) respectively.
After the First World War, Hardy & Littlewood (1920, 1921)
turned their attention to Waring's problem. Unfortunately, when
am = mk with k > 3, they could only show that the expansion
corresponding to (1.4) holds when
q^n1/k £ and
a
^q-in1/k-£
and this only accounts for a small proportion of the points z on #.
Since q~ 1S(q, #)—►() as g—»00 (for (a, q) = 1) one might hope that at any
rate F is small compared with the trivial estimate (1 — p)~1/k = n1/k on
the remaining z, a hope reinforced by the fact that (cank) is uniformly
distributed modulo 1 when a is irrational. Indeed, Hardy and

The Hardy-Littlewood method 5
Littlewood were able to show that F is appreciably smaller than n1/k
on the remainder of ^ by an alternative argument having its origins in
Weyl's (1916) fundamental work on the uniform distribution of
sequences, the consequent statement about the size of F often being
called Weyl's inequality. They further introduced the terms major arcs
and minor arcs to describe the parts of ^ where they used the analogue
of (1.4) and Weyl's inequality respectively.
Later Vinogradov (1928a) introduced a number of notable
refinements, one of which was to replace F(z) by the finite sum
M= I e(amk) (1.6)
m
where
N = [n1/k~]. (1.7)
sn
Now
/(a)s = Yj Rs(m> n)e((xm)
m = 1
where Rs(m,n) is the number of representations of m as the sum of s /cth
powers, none of which exceed n. Thus
Rs(m, n) = Rs(m) (m ^ n).
Then a special case of Cauchy's integral formula, namely the trivial
orthogonality relation
M , , fl when h = 0
e(*h)dz = \ (1.8)
o (0 when h f= 0
gives
n
f(oL)se(-(xn)doL = Rs(n). (1.9)
o
It is clear from the discussions above that g(k) is determined by
the peculiar demands of a few relatively small exceptional natural
numbers. Thus the more interesting problem is that of the estimation
of the number G(/e), defined for k ^ 2 to be the least s such that every
sufficiently large natural number is the sum of at most s /cth powers of
natural numbers. It transpires that G(k) is much smaller than g(k)
when k is large and this naturally makes its evaluation much more

6 Introduction and historical backround
difficult. In fact the value of G(k) is only known when k = 2 or 4,
namely
G(2) = 4, G(4) = 16,
the latter result being due to Davenport (1939c). Linnik (1943a) has
shown that G(3) ^ 7 and Watson (1951) has given an extremely
elegant proof of this. When k> 3 all the best estimates available at
present for G(k) have been obtained via the Hardy-Littlewood
method. Even when k = 3 the Hardy-Littlewood method can be
adapted to give G(3) < 7 (Vaughan, 1986c). Chapters 2,4, 5, 6, 7 and
12 are devoted to the study of G(k).
1.3 Goldbach's problem
In two letters to Euler in 1742, Goldbach conjectured that every even
number is a sum of two primes and every number greater than 2 is a
sum of three primes. He included 1 as a prime number, and so in
modern times Goldbach's conjectures have become the assertions
that every even number greater than 2 is a sum of two primes and
every odd number greater than 5 is a sum of three primes.
Hardy & Little wood (1923a,b) discovered that their method could
also be applied with success to these problems, provided that they
assumed the generalized Riemann hypothesis. Thus they were able to
show conditionally that every large odd number is a sum of three
primes and that almost every even number is a sum of two primes.
In 1937, Vinogradov was able to remove the dependence on the
generalized Riemann hypothesis, thereby giving unconditional
proofs of the above conclusions. This line of attack on Goldbach's
problems is investigated in Chapter 3. However, the nature of the
primes, and in particular the problem of their distribution in
arithmetic progressions, means that the further refinements of the
method (see Montgomery & Vaughan, 1975) are better viewed in the
context of multiplicative number theory and have therefore been
omitted from this tract.
For many generalizations of the methods described in Chapter 3
see Hua's (1965) monograph.

Exercises
1
1.4 Other problems
The last thirty years have seen a large expansion and diversity of the
applications of the method, and in Chapters 8, 9, 10, 11 a number of
topics have been chosen to illustrate this development. The
applications described there, particularly in Chapters 9 and 11 to general
forms and inequalities respectively, cover only a small part of the
work which has been undertaken in these areas, and should be viewed
as an introduction to the original papers listed in the Bibliography.
1.5 Exercises
1 Show that the number p(n) of solutions of the equation
x 1 + . . . + xs = n
in non-negative integers xl5 . . ., xs is (— 1)"(~J).
2 Show that the sum of the divisors of rc, o(ri) = £m|„ra, satisfies
n2 °°
a{n) = ~-nYJq 2cq(n)
where cq(n) is Ramanujan's sum, i.e.
q
cq(n) = Z e(an/q\
a= 1
(a, q) = 1
3 Let P,Q denote real numbers with P > 1, Q > IP. Show that the
intervals
{oi'.\oi-alq\^q-lQ-1}
with q ^ P and (a,q) = 1 are pairwise disjoint.

The simplest upper bound for G(k)
2.1 The definition of major and minor arcs
The introduction of various refinements over the years, most notably
by Hua (1938b) has led to a simple proof that G(k) ^ 2k + 1 which
nevertheless illustrates many of the salient features of the Hardy-
Littlewood method.
There is a good deal of latitude in the definition of major and minor
arcs, and the choice made here is fairly arbitrary.
Let n be large, suppose that N is given by (1.7) and that
v = 7^-, P = N\ (2.1)
100 v
and let S denote a sufficiently small positive number depending only
on k. When 1 < a < q < P and (a,q) = 1, let
sjJl(q, a) = {a : |a - a/q\ < Nv ~ *}. (2.2)
The sJJl(q, a) are called, for the historical reasons outlined above, the
major arcs, although in fact they are intervals. Let Wl denote the union
of the yJl(q, a). It is convenient to work on the unit interval
^ = (NV_\ 1+Nv_fc] (2.3)
rather than (0, 1]. This avoids any difficulties associated with
having only 'half major arcs' at 0 and 1. Observe that 9Jt c <?/. The set
m = °ll \ Wl forms the minor arcs.
When a/q =/= a'/q' and q, q < Nv, one has
a a! 1
Thus the Wl(q, a) are pairwise disjoint.
By (1.9) (for brevity the suffix s is dropped)
R(n) =
f(afe( - <xn)d(x (2.4)
m
f(oLfe( — an)d(x +
aw
where /(a) is given by (1.6). Before proceeding with the estimation of
these integrals it is necessary to establish some auxiliary lemmas.

Auxiliary lemmas 9
2.2 Auxiliary lemmas
The method for treating/(a) when a em can be outlined as follows.
When k = 1,
N
m= 1
is trivial to estimate. In the general case, an argument based on the use
of the forward difference operator enables/(a) to be estimated in terms
of sums in which mk is replaced by a polynomial of degree k — 1. Then
successive applications of this argument reduce the degree to 1.
Lemma 2.1 (Dirichlet) Let (x denote a real number. Then for each real
number X ^ 1 there exists a rational number a/q with
(a, q) = 1, 1 ^ q ^ X and
\*-a/q\^l/(qX).
Proof It suffices to prove the result without the condition
(a, q) = 1.
Let m = [X]. The m numbers Pq = aq — [ocq] (q = 1, 2, . . ., m) all
lie in [0, 1). Consider the m + 1 intervals
r-\
(r= 1,2,..., m + 1).
Br =
m + 1 m + 1
If there is a fiq in B x or Bm +1, then the proof is finished. If not, then one
of the m — 1 boxes £,. with 2 < r ^ m contains at least two of the /?4, say
Pw Pv with w < v. Take q = v — u,a = [olv\ — [aw].
Lemma 2.2 Suppose that X, 7, a are rea/ numbers with X > 1, 7^ 1,
and t/iat |a — a/gj ^ q~ 2 with (a, q) = 1. T/ien
X min(X7x"1, 1^^11^)^^7(- + ^ + -^-)^(2^)
vv/zere ||jS|| = min |/? — y\.
yel
Proof Let
S= ^ mm{XYx-\\\oix\\-1).

10 The simplest upper bound for G(k)
Clearly
q * XY _1
s^ ^ Z min ~rr~' llato' + r)ll
0 ^ j ^ X/q r = 1 \<Z/ + V
For each 7 let y} = [a/g2], and write 6 = q2cc — qa. Then
cc(qj + r) = (^ + ar)/q + {ccjq2}/q + Qrq~2.
When 7 = 0 and r ^ \q,
II a(g/ + r) II > II ar/^ || - 1 /(2g) 2* |1| ar/q ||.
Otherwise, for each j there are at most 0(1) values of r for which
lla(^7 + r)ll ^ ilK^j+ tfr)A?ll foils to hold, and moreover qj + r^>
q(j + 1). Therefore
S< I War/qr1
+ I (77^+ I 11^ + ^-1
0 < j< X/q VfV ^ L) r=l
q\y3 +ar
W4-1 I ^ + (^- + 1) ^ I
0 < j < X7 "r l 1 ^ h^q/2 n
and the lemma follows easily.
Let Aj denote the 7th iterate of the forward difference operator, so
that for any function 0 of a real variable a
A1(0(a);j8) = 0(a+ )8)-0(a),
A,-+i(0(a);/»i, • • • ,PJ+i) = AX(A.(0(a);)8,,. . . ,/?•); /J,+ 1).
Then it is an easy exercise to show that
A/ak; j8l9 . . . , /?,) = /?, . . . PjPj^Px, ..., )8,-)
where py is a polynomial in a of degree k—j which has leading
coefficient kl/(k —j)l
The following lemma is an intermediate step in the proofs of both
Lemmas 2.4 and 2.5 below.
Lemma 2.3 (Weyl) Let
T{<f>) = X e(4>(x))
x= 1

Auxiliary lemmas 11
where <p is an arbitrary arithmetical function. Then
17(4.)1^(20^-1 £ ... X Tj
\K\<Q \h}\<Q
where
7}= £ e(A/0(x);/i1?..., h,))
and the intervals Ij = Ij(h1, ..., hj) (possibly empty) satisfy
/i(/ii)c:[l,Q], Ij(hu . . . ,hj)cilj_1(hl, . . . , /ij- i).
Proo/ By induction on 7. For brevity write A/x) for
A/0(x); /i1? . . . , /ij). Obviously
|T{</>)|2 = I I* efAJx))
X = 1 /li = 1 — X
= t Z e(A,(x))
/il = 1 - Q X6/,
where /x = [1, Q] n[l - /i1? Q - fcj.
Now if the conclusion of the lemma is assumed for a particular
value of j, then by Cauchy's inequality,
|T(0)|2J+,^(202Jt'-^-2(20J' £ |T/
/ii,... ,hj
and obviously
\tj\2= I I e(A/x + /i)-A/x))
i/ii<gx6/,+1
with Ij+ != Ij<^{* 'x + nGIj}-
Lemma 2.4 (Weyl's inequality) Suppose that (a, q)= 1,
I a — a/g| ^q~2, </>(*) = ax* + ajX*-1 + . . . + 0^.^ + 0^
and
(2
7((/))= ^ *(0W).
x=l
T(0)«e1+t(«"1 + e_1+«e"*),/K
where K=2k~\

12 The simplest upper bound for G(k)
Proof By Lemma 2.3 with j = k — 1 (and Exercise 2.1),
iT«>)iK^(2ef-'
X
Z ••• Z Z ^i •• A-iPfc-1(*;^1,..-,fc*-i))
\hj\*Q k l
with
Pfc-i(*; V- • . ,/ik-i) = fe!a(x+i/i1 + ... +i/ik_i) + (fe- l)!a!
The terms with /ix... hk_ 1 = 0 contribute <Qk~l. Hence
\TmK<(2Qf-k(&-l+Q< £ min(2, lla/iir1)")
<QK-k+efQk-1+ £ min^fcfc-ijiafcH-i)
By Lemma 2.2, when q^Qk this is
The proof is completed by observing that the result is trivial when
q>Qk-
Lemma 2.5 (Hua's lemma, 1938b) Suppose that 1 <y < k. Then
\f(a)\2Jd(x<N2J-j+E. (2.5)
o
Proof By induction on). The case; = 1 is immediate from Parseval's
identity.
Now suppose that (2.5) holds and that 1 < j < k — 1. By Lemma 2.3
with (f)(x) = ocxk,
Ifia^^W-J-'Y, ••• T.,Le(ah1...hjpJ(x;h1,...,hj))
where pj(x;h1,..., h}) is a polynomial in x of degree k—j with integer
coefficients. Hence
\f(a)\2i<{2N)2'-'-'Y,che{oih) (2.6)

Auxiliary lemmas
13
where ch is the number of solutions of the equation
h1 . . . hjPj(x; hl9. . . , hj) = h
with \ht\ < N and xely Clearly
c0<N\ ch<N£(h^0).
By writing
|/(ar = /(«)2J"7(-«)2J"'
one also obtains
I/W2J' = IV(-^)
where bh is the number of solutions of
x i i . . . i x y j _ j J/^ .
-^-.=¾
with xh yt < N. Thus
Ifc„ = /(0)2J = N2'
and, on the inductive hypothesis,
K =
\f(a)\2Jda<N2J-j + e.
o
By (2.6), Parseval's identity, and (2.7),
|/(«)|2Jt,da«(2N)2y--'"-1XcA.
0
Moreover,
/i /1=/=0
^NjN2J~J + e + N£N2\
which gives the desired conclusion.
(2.7)
Lemma 2.6. Let c1? c2,.. . be any sequence of complex numbers and
suppose that F has a continuous derivative on [0, X]. Then
rx
X cmF(m) = F(X) X cm -
m===5 A" m^ A" »/
F'(y) £ cmdy
0 m^y
^^
Proof The lemma follows at once by writing F(m) = F(X) —
and interchanging the order of summation and integration
F'(y)dy

14 The simplest upper bound for G(k)
2.3 The treatment of the minor arcs
Theorem 2.1 Suppose that s > 2k. Then
»
|/(a)|sda^tts/fc-1_<3.
m
Proof An amount n~ 1 ~d has to be saved on the trivial estimate ns/k.
Hua's lemma with j = k saves nE~ 1, and Weyl's inequality is used to
save the rest.
Obviously
-2k
|/(a)|'da« sup|/(a)|
m \aem
n
0
|2*
|/(a)|2"da. (2.8)
Consider an arbitrary point a of m. By Dirichlet's theorem
(Lemma 2.1) there exist a, q with (a, q) = l and q^Nk~v, and such that
|a-a/^|^^_1Nv-fc. Since aem d(Nv"fc, 1 - Nv~k) it follows that
1 < a < q, whence q> Nv (for otherwise a would be in SD1). Therefore,
by Weyl's inequality,
fiocXN'+^q-1 +N-1 +qN~k)m <Nl + E~v/K.
This, with (1.7), (2.8) and Hua's lemma, gives the theorem.
2.4 The major arcs
The first step is to obtain a suitable approximation to the generating
function f on sM(q,a) by the auxiliary functions
v(P)= £ \m"k-'e{Pm\ (2.9)
m= 1 ^
q
S(q,a) = ^ e(amk/q). (2.10)
m= 1
The function v is that obtained from/by replacing the characteristic
function of the /cth powers by the probability that m is a /cth power.
The sum S(g, a) is an extra factor that has to be introduced, when a is
close to a/q, because the /cth powers, in general, are not uniformly
distributed modulo q.
Lemma 2.7 Suppose that 1 < a < q ^ Nv, (a, q) = 1, ae9Jt(g, a). Then
/(a) = g- ^(¢, a)v(a-alq) + 0(N2v).

The major arcs 15
Proof For Y> 0,
X e(amk/q) = J e(arfcAz) I l = Yq-lS(q,a) + 0(q\
m<Y r=l m<Y
m = r(mod q)
and
I 1
Let
/»yk
m1'*"^
1
, a1/fc " Ma + 0(1) = Y + 0(1). (2.11)
i *
e(am/q) — q 1S(q,a)-m1/k 1 when m is a /cth power
k
m , j
— g 1S(q<a)-m1/k * otherwise,
and take Y=y1/fc. Then
1^4 (y>0).
Hence, by Lemma 2.6 with F(y) = e(/ty),
m< X
Taking X = n, /? = a — a/g establishes the lemma.
The function v that occurs in Lemma 2.7 is not the only possible
choice. Both
vl{P) =
0
and
- r(/i + i/fc)
h=o h\k
would serve equally well. There are arguments for and against each of
v,vl9v2. It is easier to investigate the analytic behaviour of v1 than that
of v or v2, and the use of v2 would avoid some of the technical
complications in the evaluation of J(n) below. However v2 is
somewhat artificial and the evaluation of J(n) when v is replaced by v1
requires Fourier's inversion formula.
That v and v1 behave in much the same way when /? is fairly small

16
The simplest upper bound for G(k)
can be inferred from (2.11) and Lemma 2.6. Thus
v(p) = e((]n)nllk-2nip
n
e(Py)y1/kdy + 0(l+n\p\)
o
o
e{Py)jyllk-l&y + 0{\+n\p\)
k
= vl(P) + 0(l+n\P\).
Let
K(a, q, a) = q 1S(q, a)v(cc — a/q).
Then, by Lemma 2.7, when aesJJi(g, a),
/(a)s - K(a, q, of <Ns~l |/(a) - K(a, 4, a)| ^ AP " l + 2v,
Therefore
(2.12)
z z
q ^ Nv a = 1
(a, 4)= 1
|/(a)s-K(a, <?,a)s|da^N
s - fc - 1 + 5v
a»(«,a)
Hence there is a positive number (5, depending only on /c, such that
f(a)se( - an)d(x = R*(n) + 0(n
s/k - 1 - d
)
(2.13)
a«
where
**(«) = i z
q< Nv a = 1
K(a, g, a)se( —an)da.
2R(4,fl)
(fl,4)= i
By (2.2) and (2.12), R*(n) factorizes as
R*(n)= S(n, N")J*(n),
where
®(",G) = Z Z to" ^(¢,^(-^/4)
(2.14)
(a,q)= 1
and
J*(n) =
N'~k
v(P)se(-Pn)dp.
(2.15)
N
\ -k
The series <3(tt,()) and integral J*(n) are best evaluated by first
completing the series to infinity and then replacing the interval of
integration by a unit interval.

The major arcs 17
Let
S(q)= I (q-lS(q9a)Ye(-an/q). (2.16)
a=l
(a,q)=l
A+e-l/K
By Weyl's inequality, S(q, a) <ql+e~l/* provided that (a,q)=l.
Therefore
S(q)<q{*-llK)s + l<q-l-2-k (2.17)
when s > 2k + 1 and e is sufficiently small. Therefore
00
S(n)= ^ %) (2.18)
4=1
converges absolutely, and uniformly with respect to n, and
&(n,Nv)-&(n)<n-\
Hence, by (2.14),
R*(n) = (S(n) + 0(n-*))J*(n) and S(n) < 1. (2.19)
In order to extend the interval of integration in J*(n), as mentioned
above, it is necessary to estimate the rate of decay of v(fi) as |jS|
increases from 0 to \.
Lemma 2.8 Suppose that |jS| ^ \. Then
v(P)<mm(nlt\\P\-llk).
The same conclusion holds for the functions vx and v2 discussed
above, the proofs being similar in each case, and the result for vx
holding for all real /?.
Proof This is by Abel summation. By (2.11),
£ -rllk-l=mllk + 0{\\
r = i K
and the lemma follows at once when |j3| < l/n.
Now suppose that |jS| > l/n and let M = [|/?| _1]. Then the terms in
m= 1 ^
with m ^ M contribute <^ M1/fc ^ | jS| "1/fc. To estimate the remaining

18 The simplest upper bound for G(k)
terms, let
m
Sm = Z e(M, Cm =
__™l/fc- 1
m
Then
n 1 n
£ -m1//c le{pm) = cn+lSn-cM+iSM+ £ (c»».-Cm+i)S,
= M + 1 ^ m = M+ 1
Since |SJ ^ 1/(21/?|) and cm is a decreasing sequence one has
t \mi'k-1e(pm)<clt+1\p\-1<\p\-1"'
m = M + 1 &
as required.
Let
f 1/2
J(n) =
Then, by (2.15) and Lemma 2.8,
J(n)<
and
v(P)se(- j8n)dj8. (2.20)
- 1/2
'•oo
min(tts^/rs/fc)djMtts/fc_1
o
f* 00
J *(n) - J (n) <
P~s/kdp<ns/k~1~d
tfv-fc
provided that s> k. Hence, by (2.17),
R *{n) = ®(n)J (n) + 0(ns/k~' ~d). (2.21)
This coupled with (2.4), Theorem 2.1 and (2.13) gives
Theorem 2.2 Suppose that s > 2k. Then
R(n) = &{n)J (n) + 0(ns/k~ ^3).
2.5 The singular integral
The singular integral is estimated by induction on s. The following
lemma has the dual role of sparking off the inductive process and
providing the inductive step.

The singular integral 19
Lemma 2.9 Suppose that a, ft are real numbers with a^ ft > 0,
)8< 1. Tten
"S m'" Hn ~ mf- 1 = n^~ {^^ + 0(i.-')\
m=i \r(j8 + a) /
w/iere t/ie implicit constant depends at most on a and /?.
Proof Consider the function
On (0,n), (j) has at most one stationary point. Thus (0,n) can be divided
into two intervals {0,X\ (X,n) (one of which may be empty) such that 0
is increasing on one and decreasing on the other. Therefore
n- 1 <*'
Z 0M =
m= 1
4>(y)dLy + 0(rf-l+rf + *-2)
0
= r(^)r(«)
Theorem 2.3 Suppose that s ^ 2. T/ien
j(n) = rn+^jr(|j ns/k-^l + ofa-1")). (2.22)
Froo/ By (2.9) and (2.20),
J(n) = Js(n)= t ••• t k-s{m,...msYlk-\
m i — 1 ms — 1
mi + ... + ms = n
When s = 2, Lemma 2.9 gives the theorem at once. Suppose the
theorem holds for some s ^ 2. Then
Js+M="l \milk-lJs{n-m)
m= 1 ^
= r(l +lYr(r) V 'V w1'"- '(n-mf" '
+-o(X wl/"" > -m)(s" 1)/l" l J.
Lemma 2.9 now gives the case s + 1.

20 The simplest upper bound for G(k)
2.6 The singular series
The singular series reflects the distribution of the /cth power residues
modulo q. Before investigating the properties of &{n) it is necessary to
examine S(q, a) and S(q).
Lemma 2.10 Suppose that (a, q) = (b, r) = (q, r) = 1. Then
S(qr, ar + bq) = S(q, a)S{r, b).
Proof By Euclid's algorithm, each residue class m modulo qr can be
represented uniquely in the form tr + uq with 1 ^ t ^ q and 1 < u ^ r.
Therefore, by (2.10),
q r
S(qr, ar + bq)=Y, Z e(atfcrfc/<7 + bukqk/r)
¢=11(=1
Moreover tr and uq run over complete residue classes to the moduli q
and r respectively. Hence the lemma.
Lemma 2.11 The function S(q) is multiplicative.
Proof Suppose that {q, r) = 1. Then, by (2.16) and Lemma 2.10,
S(qr) = X I q~ sr~ sS(qr, ar + bq)se(- (ar + bq)n/(qr))
a=1 5=1
(a, q) = 1 (b, r) = 1
= S(q)S(r).
For each prime p, define formally
00
Tip) = X S(p"). (2.23)
/i = 0
Theorem 2.4 Suppose that s > 2k. Then T(p) converges absolutely, so
does \\ T(p\ and
v
S(n) = [I Tip).
P
Moreover there is a positive number C, depending only on /c, such that
\< n Tip)<i

The singular series 21
Proof This follows easily from (2.17), Lemma 2.11 and the elementary
theory of series of multiplicative functions (see Theorem 286 of Hardy
& Wright, 1979). Note that, by (2.16) and (2.10), replacing a by -am
the definition of S(q) gives S(q) = S(q). Thus S(q\ and so T(p), is
real.
It remains to treat T(p) when p^C. There is a close connection
between Tand the number Mn(q) of solutions of the congruence
m\ + . . . + m\ = n (mod q)
with 1 ^ rrij ^ q.
Lemma 2.12 For each natural number q,
YJS{d) = q'-*Mn{q).
d\q
Observe that if q = p\ then the left-hand side is
£ s(p")
h = 0
and thus, by (2.23),
T(p)=limpl{1-s)Mn(pl)
/-♦00
whenever either this limit or the limit in (2.23) exists.
Proof The orthogonality relation
l-Ye(hr/q) = \l ^'
«r^i 10 qjh,
implies that
M.(^) = - Z Z ••• Z <KrK + . . . + mks - n)/q).
1 r=l «1=1 ms=l
Now the sum over r is rearranged into subsums according to the value
of {r,q). The general term in each subsum is a periodic function of nij
with period q/{r, q) = d, say. Hence
MM = -T. £ (fi £ ••• £ e(a(m\ + ... + wt - n)/d)
Q. d\q a=l \aJ mi = 1 ms = 1
(a, <0 = 1
and the lemma follows from (2.10) and (2.16).

22 The simplest upper bound for G(k)
Before proceeding further it is useful to summarize some
consequences of the theory of the multiplicative structure of the reduced
residue classes modulo p*. For an exposition of this theory, see
Chapter 6 of Vinogradov (1954), or Chapter 10 of Apostol (1976).
The number of /cth power residues modulo p\ i.e. residues of the
form xk with p][x, is (/>(//)/(&, 0(//)) when p is odd or t = 1 or k is odd,
and 2t~2/(k,2t~2) when t^2 and p and k are both even. (Here <\>
denotes Euler's function.) Thus when p divides k to a high power the
/cth power residues modulo pl are comparatively scarce, and so Mn{pl)
is relatively difficult to estimate. It is convenient, therefore, to define
t = i(p) to be the highest power of p that divides /c,
Px\\k (2.24)
and to write
ft + 1 when p > 2 or when p = 2 and t = 0, ^ _
(j + 2 when p = 2 and t > 0.
Thus the number of /cth power residues modulo py is
4>{px + *)/(&, 0(pT + *)), and the number of solutions of the congruence
xk = a (mod py)
when p\a is 0 or p7"1" 1{k,4>(pT+ *)). Also, if a is a /cth power residue
modulo py, then it is a /cth power residue modulo p* for every t.
Let M*(g) denote the number of solutions of the congruence
x\ + . . . + xks = n(mod q) (2.26)
with (x^q) = 1.
Lemma 2.13 Suppose that M*(py) > 0 and t ^ y. Then
Mn{pl)^p{t-y){s-l\
Proof Consider any solution of
x\=n — x\— ... — xks (mod py)
with p\xv Then p{t ~y){s~ x) solutions of
y\ = n-yk- ... -yk{mod pl)
can be constructed by choosing y2, • - -, ys so that y, = Xj (mod pv).

The singular series 23
Then n — y\ — ... — y\ is a /cth power residue modulo py, and hence
also modulo p*.
The following lemma is useful in establishing the solubility of (2.26).
Lemma 2.14 (Cauchy, 1813; Davenport, 1935; Chowla, 1935a). Let
.a/, 3 respectively denote sets of r, s residue classes modulo q. Suppose
further that OeJ# and that for every beM with b ^O(modg) one has
(b, q) = 1. Let srf + 3 denote the set of residue classes modulo q of the
form a + b with ae,c/ and beM. Then
card(,£/ + 3) ^ min(g, r + s - 1).
Proof It can be supposed that r + s — 1 ^ q, for otherwise one can
simply remove s — {q — r + 1) elements from 3. The case r = q is
trivial so it may be assumed further that r < q. The proof now
proceeds by induction on s. The case .s = 1 is trivial. Suppose that
s > 1 and that the conclusion holds whenever card 3 < s. Now there
exist cesrf,be$ such that c + b$:stf, for otherwise for each be3, a + b
would range over s/ as a does, in which case
£ (a + b) = Yj a (mod q), rb = 0 (mod q\
aes/ aes/
Let <$ = {b :be&c + b4s/}, .&!= s/v({c} + «), J8X = 3\«. Then
1 < card «#! < s, card j^ + card Mx = r + 5, and
Lemma 2.15 Suppose that s > (/c, pT(p — 1)) wfcen 7 = x + 1, £/zaf
p- 1
s ^ 2T + 2 w/zerc 7 = t + 2 and /c > 2, and fto s > 5 wfen p = k = 2. Then
M*{py) > 0 /or every n.
Proof When 7 = 1 + 1 the lemma follows by repeated application of
Lemma 2.14. When p = 2 the result is trivial, for when k > 2 one has
s ^ 2y and the congruence can be satisfied by taking the Xj to be 0 or
1, and when k = 2 the congruence x\ + .. . + x\ = n(mod 8) is
easily seen to be soluble with 2\xv

24 The simplest upper bound for G(k)
Collecting together the conclusions of Theorem 2.4 and Lemmas
2.12, 2.13 and 2.15 gives
Theorem 2.5 Suppose that s > 2k. Then 3{n) > 1.
2.7 Summary
By (2.19) and Theorems 2.2, 2.3 and 2.5 one has
Theorem 2.6 When s > 2k the number of representations, R(n), ofn as a
sum of s kth powers of natural numbers satisfies
R(n)=r(\ +M v(*A n^-'Sin) + 0(^-1^) (2.27)
where 3(n) > 1.
Corollary G(fc)<2* + 1.
The asymptotic formula (2.27) probably holds whenever s ^ k -f 1
(and k ^ 3). It has been established for s = 2k for such k by Vaughan
(1986c,rf,£?),fors ^ |2fc + 1 when k > 6 by Heath-Brown (1988, 1989)
and for s = |2fc for the same range of k by Boklan (1994). There are
even better bounds when k > 10. For this see Chapters 5 and 12. It
would be of great interest to show that
N
Z e(«*3)
JO A" = 1
dy.<Nll2~d (2.28)
and Theorem 2.6 with s = 8, /c = 3 would then follow easily.
There is a conjecture that
/. i
|/(a) 12sda ^ N£max(;Vs, N2s ~k) (2.29)
o
and this would imply that (2.27) holds whenever s > 2fc + 1.
Let fc > 2. Hardy & Littlewood (1922) define T(/c) to be the least s
such that for every prime p there is a positive number C(p) such that
T(p) ^ C(p) uniformly in n. In a later paper, Hardy & Littlewood
(1925), they show that S(n) > 1 whenever s > max (r(/c),4).

Exercises 25
If one defines T0(/c) to be the least s such that for every q and n the
congruence
x\ + ... + xks = n (mod q)
is soluble with (x1? q) = 1, then the proof of their Theorem 1, Hardy &
Littlewood (1928), shows that V0(k)= V(k). They conjecture that
T(/c)—>oo as /c—>oo, but it is still not even known whether
liminfr(/c)^4.
k -* oc
2.8 Exercises
1 Show that for 1 ^/ ^ k the /th iterate Aj of the forward difference
operator satisfies
A,(a"; /?,,..., pj) = ^ nrr-7Ta'°# ■ • • #
/0,/1,. •• Jj 'o!'i! • • • lj'
/o > 0,/j ^ 1,...,/, ^ 1
/o + /1 + ... + I j = k
= Pi . . ./?kP;(a; 0!, . . .,0,.)
where p, is a polynomial in a of degree k—j and having leading
coefficient k\/(k— j)\.
2 Suppose that (a,q) = 1 and 0 < |a — a/q\ ^q~2. Show that
there are r,b with (b,r)=\ such that j^r\(xq — a\^2 and
21 ar — b | ^ | ccq — a |. Deduce that the conclusion of Lemma 2.4 can
be replaced by
T(4>) < Ql +£(/~l + Q~l +/*Q~k)l,K
where / = q + Qk \ <xq — a |.
3 Show that, when k > 2, G(/c) ^ max(/c + 1, ro(/c)).
4 Show that every large natural number is the sum of one square
and seven cubes.
5 Show that for s ^ 2
s-k ATS/2
|/(a)|sda > max(Ns-\Ns/z).
0
6 Show that the number R of solutions of
*i + y? + )1 = *2 + V3 +) 4
with x^n1/2, >,^n1/4 satisfies R<^n1+e. Obtain an asymptotic

26
The simplest upper bound for G(k)
formula for the number of representations of a number as the sum of
two squares, four fourth powers and a /cth power.
7 Let
e(pf)dy, v2(p) = £ ^+//^/,).
0
/i = 0
h\k
Show that
00
— oc
v^PYei-PriflP and
v2(pfe(-pn)dp
0
- 1
both equal r( 1 +- J Tl - ) ns/fc x asymptotically as n-> do.

Goldbactfs problems
3.1 The ternary Goldbach problem
Vinogradov's attack on Goldbach's ternary problem follows the
pattern of the previous chapter, but this time with
M= I OogpMap). (3.1)
The poor current state of knowledge concerning the distribution of
primes in arithmetic progressions demands that the major arcs be
rather sparse. The principal difficulty then lies on the minor arcs and
the establishment of a suitable analogue of Weyl's inequality.
Let B denote a positive constant, and for n sufficiently large write
P = (log n)B. (3.2)
When 1 ^a ^ q < P and (a, q) = 1, let
Wl(q, a) = {<*:{<*-a/ql^Pn'1} (3.3)
denote a typical major arc and write Wl for their union. Since n is large,
the major arcs are disjoint and lie in
^ = (Pn"1,l+Pn"1].
Let m = #\9W. Then, by (3.1),
R(n) =
f((x)3e( — wa)da
f(ct)3e( — n(x)doL +
2R
f((x)3e( - na)da (3.4)
where
R(n)= X Oogp1)(logp2)(logp3). (3.5)
Pi + P2+ Ps = n
The treatment of the minor arcs rests principally on the following
theorem
Theorem 3.1 Suppose that (a, q) = 1, q < n and |a — a/q\ ^ q~ 2. Then
f(0L)<(\ognf(nq-1/2+n*/5+n1/2q1/2).

28 Goldbach's problems
Proof Let
Tx= Z M<0
rf|x
where [i is Mobius's function. Then taking X = n2/5 and
/l(x, y) = \(y)e(ccxy) in the identity
Z A(i,y)+ Z Z M(*,y)
X < y < n X<x^nX<y^ n/x
d^XX<y<n/dz^ n/(yd)
gives
where
/(aH^^^ + Odi1'2),
5i = Z Z M*)(iog y) ^(°^
x< X y < n/x
S2= Z Z cxe(ccxy) with cx = Z Z Md)A(y),
x^X2 y^n/x d^ X y^ X
dy = x
S3= Z Z TxAfrMa*)')-
x>X y>X
xy < n
Here A is von Mangoldt's function, and the identity follows by
observing that tx = 0 (l<x^X) and inverting the order of
summation.
The inner sum in 5X is
(*n/x
fi(x)
and cx <^ log x. Hence
2^ e(ccxy)—,
1 y < y < n/x /
S1,S2<(\ogn) Y, min(n/x, ||ax|| 1).
x< x*
Therefore, by Lemma 2.2,
Sl9S2<(\ogn)\nq~l +nA,s +q).
Thus it remains to estimate 53.
Let stf = {X, 2X, 4X, . . ., 2kX: 2kX2 < n ^ 2k + * X2}.

The ternary Goldbach problem
29
Then
S3= £S(Y)
Yes*
where
Y <x^2Y X<y^n/x
By Cauchy's inequality,
\S(Y)\2<[ £ d(xf) I
,x<2y /Y<x^2Y
Z A(y)e(ax3;)
X < y < nix
It is easily shown that ]T d(x)2 <^ Z(log 2Z)3. Hence
\S(Y)\2<Y(logn)5 ^ Z min^lMy-z)!!"1).
Thus, by Lemma 2.2,
|S(7)|2 ^n(log nf{nq-1 + Y+ n/Y+q)
which gives
S3< X (\ogn)3(nq-112 +nll2Y112 +nY~1{2 + n1/2q112)
Yes*
< (log nf(nq- 1/2 + n4'5 + n1'V/2)
as required.
To estimate
f(cc)3e( — ccn)dcc
m
it is now only necessary to make two observations. First that
Parseval's identity and elementary prime number theory together
give
|/(a)|2da= £ (log/?)2 < n log n.
0 p^n
Second that, by Theorem 3.1 (cf. the proof of Theorem 2.1),
sup|/(a)| <n(\ogn)
4-B/2
aem
Thus

30 Goldbach's problems
Theorem 3.2 Suppose that A is a positive constant and B ^ 2 A + 10.
Then
|/(a)|3da^2(logn)
- A
m
The treatment of the major arcs, although straightforward, requires
an appeal to the theory of the distribution of primes in arithmetic
progressions.
Lemma 3.1 Let
v(P)= t e(M. (3.6)
m=l
Then there is a positive constant C such that whenever 1 ^ a ^ q ^ P,
(a, q) = 1, cteyjl(q, a) one has
fW = ^v(a-a/q) + 0(nexp(- C(log<2)).
¢(4)
Proof Let
/x(«) = Z (log pWp\
Then
fx(a/q)= t e(ar/q)9(X9q9r) + 0((logX)(logq))
r = 1
(r,q) = 1
where
S(X,q9r)= X log/>-
p^ x
p = r(modq)
By Theorem 53 and (40) of Estermann (1952) it follows that
whenever x/n < X ^ n one has
fx(a/q) = ~^~ t e{arlq)^0{n^V{-Cl{\ognY'2)). (3.7)
</>(<?) r=l
(r,q)=l
Observe that this is trivial when X ^sfn. Also, by Theorem 271 of
Hardy & Wright (1979)
q
X e(ar/q) = fi(q).
r = l
(r,q) = l

The ternary Goldbach problem
31
Hence, by (3.1), (3.6), (3.7) and Lemma 2.6 with X = n. F(m) = e(Pm),
P = a — a/q,
e(am/q) log m — fi(q)/(t>(q) when m is prime,
cm =
- v(q)/<t>(q)
otherwise,
one has
Kq)
/(a) —-—v(cc —a/q) <^ (1 + n|a — a/q\)nexp{ — C\(log n)1/2).
</>(<?)
With (3.3) and (3.2) this establishes the lemma.
Let (x.e3R(q,a). Then, by the above lemma,
<t>(q)
Now integrating over 90¾ gives
I I
q^P a=l J
(a,q)=l
*/(a)3" Tr\3v^ ~ fl/fl)3M - a")da
<<PVexp(-C(logtt)1/2).
Therefore, by (3.3),
/*p/
/(a)3e(-ow)da = S(n, P)
SR
P/n
v(P)3e(-Pn)dP
- P/n
1/2'
+ 0(P3n2exp(- C(logn)1/2)) (3.8)
where
®(^)= I Z 1¾ *(-*"/«>•
(3.9)
q^P «=i </>(<?)'
By (3.6), when jS is not an integer,
viPXWPW'1. (3.10)
Hence the interval of integration [ — P/n, P/n] can be replaced by
[ — 2» 2] w^tn a total error
< Z <t>(q)-2n2P
q^P
- 2
Therefore, by (3.2),
2B>
f((x)3e( - an)d(x = S(n, P) J(n) + 0(n2(log n)~ ia) (3.11)
an

32
Goldbach's problems
where
C 1/2
J(n) =
v(P)M-Pn)dP-
1/2
By (3.6), J(n) is the number of solutions of m1 +m2+m3 = n with
1 ^ nij ^ n. Thus
Also, by (3.9),
where
J(n) = %n-l)(n-2).
S(n,P) = S(n) + ofj]^)- 2)
S(n) = Z T7~^ Z e( - fln/fl)-
(3.12)
(a,q) = l
(3.13)
Hence, by (3.1), (3.11) and Theorem 327 of Hardy & Wright (1979)
»
f((x)3e(-(xn)d(x = <5(ri)J (n) + 0(n2(\ognyB/2).
By Theorems 67 and 272 of Hardy & Wright (1979) Ramanujan's
sum,
q
C«M = Z e( ~ ^1°)
a=l
(a,q) = l
is a multiplicative function of q and satisfies
/%/(<?, n))4>{q)
c„(n) =
¢((1/((1, n))
(3.14)
Hence, by (3.13),
^n) = (l\(l+{p-l)-*)\\{l-(p-ir2). (3.15)
\pln Jp\n
This establishes
Theorem 3.3 Suppose that A is a positive constant and B^2A. Then
f((x)3e(- an)d(x = ±n2&(n) + 0(n2(logn)~A)
where 3(n) satisfies (3.15).

The binary Goldbach problem 33
Note that S(n) > 1 when n is odd and <3(n) = 0 when n is even.
When coupled with Theorem 3.2 and (3.4), Theorem 3.3 yields
Theorem 3.4 Suppose that A is a positive constant and R(n) satisfies
(3.5). Then
R(n) = \n2S{n) + 0(n2(\og n)~ A)
where S(n) satisfies (3.15).
Corollary Every sufficiently large odd number is the sum of three
primes.
3.2 The binary Goldbach problem
In the binary Goldbach problem it is not possible to obtain an
asymptotic formula in the same manner as in §3.1. However, a non-
trivial estimate can be obtained for
£ {R^-m&M))2
m=l
where
Ri(m)= X (logP!)(logp2)
Ply P2
Pi + p2 = m
and S^m) is the corresponding singular series. This is because the
above expression corresponds to a quaternary problem, rather than
to a binary problem. It leads to the less precise conclusion that almost
every even number is a sum of two primes.
Let
R1(m) = R1(m,n)= I I (\ogPl)(\ogp2). (3.16)
P\ ^ n pi < n
P i+ P2 = m
Then
R1(m) = R2(m) + R3(m) (3.17)
where
R2(m) =
2
/(a)2e(-am)da (3.18)
aw

34
Goldbach's problems
and
R3(m) =
f(oc)2e( — am)da.
(3.19)
m
Here/, $R, m are as in § 3.1.
Now R3(m) is the Fourier coefficient of the function which is/(a)2 on
m and 0 elsewhere. Hence, by Bessel's inequality
n
E \R3(m)\2^
m=l
l/(«)l4da.
(3.20)
m
Theorem 3.5 Suppose that A is a positive constant and B > A + 9. Then
t \R3(m)\2^n3(\ogn)-A.
m= 1
This can be deduced, via (3.20), in a similar manner to Theorem 3.2.
Let
q fi(q)2
®i(w,P)= z z
■e(-flm/g).
(3.21)
Then by making only trivial adjustments to the argument that gives
(3.8) one obtains
*P/n
K2(m) = 8^^, P)
v(P)2e(-Pm)dp
- P/n
+ 0(P3nexp(-C(logn)1/2)).
Moreover, by (3.10),
'1/2
P/n
\v(P)\2dp<$nP-\
Hence, by (3.21) and the elementary estimate Z,<p</>(g) 1 <^logn,
one has
R2(m) = S^m^J^m) + 0(n(\og n)1 ~ B)
where
f 1/2
AW =
v(P)2e(-Pm)dP-
-1/2
By (3.6), Ji(w) is the number of solutions of m1 -\-m2 = m with
1 < m,- < n. Hence, when m^n, one has J^m) = m — 1. Therefore, by
(3.21),
K2(m) = mSj (m, P) + 0(n(log n)1" B) (1 ^ m ^ n). (3.22)

The binary Goldbach problem 35
By (3.14), one has
L T732 2. ^-^14)=1.-^ L T7-T2
(a,q) = 1 (q,m) = 1
^S-l'1 ,3231
using the elementary fact that
X 4>(qr2<z-K
q>Z
Hence
siM= Itt^i E 4-™/«) (3-24)
q=l <K4) «= 1
(«,«)= 1
converges,
(^(rajP) — S^m) <^ logm
and
t lei^^-SiHI^Oogn)! ^^minf^l
m=l d<„ 0(«)« Vp
d<„ 0W
^n(logn)2P_1.
Hence, by (3.2) and (3.22),
«
X |^2(w)-wS1(w)|2«n3(logn)2-B. (3.25)
m=l
By (3.14),
S1(m) = (n(l-(P-D"2))ri(1+(P-1)"1)- (3-26)
\plm / p\m
Now, by choosing B suitably one obtains
Theorem 3.6 Suppose that A is a positive constant and B ^ A + 2. T/i^n
£ |R2(m) — mS^m)!2 <^ n3(logn)_y4
m=l
vv/iere Si(m) satisfies (3.26).
Combining (3.17) and Theorems 3.5 and 3.6 establishes

36 Goldbach's problems
Theorem 3.7 Let A denote a positive constant. Then
n
Y, l#i(w) — raS^ra)!2 <n3(logn)~A
m= 1
where R1 and (B1 satisfy (3.16) and (3.26) respectively.
Note that S^m) > 1 when m is even and S^m) = 0 when m is odd.
Corollary The number E(n) of even numbers m not exceeding n for
which m is not the sum of two primes satisfies
E(n) < n(\og n)~A.
Proof By (3.16) and (3.26), for each m counted by E(n\
m~2\R2(m) — mS^w)!2 = S^m)2 > 1.
Hence
n
E(n)<£ Yj m~2\^2(m\ ~ wS^ra)!2.
m=l
The conclusion now follows from Theorem 3.7 by partial summation.
3.3 Exercises
1 Show that every large natural number can be written in the form
Pi + Pi + **•
2 Suppose thata1?... ,a4are fixed non-zero integers witha1?a2,a3
not all of the same sign. Show that
*(")= Z Z Z (logPl)(logp2)(logp3)
P\ < n P2 < n P2, < n
a\P\ + a2P2 + a3p3 + a4= 0
satisfies
R(n) = J (n)S + 0(n2(log n)~ A)
where J(n) is the number of solutions of
aim1 +a2m2 +a3m3 +a4 = 0
with mj ^ n and
oo 4
®= Z *fa)~3rW*A
4=1 J'=l
Show that if (al9 a2, a3)\a4, then J(n) > n2 for large n.

Exercises 37
3 In the notation of the previous exercise show that a sufficient
condition for S > 1 to hold is that
[0,2-, #3? CI4.) = (^1? ^3? ^4/ = 1^1» &!•> ®a) = V^l? ^2? ^3/'
ax + a2 + «3 + fl4 = 0(mod 2(a1? a2? ^3? #4))-
Show that this condition is also necessary, and that, if it fails, then
8 = 0.

4
The major arcs in Waring*s problem
4.1 The generating function
The theory of the major arcs in Waring's problem can be refined
considerably over that contained in Chapter 2. The intention here is
to obtain a relatively good error term for the approximation V(ol, q, a)
to the generating function/(a) on each major arc whilst making the
major arcs as wide and numerous as possible.
Let
S(q9 a,b)= X <(axk + bx)q~1). (4.1)
x=l
Lemma 4.1 (Hua, 1957a) Suppose that (q,a)= 1. Then
S(q,a,b)^q1/2+£(q,b).
The proof uses a deep theorem of Weil (see the reference to Schmidt
below). There is a more elementary theorem of Davenport &
Heilbronn (1936b, 1937a) in which the exponent \ is replaced by
f when k = 3 and f when k>4. Also Theorem 7.1 below gives 1 — l//c
in place of \. Indeed the argument of Mordell used to prove
Theorem 7.1 in the case when q is prime can be adapted so that when
combined with the argument below it gives the Davenport-
Heilbronn theorem.
Proof When (q1, q2) = 1 one has, cf. the proof of Lemma 2.10,
S(q1q2, a, b) = S(qu aqk2~ \ b)S(q2, aq\ ~ \ b).
Thus it suffices to show that for each prime power pl with p\a
S(p\ a, b) < pll2{pl, b). (4.2)
When /=1, (4.2) follows at once from Corollary 2F of Chapter II of
Schmidt (1976). Thus it can be supposed that
/> 1.
If b = 0, or b ^ 0 and the highest power of p, pd, dividing b satisfies

The generating function 39
6 ^ //2, then (4.2) is trivial. Similarly if the highest power of p, p\
which divides k satisfies t > //2, then (4.2) is trivial. Thus it can be
further assumed that
b ± 0, t < ^/, 0 < \\.
Let
Then 3/ — 3v ^ /. In the definition of S(pl, a, /)), (4.1), each x, modulo
p\ can be written uniquely in the form zpl~v + _y with 1 ^y ^pl~ v,
1 ^ z ^ pv. Hence, by the binomial theorem,
S(p\ a, b) = "X X *( (fl/ + by)p~l + (fai/ " ' + &)zp" v
y= 1 z = 1 V
Suppose first that / is even or p\(k2). Then (k2)pl~2v is an integer, and
hence, by (4.3),
\S(p\ a, b)\ ^ pvN
where N is the number of solutions of the congruence
kayk ~1 + b = 0 (mod pv) (4.4)
with 1 ^ y <: pl ~v. Recall that max (0, t) < //2 ^ v. Thus the
congruence is insoluble unless 6^ z and 0 — x is a multiple of /c — 1. If
(4.4) is insoluble, then (4.2) is immediate. In the contrary case
let X = (0 — z)/(k — 1). Then N is the number of solutions of
(kp ~ x)awk ~ 1 + (bp ~e) = 0 (mod pv ~e)
with 1 ^ w ^ pl ~v" A. Note that A < 0 < / - v. When /-v-a^v-0,
one has N <^ 1, so that
|5(pz,a,/7)|^pv.
When /-v-A>v-0, then N <$ pl + e" 2v" A, so that
|S(pz, a, &)| < pl ~ vpe.
In either case
\S(pf,a,b)\<pv(pf,b). (4.5)
When / is even, v = [(/+ 1)/2] =//2, and when p\(k2\
pv ^ p1 +l/2 <^ pz/2. Hence (4.2) follows from (4.5).

40 The major arcs in Waring's problem
It remains to consider the case when I is odd and pl{\). Then
v =^(/+1), v^2.
Each z in (4.3) can be written uniquely, modulo pv, in the form rp + w
with 1 ^ r ^ pv ~ 1 and 1 < w < p. Moreover
Wvfc " 2z2 = I \ayk ~ 2w2(mod p).
Thus the sum over r is zero unless kayk ~ 1 -\-b = 0(mod pv ~ 1). Hence
/ — V
S(p\a,b) = f-'PZ e((ayk + by)p-1)
y= l
x t e(((2y"v+y7j(4^
with y and v satisfying
kayk ~ x + b = 0(mod pv" x) and i; = (lea/ " x + b)/?1 _v. (4.7)
First consider the contribution S1 from those terms with p\yk~ 2.
Then /c > 2 and the innermost sum is zero unless p\v. Thus, by (4.7),
S1 <^ pvN
where N is the number of solutions of the congruence
fopfc-ij/c-i + fo = 0(modpv)
with l^u^pl~v~1. In a similar manner to the treatment of the
previous case one obtains N = 0 unless 0 = k — 1+t+(/c— 1)/1 with
A > 0, in which case N is the number of solutions of the congruence
(kp~ x)ayk ~ ' + (bp~ °) = 0(modpv ~e)
with 1 < y < p1" v" 1 ~ \ Note that 0 > k - 1 > 0. When
Z-v-l-A^v-fl one has N<$ 1, and so S! ^pv<pv"1+a<
pll2(pl,b). When / - v - 1 - A > v - 6/, then N ^pz_v-x " A_(v-0),
so that
once more.
It now remains to estimate the contribution S2 from those terms in
(4.6) with plyk ~ 2. Then the innermost sum is easily seen to be < p1'2
(cf. the case k = 2 of Theorem 4.2 below). Thus
S2<pv~ll2N

The generating function
41
where N is the number of solutions of the congruence
ha/'1 +b = 0(modpv~1)
with l^y^pl~v. Note that v-\ = \U I - v = v - 1 =\{l - 1) > 8
and I > 3. If 8 = \{l — 1), then at once
S2 < Pl/2(p\ b).
If 0 < W - 1), then as before either N = 0, or 8 - x = X{k - 1) with
A>0, and so N <p{l~ i)l2~k~{{l~X)l2~0) ^p°. Thus in this case also,
S2 < Pll2{p\ b).
The next lemma is often the igniting spark for the estimation of
exponential sums. It is a truncated form of the Poisson summation
formula.
Lemma 4.2 Suppose that X < Y, F" exists and is continuous on
[X, Y] and F' is monotonic on [X, Y~\. Let H1, H2 denote integers such
that H1 ^ F'(a) < H2 far every a in [X, Y]. Then
H
CY
I e(F(x))= £
where H = max(|JL/1|, \H2\).
e(F((x) - (xh)da + 0(log(2 + H))
Proof For a differentiable function i//(a) with if/' continuous, the
Euler-Maclaurin summation formula gives
rY
I Mx) =
X < x ^ Y
i//(cc)d(x -|>(a)(a - M -i)]£
+
ijj'((x)((x — [a] — ^)da.
(4.8)
Therefore
rr
I e(F(x)) =
X < x < Y
+
e(F(a))da
27ciF'(a)e(F(a))(a - [a] -£)da + 0(1).
Now recall the Fourier expansion
oc
a - M - 2 = X
/i = — oo
/i f 0
e(— a/i)
2nih

42
The major arcs in Waring's problem
This is boundedly convergent for all real a. Hence the second integral
above becomes
00
z
1
ry
h= - oo h „
h f 0
F'(a)e(F(a)-afc)da.
When h>H2 or /i < Hl9 F'(a) — h is monotonic and non-zero on
\_X, Y]. Hence F'(ol)/(F'((x) — h) is also monotonic on \_X, Y]. Thus
integration by parts gives
ry
F'(<x)e(F(<x)-h(x)doL<
F'(Y)
F'(Y)-h
+
F'(X)
F'(X) - h
Therefore
oo i r y
h = H2+ 1 n J
hfO
F'{(x)e(F((x) — hcc)d(x
00
< I
IH;
+
|H,|
H = H2 + i\\h\(h-H2) Mh-H.l
hfO
H + 1 1
h= 1 n
and similarly for the sum over h^Hl — 1. Integrating the remaining
terms by parts gives
rY
Z e(F(x)) =
X <x< Y
H
rY
e(F((x))d(x+ ^
h = Wi
hf 0
e(F(cc) — cch)dcc
+ 0(log(2 + tf)).
If #! ^ 0 ^ H2, then the proof is complete. If 0 < H1 or H2 < 0, then
|F'(a)| ^ 1, and so
ry
e(F(cc))d(x =
e(F(0L))
2niF'((x)
Y (*Y 17//
X „
F"{a)e{F{oi))
x F'{olY 2%\
da
<h
which can be absorbed in the error term.
Let
/(a)= Z *(aA
1
S(q,a)= X e(amk/<l\
(4.9)
(4.10)
m= 1

The generating function 43
v(P)= I V'-M/fr), v1{fi =
•v/k
<-fc
x 5* n
e{Pyk)&y, (4.11)
0
V(cc, q,a) = q S (q, a)v (a — a/q). (4.12)
Theorem 4.1 Suppose that (a,q) = 1 and a = a/g + /?. T/i^n
/(a)- K(a,g,a)<M± + e(l + n|j8|)±. (4.13)
If further \p\ < (2fcg) " V/fc " \ ffcen
/(a)- K(a,<j,a)<<j* + e. (4.14)
,4/so f/i? sam? conclusions hold when v(fi) is replaced by v{(P).
The proof uses Lemma 4.1. If any of the weaker results mentioned
in the remark after that lemma are used instead, then Theorem 4.1 is
obtained with the exponent \ of q replaced by the corresponding
larger exponent.
It would be very interesting to decide whether (4.13) holds with the
right hand side replaced by (q -f qn | /? | f with the exponent 6 smaller
than \, perhaps as small as \.
Proof For brevity, put X = n\ By (4.1) and (4.9),
/(«)= £ e(Pxk) f e(amk/q)
x^X m= 1
m = x(modq)
= <T' Z S(q,a,b)F(b) (4.15)
-q/2 <b^q/2
where
F(b) = I e(Pxk - bx/q).
x ^ X
When — q/2 < b ^ q/2 the expression fikyk ~ l — b/q is a montonic
function of y on [0, X~\ and lies between — | jS | kXk ~ x — \ and
| /? | kXk ~ { + \. Thus Lemma 4.2 can be applied with Hx = — H2 —
- H where H = [ | /? | kXk ~ l + f], whence
F(ft)= X /(ft + hq) + O(log(2 + H))
h= - H

44
The major arcs in Waring's problem
where
1(c) =
x
e((]yk - cy/q)dy.
o
Moreover, q
4.1,
1 £g= x(q, b) ^ d(q). Therefore, by (4.15) and Lemma
fM-q-'Sfaa^iP)
= q~l ^ S{q,a,b)I (b) + 0(q- + £\og(2 + H)) (4.16)
-B<b^B
where B = (H + \)q.
First of all, (4.14) is dealt with. Thus | p | ^ (2kq) ~ V//c ~ l. Hence,
when b # 0 and 0 ^ y ^ X one has | /2/ey*" l — b/q \ > j\b/q\, and
// = 1 or 2. Therefore, by integration by parts, 1(b) <^ \ b/q \ ~ l. Thus,
by Lemma 4.1, the right hand side of (4.16) is
<q~l Z q± + e(qMqb-1 <q
b= 1
i + 2£
and so gives (4.14) with v(fi) replaced by v^P).
Let
G(Y)= I --^-1
m
m^Y
The Euler-Maclaurin summation formula, (4.8), gives
G(Y)= Y1/k + Ck + 0[Yllk-x).
Hence, by Lemma 2.6 and (4.11),
v(P) = G(Xk)e(pXk) - 2niP
x*
G(y)e(Py)dy
= {X + Ck)e(pXk)
^x«
(y1/fc + Ck)e(Py)dy + 0{Xl~k + \p\X).
— Inifi
Thus, by integration by parts and a change of variables,
v(P) = v1(P) + 0(l+\p\X).
(4.17)

The exponential sum S(q,a) 45
Hence (4.14) follows from the corresponding result with i;(0) replaced
by vtiP).
To treat (4.13), one follows the above model as far as possible.
Begin by observing that the error term in (4.16) is acceptable.
By integration by parts the contribution to 1(b) from those y for
which
\kbck~' -b/q\ >%\b\/q
is <q/\b\ and this contribution to the total in (4.16) is
<^ q* +£log(2B) <^ q* + 2E(l + n|0|)*. For the remaining y one has
\kf$yk~ l — b/q | ^ \\ b\/q. Such y can only arise when \\b\/q
^ k |0 | yk" 1 ^ 11 b | /q and so | b | ^ 2kq | 01 Xk" K For such a b let
5 = |j8|1/(2fc-2)(|&|/^)(k-2)/(2fc-2). (4.18)
By integration by parts the contribution from the y with | kfiyk ~ 1
— b/q | ^ S is ^3-1. Moreover the remaining y with | /e0yfc ~ 1
— b/q\^S lie in an interval [7i,72] which satisfies
k\P\\y2-y1\(y\-2 + y\-3y2 + '-' + yk2-2)^2d. Thus \y2-y,\
|0|(|fr|/(4l0|))(fc~2)/(fc_1)^(5, whence by (4.18) \y2-y^\ <S~K
Therefore the total contribution from the remaining y to (4.16) is
< I qE" Hi, b)\p\~l/i2k" 2)(«/&)(k"2)/(2/c"2)
0 < h < 2/«/ I p I Xk " »
Thus (4.13) holds with v replaced by vl.
To complete the proof of (4.13) one adverts to (4.17). Note that
(4.13) is trivial unless | 0| ^ 1, in which case the additional error is
<^1+|0|X<^(1+|0| n)*. This completes the proof of Theorem
4.1.
4.2 The exponential sum S(q,a)
Lemma 4.3 Suppose that p\a. Then
S(p,a)= X xia)r(x) (4.19)

46 The major arcs in Waring's problem
where srf denotes the set of non-principal characters x modulo p for
which xk is principal, and t(x) denotes the Gauss sum
p
X X(*)e(x/p)-
x= 1
Also \t(x)\ = p1/2 and card s/ = (k, p — 1) — 1.
Proof Let g denote a primitive root modulo p. Then s# is the set of
characters Xh °f tne form
Xh(x)=e(- 7- ind^x ) (p/x)
with 1 < h< (k, p - 1). Thus
1 + Z X(x)
is the number of solutions in y of the congruence yk = x (mod p).
Hence
S(p,a)= X e(ax/p)ll+ X X(x)\
X = 1 \ Z6J2/ /
which gives (4.19). The remaining assertions are trivial.
Let t and y be as in (2.24) and (2.25). Note that
y < k unless k = p = 2 in which case y = 3. (4.20)
Lemma 4.4 Suppose that p\a and I > y. Then
S(p, a) =
pl 1 when I ^ /c,
pk ~ ^Sip1 ~ k, a) when I > k.
Proof Recall that a reduced residue modulo pl is a /cth power residue
if and only if it is a /cth power residue modulo py. Thus
Stf,a)= £ "l e(a(zp> + /)p-') + PE e(ap"-y).
y = 1 2 = 1 y=l
The innermost sum in the double sum is 0, and the sum on the far right
is pl ~ 1 when I < k and
pk-1S(pl~k,a)
when / > k.

The exponential sum S(q,a) 47
Lemma 4.5 Suppose that (q, r) = (qr, a) = l. Then
S(qr, a) = S(q, ark ~ ^(r, aqk ~ *).
Proof See Lemma 2.10.
Theorem 4.2 Suppose that (q,a)= 1. Then
S{q,a)<q'-'lk.
Proof When k = 2,
\S(q, a)\2= t t e(a(y2 - x2)/q)
x = 1 y = 1
q q
= Z Z e(a(z + 2x)z/g)
X= 1 2=1
= «Z e(<^2A?)
= = 1
«12 =
<2fl.
Hence it can be supposed that k > 2. Write l = uk + v with 1 ^ 1; ^ /c,
w > 0 and suppose that p\a. By Lemma 4.4 and (4.20),
S(p\a) = p{k-1)uS(pv\a). (4.21)
Consider first the case v>\. If p > /c, then y = 1, so that, by
Lemma 4.4,
S{p\a) = pv-\
If p ^ /c, then trivially
15(^,0)1^^-1.
Hence, by (4.21),
Pl ~l/k (P > k\
kpl ~l/k (p ^ k).
Now consider the case v=l. By Lemma 4.3,
\S{pv,a)\<kpli2 <kp~ li6pl~ l,k.
Thus, by (4.21),
\Stf,a)\ < ^ , _ l/k :^,: (4.22)
w^)i^^-- 1^ n (423)

48 The major arcs in Waring's problem
By (422), (4.23) and Lemma 4.5,
\S(q, a)\^q'~llk f] K
which gives the theorem.
Lemma 4.6 Suppose that (q, a) = 1. Then
V(a/q + jS, 4, a) < (<T 1 min(n, || j9|| " ^K
Proof At once by (4.12), Theorem 4.2 and Lemma 2.8.
4.3 The singular series
For each integer h, let
Sh{q)= t (S(q,a)q-1ye(-ah/q). (4.24)
a = 1
(«,«) = 1
Thus, in the notation of (2.16) and (2.18),
00
S(q) = SH(q), S(")= Z $„(*). (4.25)
q= 1
Lemma 4.7 Suppose that s > 1 and l = uk + v with 1 < i; < /c. T/ierc
k fp" s/2(p1/2(p' " S h) + (p', h)) wten / = 1 (mod k\
P fciP ) lp~s(p', Ai) wten / # 1 (mod fc).
Moreover, when X = l — max(/c, y) satisfies X > 0 arcd pA//i, t/ierc
S„(p') = 0.
Proof Suppose first that p > /c, so that y = 1. Write l = uk + v with
1 < i; < /c. Then, by Lemma 4.4,
plssh(Pl) = (Pu{k ~ 1]y t s(pw, «M - *hP~ ')• (4-26)
a= 1
Each a can be written uniquely in the form a = xpv + _y with
0 ^ x < pl ~ v, 1 ^ y ^pv, ply. The sum over x is 0 unless pl ~ v\h, in
which case it is pl ~v. In the latter case, the sum over y, when

The singular series 49
v> 1, by Lemma 4.4, is
ps(v-i) £ e(-yhp-1)
y= i
ply
and in modulus this does not exceed ps{v~ 1](pv, hpv~l). Thus
\Sh(pl)\<P~us~s(pl,h) (Z#l(modfc)).
On the other hand, when v=l, the sum over y, by Lemma 4.3, is
p
Z ••• Z T(xi) ■••*(&) Z Xi ■■ xs(yM-yhp~ll
When xx . . . xs ls non-principal, the sum over y here is
Xi • • - XsfrP1 ~ l)*(Xi .'-Xs\
and when X\ • ■ • Xs *s principal, it is — 1 when p'J^ and p — 1 when pl\h.
Hence, by Lemma 4.3,
Sh(pl)<p-us-s/2(p1/2(pl-\h) + (p\h))
as required.
Suppose now that p < /c. When /<max(y,/c) the conclusion is
trivial. Hence it can be assumed that I > max (7, k). Write I = uk + v
with max (7, /c) — k < v ^ max (7, /c). Then, by Lemma 4.4, (4.26) holds.
Moreover, as above, Sh(pl) = 0 unless pl ~ v\h, in which case
plsSh(pl) = p^k ~ 1V - - £ S(^, y)se( - yfcp" ')
y= 1
'Zprw-vtf, h),
since p^k. Thus
s*(p'Mp",w~,,V,'0
which is more than is required.
Theorem 4.3 Suppose that s ^ 4. Then
00
S(n)= Z 5,to
converges absolutely and S(n) ^ 0. Also, when s ^ max (5, k + 2) oh£ /ias
S(n) <^ 1, and when max(4, k) ^ s < max(5, /c + 2), one has &(n) < n£.

50 The major arcs in Waring's problem
Proof By Lemma 2.11 and (4.25), Sn(q) is a multiplicative function of
q. By Lemma 4.7,
00
X \Sn(pl)\<np-(s-1)/2<np-3/2.
i = l
Hence
X \S„(q)\< I! d + ClP-3/2)"<C"2
where Cx and C2 depend only on k and s. This gives the absolute
convergence of <3(n). That <3(n) is non-negative follows from
Lemma 2.12.
Let p9 denote the highest power of p dividing n and let / = uk + v
with 1 < v < /c. Then, by Lemma 4.7,
S„(p') ^ P™ when / < 0 + max (/c, y), j 27
Sn(pl) = 0 when / > 0 + max(fc, y), j
where
f — ^s when / < 0 and 1; = 1,
co + us — min(/, 0) =< — |(s — 1) when I > 0 and v = 1, (4.28)
/ — s when 1; =£ 1.
Hence
00
I |S„(p')|
1 = 1
is 0(p~ 3/2) when 6/ = 0, or 0 > 1 and s > max(5, k + 2), and is 0(0)
when 0^1 and s ^ max (4, /c). Therefore in the former case &(n)<l
and in the latter S(n) <^ d(nf where C depends only on s and /c. Hence
result.
Lemma 4.8 Suppose that s ^ max (4, k + 1). Then
X 41/fcis„(<z)N(rcer.
Proof By (4.27) and (4.28),
(pl)1/ksn(pl)
is 0(0), 0(/7-1) or 0 according as Z < 0, 0 < / ^ 0 + max(/e, y) or
/ > 0 + max (/c, y). Hence

The contribution from the major arcs 51
X q»k\S„(q)\< FI (l+ I (p')1/k|S„(p')l
^(n)cn (i+c/p)
where C depends only on s and k.
4.4 The contribution from the major arcs
Let n denote a large natural number
N = [n1/k], (4.29)
P = iV/(2fc), (4.30)
Wl(q, a) = {a: \(x-a/q\ < Pq~ 'n~ '} (4.31)
and write $R for the union of the yjl(q,a) with 1 ^ a ^ q ^ P and
(a,g) = 1. Then the W{q,a) are disjoint and contained in
W=(Pn~\ 1+Pn"1]. (4.32)
Let
R^(m) =
f(a)se(-(xm)doL. (4.33)
Lemma 4.9 Suppose that t ^ max (4, /c) and that X = 0 w/zen t> k+ 1
and X = l//c w/zen t = /c. Let
ST(q)= t ISfeaJq-1!' (4.34)
a= 1
(a, <*) = 1
X <TAs*(<zMe£.
Proo/ In the same manner as for Sn(q) and with the same notation as
in Lemma 4.7 it can be shown that Sf(q) is multiplicative and
Put-ls*(Pl)
is 0(p~t/2) when /=1 (mod/e) and is 0(/7-') when /^1 (mod/().
Thus
and

52
The major arcs in Waring's problem
00
00
1=1 u=0
- ukk + uk - ut I -.1 - A-f/2 , y -vk + v-t
v = 2
provided that X ^ max((1 + k - t)/k, 2 - k).
Theorem 4.4 Suppose that s ^ max (5, k + 1). Then there is a positive
number S such that whenever 1 ^ m ^ n, one has
lxs
- l
Rw(m) = r 1 + - r - mslk ~ l<B{m)+ 0(ns/k ~ l ~d)
,s/fc - 1 - d)
Proof Let ae^l{q, a). Then, by Theorem 4.1, (4.29), (4.30) and (4.31),
f(x)-V(0L,q,a)<q1/2 + E.
Therefore
/(a)s - V(ol, q, a)s < (q1/2 + J + qm + £|F(a, q, a)\
Hence, by (4.12), (4.31) and (4.34),
s- 1
I
a= 1
(a, q) = 1
(/(a)s — V(<*>, & a)s)e(— am)da
2« (q, a)
^Pn-V/2 + £)s + 41/2 + £Ss*-i(4)
ri/2
- 1/2
MP)?-1 dp
Therefore, by (4.33) and Lemma 2.8,
Rm(m)= I Z
4<P fl=1
(a, q)= 1
F(a, g, a)se( — am)da + £ (4.35)
9W (q, a)
where, taking into account the possibility s = k + 1,
E<^P2n-1(P1/2 + E)s
+ p3/4 + £ X «"1/4ss*_^^-^-^8.
4<p
By Lemma 4.9, (4.29) and (4.30),
E<nslk~l-\ (4.36)
for a suitable positive number S.

The congruence condition
53
Let 9l(q, a) = {(x: P/(qn) < |a - a/q\ < \}. Then, by (4.12), (4.24) and
Lemma 2.8,
I
V(oc, q, dfe{ - am)da < \Sm(q)\
a = 1 »/
(a, q) = 1
91 (q, a)
'00
P ~ slk<\P
P/(nq)
<(nq/P)s/k-1\Sm(q)\.
Hence, by Lemma 4.8,
I I
q^P a=1
(a, q)= 1
F(a, ¢, a)se( - am)da < ns/k ~ 1 ~ d.
9l(q,a)
(4.37)
Therefore, by (4.35), (4.36) and (4.24),
Rm(m) = <S(m, P)I(m) + 0(nslk ~ l " *)
where
S(m,P)= ^ SJq), I(m)= \ v{fSfe(-fimW-
q^P J- 1/2
By Lemma 4.8,
Q <4< 2Q
Hence, by (4.29) and (4.30),
I \SJq)\<n-*
q>P
so that, by (4.25),
S(m, P) = S(m) + 0(n"'). (4.38)
Finally, by (4.11), (2.20) and Theorem 2.3, when 1 ^ m ^ n,
I(m) = J(m)
= H 1 + jH r(|) ms/k" x(l + 0(m~ 1,k)).
The theorem now follows from Theorem 4.3, (4.37) and (4.38).
4.5 The congruence condition
Let Mn(q) and M*(q) be as in § 2.6.
Theorem 4.5 Suppose that s > max (4, k + l)arcd M*(py) > 0 for every
prime p. Then ®(n)> 1.

54 The major arcs in Waring''s problem
Proof By Lemmas 2.12 and 2.13,
00
/1 = 0
the absolute convergence of the infinite series being ensured by that of
®(rc), cf. Theorem 4.3.
It now suffices to show that when p > k one has
00
^S^^-Cp-312 (4.39)
I = 1
Note that y = 1. The argument of Lemma 4.7 shows that if I = uk + v
with 2 ^ v < /e, then
p(-+i).-«S(p.)=1_if_ij0
P P
according as p'|w, p*" x || w, pl ~ 1j[n (4.40)
and that if / = 1 (mod/c), then S„(p') = 0 unless pl~ l\n in which case
p- [i/*](*-»)S|i(pi)
p- i
= P"S Z ••• Z ^i)---^(l)Z Zi-.-z^aM-fliip-1). (4.41)
Choose 0 so that pa||n. Then, by (4.40),
X S„(p')^-//
/#l(mod/c)
where /I = [^//c](/c — s) + 1 — s. It is readily seen that X ^ — 2.
By Lemma 4.3, the terms in (4.41) with Xi • • • Xs ^ Zo contribute
<^p(s+1)/2 and if pjfnp1'1, then those with Xi...xs = x0 contribute
<^psl2. Hence
X s„(P') = I sn(Pl) + o(p-il2).
/=l(mod/c) /=l(mod/c)
If s ^ 5, then, by Lemma 4.3 and (4.41),
sn(pl)<P[l/k]ik-s)-3/2
so that
X s„(P')« p" 3/2.
/ = 1 (mod /c)
P1!"

Exercises
55
Hence it remains to consider Sn(pl) when pl\n and s = 4.
By (4.41),
p-u/kw-s)Sn{pi) = Snpi i{p) = Sp{p)
It therefore suffices to show that Sp(p) ^ 0.
By (4.24),
Sp{p) = V (S(p, a)p~ r.
a= 1
The proof is completed by showing that when k = 2 or 3 and p > /e,
S(p, a) is real or purely imaginary. Observe that when s = 4, one has
/c = 2 or 3.
When k = 2, Lemma 4.3 gives
where / is the Legendre symbol. Thus
S(p, a) = S(p, -a) = x(-a)r(x) = x(~ l)S(p, a),
so that S(p, a) is real or purely imaginary according as /( — 1)= 1 or
x(-i)=-i.
When /c = 3, one has (— x)* = — xk. Thus
3(p, a) = S(p, - a) = S(p, a),
so that S(p, a) is real.
Theorem 4.6 Suppose that s ^ 5 w/zen /c = 2, s > 4/c w/zerc k is a power of
2 with k > 2, and s^\k otherwise. Then &(n) > 1.
Proof At once from Lemma 2.15 and Theorem 4.5.
4.6 Exercises
1 (Vaughan, 1983) Use Theorem 4.1 to deduce the special case
4>(x) — ax3 of Lemma 2.4 (Weyl's inequality).
2 Consider the statements
(i) s^4 and &(n)> 1,
(ii) Mn(q) > 0 for every n and for every large q,
(iii) M*(q) > 0 for every n and for every large q.
Show that if (i) holds for every n, then (ii) holds, and that if k ^= 2 or 4,
then (ii) implies (iii).

56 The major arcs in Waring s problem
3 Suppose that s0(k) is given by the following table.
k 3 4 5 6 7 8 9 10 11 12 13 14 15 16
s0(fc) 4 16 5 9 4 32 13 12 11 16 6 14 15 64
Show that when s ^ s0(k) one has S(n) 5> 1 for every n.
4 Show that Theorem 4.4 holds when s ^ 2/c, 6 is any number less
than ^ + 2 and $R is replaced by the union of the intervals
{a : | ccq — a | ^ ne~ 1} with 1 ^ a ^ q ^ nd and (a,q) = 1.

Vinogradov's methods
5.1 Vinogradov's mean value theorem
When k is small, i.e. less than 11 or 12 or so, Lemmas 2.4 and 2.5, the
essential ingredients for the estimation of the minor arcs in Chapter 2,
have only been strengthened to the extent described in the remarks
after those lemmas. However, for larger /c, significant improvements
can be obtained via Vinogradov's mean value theorem. This theorem
is also of importance in the theory of the Riemann zeta function.
In order to describe the theorem it is necessary to introduce some
notation. Let °Uk denote the /e-dimensional unit hypercube (0, l]fc and
write
f{&)= Yj eipi^x + (x2x2 + ••• + ukxk). (5.1)
Y < x ^ Y + X
For each /c-tuple h = (h1,..., hk) of integers hj9 let Jf ](X, Y, h) denote
the number of solutions of the k simultaneous equations
£ (X; - y{) = hj (1 <; < k) with Y<xr,yr^ Y + X. (5.2)
r = 1
Then
J«\X,Y,h) =
\f(a)\2se(a.h)d<x (5.3)
where oc.h denotes the scalar product oc1h1 + • • • + ockhk. Trivially
J^(X,Y,h)^Jf\X, F,0). (5.4)
By writing xr = M + ur, yr = M + vr and applying the binomial
theorem, it is easily seen that
Jifc)(N,M,0) = Jifc)(N,0,0). (5.5)

58 Vinogradov's methods
For brevity write
Js(X) = Jf\X,0,0). (5.6)
By (5.2), JS(X) is the number of solutions of
t (4 -yi) = 0(l^j^ k) with 0 < xr,yr sC X. (5.7)
r = 1
A non-trivial estimate for JS(X) is known as 'Vinogradov's mean
value theorem'.
All known methods for estimating JS(X) when k is large depend on
a reduction which relates JS(X) to Js_k(X/p) where p is a suitable
prime number. The method adopted here is a refinement of an
argument of Karatsuba. Later, in §5.5, an improved treatment due to
Wooley (1993a) is presented. At this point only the case d = 0, due to
Linnik (1943c), of the following lemma is required, but the general
case is required later in §5.5 and the proof is essentially identical to
that in the case d = 0.
Lemma 5.1 Suppose that d ^ 0 and p is a prime number with p > k. Let
A(p, h) denote the number of solutions of the k simultaneous congruences
r = 1
with nr ^ pk + d and the nr distinct modulo p. Then
A(p,h)^k\pk{k-1)/2.
Proof Let B(g) denote the number of solutions of
k
Xn;ES.(modpk + d)(Ui$fc) (5.8)
r= 1
with nr ^ pk + d and the nr distinct modulo p. Then A(p, h) is the sum of
all the B(g) with g. = hj (modpj + d) and 1 ^ g. ^ pk + d (1 < j^ fc).
The total number of possible choices for g is pk{k ~ 1)/2. Thus it now
suffices to show that
Big) < k\

Vinogradov's mean value theorem 59
and this will follow on showing that every solution of (5.8) is a
permutation of any given solution.
For a given g, suppose that n i,..., 72K is a solution of (5.8) with
nr ^ Pk + d anc* the nr distinct modulo p. Suppose that m1?..., mk is
another such solution and let
P(x) = n (* - nr). (5.9)
r= 1
Then, by Newton's formulae connecting the sums of the powers of the
roots of a polynomial with its coefficients, and the fact that p > k, it
follows that
k
P(x) = Y\ (x- mr) (mod// + d).
r= 1
Thus
P(mr) = 0 (modpk + d) (1 ^ r < k). (5.10)
Hence, for each r there is an s such that ns = mr (mod p). Also, since
the ns are distinct modulo p it follows that ns is unique. Hence, by (5.9)
and (5.10), ns = mr (modpk +d), whence ns = mr. Thus the mr are a
permutation of the nr as required.
It is convenient to state here a lemma which is required only later,
in §5.5, but whose proof uses a similar idea.
Lemma 5.2 Suppose that 0 < k ^ s and p is a prime number with p > k.
Let B(p,h) denote the number of solutions of the k simultaneous
congruences
s
Yjni=hJ(modp)(l^j^k) (5.11)
r= 1
with nr < p. Then
B(p,h)^k\ps~k.
Proof It clearly suffices to treat the case s = k. Let m1?... ,mfc be
another solution of (5.11), so that
!«;= Z mi (modp) (1 ^7 < k). (5.12)
r = 1 r = 1

60 Vinogradov's methods
As in the proof of Lemma 5.1, with the notation (5.9), P(mk) = 0
(mod p). Thus there is an I such that nt = mk (mod p). Now delete nt
and mk from (5.12) and repeat the argument. Thus once more one
finds that the m are a permutation of the #i.
Let R^h) denote the number of solutions of the simultaneous
equations
t 4 = hj(l <j<fc) (5.13)
r = 1
with 0 < xr < X and xl9...9xk distinct, and let R2(h) denote the
corresponding number with at most k — 1 of the xl9... ,xk distinct.
Then
J,(X) = £(«i(A) + R2{h)f < l^iR.ihf + R2(h)2).
h h
Hence
JS(X) ^2^+2^ (5.14)
where/, = X»*((*)2-
First of all consider I2. By (5.1)
2
^i k
2 ^ '
/, ^(':i |/(2a)2/(a)2s - 41 da
j^k
and by Holder's inequality the integral here does not exceed
/»
l/s
f*
da
l/s
f*
1/(2«) 12sda
1 - 2/s
l/(«) I 2sd«
«t / \J*
= J^)1 "V
Thus, if /2 ^ /l9 then Js(^0 ^ 4s (2) and so in any case
^(^)^4-/^ + 4^. (5.15)
It remains to treat /^ Let ^9 denote the set of k2(k — 1) smallest
primes p with p > X1/k and p > k, and let co0(n) denote the number of
different primes p > X1/k dividing n. Given xl9..., xk all distinct and
not exceeding X, put P(x) = 11^= ^11) = i + 1(xi —Xj). Then
co0(P(jc)) < (klog\P(x)\)/(logX) < \k2{k - 1). Now Ix is the num-

Vinogradov's mean value theorem
61
ber of solutions of (5.7) with x1,... ,xk distinct and y1?... ,yk distinct.
Thus, for any solution x,y counted by 1^ there is a prime peg? such
that A]j • • < j -\r. arc distinct modulo p and yl9...,yk are distinct
modulo p. Therefore
h ^ I Ii(P) (5-16)
where I^p) denotes the number of solutions of this kind.
Let
/(<*, y)= Z e(aix + a2*2 + ''' + a**fc)
x = y (mod p)
and let stf denote the set of /c-tuples a = (al9..., ak) with 0 < ar < p
and the a„ distinct. Then
/i(p) =
^*k
£/(0^).../(0^)
By Holder's inequality
2s- 2fc
Z /(<*>*)
x ^ p
2s- 2k
da.
Z /(<*>*)
x ^ p
^p
2s - 2fc - 1
I l/(«,x)
2s - 2fc
x ^ p
Hence
/i(p)<p2s"2fcmax/3(x)
(5.17)
x ^ p
where I3(x) is the number of solutions of the simultaneous equations
s-k
£ (mi - n{) = X ((pyr + %Y - (pzr + x)>) (1 <j < /c)
r = 1
r= 1
with 0 < mr, nr < Jf, with — x/p < yr, zr ^ (X — x)/p, with
m1?... ,mfc distinct modulo p and with nl9..., nfc distinct modulo p. A
simple application of the binomial theorem shows that I3(x) is the
number of solutions of the simultaneous equations
s-k
X (K - xy - (nr - xy) = X pV, - 4) (Kj^k)
r = 1
r= 1

62 Vinogradov's methods
with the variables satisfying the same conditions as before. Note that
since nr ^ X < pk, nr is uniquely determined by its residue class
modulo pk. Then, by Lemma 5.1,
/3(x) < Xkk\pk{k ~ 1)/2 max J{k)_ k(X/p, - x/p, h)
h
and so by (5.4), (5.5), (5.6), (5.15) and (5.17)
fh\2s
JS(X) < 4s()+ 4/c!/c2(/c - l)*fcmax (p2s + k{k ~ 5)/2Js _ k(l + X/p)).
(5.18)
Theorem 5.1 (Vinogradov's mean value theorem) For each pair of
natural numbers /c, I there exists a positive number C(k, I) such that for
every X > 0
Jkl(X)^C(kJ)X2lk-k{k + 1)/2 + ri
where n = \k2(\- l//c)z.
It should be observed that for applications in multiplicative
number theory it is necessary to know something of the behaviour of
C(/e, I) as I and k grow. Here, however, it is of lesser importance. Note
that the theorem is trivial when k = 1.
Proof This is by induction on /. By a similar argument to that used to
estimate B(g) in the proof of Lemma 5.1, it can be shown that when
s = k all the solution of (5.7) are obtained with the yr as permutations
of the xr. Thus
Jk(X) ^ k\Xk
which gives the case / = 1 at once.
Now suppose that I > 1 and the theorem holds with I replaced by
/-1. When X ^ kk the desired conclusion is trivial. Thus it may be
supposed that X > kk. Then, by Bertrand's postulate, when
pe&,p^ 2k2{k~l)Xltk. By (5.14) and the inductive hypothesis there is
a prime p e & such that
fk\2kl
Jkl(X) ^4k,r +Cx(k, l)Xkp2kl + fc(fc - 5^2(X/p)2kl ~ k{k + ^2 + "'

The transition from the mean 63
where 77' = \k2{\ — 1/k)1 1. The exponent of p here is k2 — 77', so that
Ju ^ 4kl(kYkl + C2(k,l)X2kl ~fc(fc + l)>2 + "'
which gives the desired conclusion.
5.2 The transition from the mean
Let
v(x) = vik)(x) = (x, x2,. . . , xk) (5.19)
and for <xeRk write
JV
/(«)= X e(v(x)-«) (5.20)
x= 1
where as usual for two elements a, p of Uk, <x-p denotes the scalar
product ccipi-\-.. . + afc/?fc. Suppose that M is a non-empty set of
integers with
^c[l,N], M = card MO. (5.21)
Then, for me,l,
JV + m
/(«)= ^ ^(v(x-w)-a)
x = 1 + m
f 1 / 2JV \ N + m
X e(v(x-m)-« + x/J) I e(-j;/J)d/J.
0 \x = 1 / y = 1 + m
Therefore, on summing over the elements of .#,
f(*)<M
pi
- i
X |c/(m,/?)| min(iV, H/jy-^d/J
0 \mei /
where
2JV
g(m,fi)= X e(v(x-m)-a + xj8). (5.22)
x= 1
Thus
/(a)^M"1(log2N) sup £ \g(m, P)\, (5.23)
and an estimate for f can be deduced from a suitable mean value
theorem.

64 Vinogradov's methods
The following lemma embodies a relationship between discrete
mean values and corresponding continuous mean values. Note that
Z \a(n)\2 =
neJf
\S(p)\2dp.
*!
A number of alternative methods have been devised for obtaining
such relationships, but all are based on similar ideas. The method
given here is suggested by the large sieve inequality, and is a
generalization of that inequality to I dimensions which has been
useful in algebraic number theory. See Huxley (1968) and Wilson
(1969).
Lemma 5.3 Suppose that Sj>0 (/= 1,...,/) and that V is a
nonempty set of points y in Ul such that the open sets
mv) = {fi:\\Pj-yj\\<8j, 0^Pj<1}
are pairwise disjoint. Let Nl5 . . . , Nt denote I natural numbers and Jf
denote the set of integer l-tuples n = (n1,. .., nt) with 1 ^ nj ^ Nj. Then
the sum
S(p)= £a(n)e(np),
where the a(n) are complex numbers, satisfies
I |S(y)|2 < £ \a(n)\2 \\ (JVj + SJ x).
yeT fie V j' = l
Proof Without loss of generality it can be supposed that 0 < <5 ■ < \.
It suffices to bound the dual form
I \T(nf
nsJ<
where
T(n)= 5>(yMiry).
yer
Since 1 - \h\/(2N) > \ whenever \h\ ^ N it follows that
2"<£ \T(n)\2 ^ ^ PW 11 (1 " WNjft
ne.V h j = l
\hj\ < 2Nj

The transition from the mean
65
On squaring out and interchanging the order of summation this
becomes
I I b(y)b(y') n ' l
yeT y'eT
2N
X e(n(y'j-yj))
f=\\2Njn=l
The innermost sum here is
< mm(Nj9 \\y'j - yj\\ " l) < JV/1 + Nj\\y'j - yj
Therefore
X |T(n)|2 < £ \b(ytf £ rK^O+^IIVJ-^ll)"2)- (5-24)
ne/ yeT y'eT j = 1
Let F(P,N,S) = N when || j8 || < <5 and JV(1 + JV || P \\ )~2
otherwise, and put
I(ol9S) = {)8: || a - )8 || < 5,0 ^ P < 1}
and F.(j8) = F(P,Nj9dj). It suffices to prove that
F(a,N,<5)<<5
- l
F(P,N,8)dp9
(5.25)
/(a,JV,<3)
for then by (5.24)
Iir(n)|2«5>(7)|2KV^r1 n
yer y'er j = 1 \ J
ne^
1(7j ~ Vj,*j)
Fj(P)dp .
Hence, by a change of variables the product here becomes
ri FjiPj - 7j)dfi
&(y') j = l
and so, on the hypothesis concerning ^(y'), the sum over /' is at most
(V-.^r1 nf I F0-yj)dp\
j = l V J o /
The jth integral here is
<
o
Njdp + JV/1 + NjP) ~ 2dp = NjSj + (1+ NjSj)
-1
and the lemma would follow. It remains, therefore, to establish (5.25).
Suppose that || a || ^ S. Then Pel(a93) implies that 1 +

66 Vinogradov's methods
N II P II ^ 1 + N( || a || + S) ^ 2(1 + N || a || ) and so F(/?,N, 5) ^
N(l + JV || 0 || )" 2 ^ iF(a,JV, d) which gives (5.25) when || a || ^ 5.
Suppose that || a || < S. Then F(a9N9S) = N. Choose fce/so that
fc — <5<a<fc + <5. When fc < a < fc + <5 and 0 ^ /? < 5 one has
F(P9N95) = N9PeI(a9S) and 5" x j/(M) F(j?,N, W > N. When
h — S < a ^ h the same conclusion follows by considering jS with
1 -£</?< 1.
The technique adopted here for estimating /(a) is to compare
I i</(m,/?)i2s
with JS(2N) by means of Lemma 5.3 and a suitable choice of M.
Let y(m) = (y 1 (ra), ..., yk _ {(ra)) where
yj(rn)=l*k(h\-rnf-J (Kj<fc-1)- (5.26)
fc = j \J /
Then, by (5.19),
v(k)(x-m)-a = v(k~1)(x)-y(m) + xkak + £ ^-(-^- (5.27)
j= i
Thus, in order to apply the lemma to the sum g9 given by (5.22), it is
necessary to discuss the spacing of the y/ra) modulo 1.
Suppose that 1 ^ x, y^N9 x^y and define
h+l\ J J y-x (5.28)
ahj = 0 (1 ^ h < j < k).
Note that ahj is an integer and a-j} = k\. Further define
Ph = oih+1(h + l)(y-x) (5.29)
and Tj = k\(yj(x) - yj(y)). Then, by (5.28),
k - 1
TJ = k\(yj(x)-yJ(y))= £ /^¾. (5.30)
h = 1
The next objective is to invert this linear transformation. Write
A _ (n )k - 1 k - 1
^ ~~ \ahj)h = 1 j = l
and B = A — k\I where / denotes the unit (k — 1) x (k — 1) matrix.
The matrix A is lower triangular, and B = (bhj) where bhj = 0 when

The transition from the mean 67
1 ^ h ^ j < k and bhj = ahj when 1 ^ j < h < k. Hence (5.30) can be
written in the form
t = PA. (5.31)
The tth power of B satisfies
B' = (£>£)
where
k- 1 k- 1
bhj = L • • • Z bhjlbjlJ2.. .bjt_xj.
jy = 1 Jr- i = 1
Therefore b^) is an integer and bty = 0 when h<j + t. Moreover, by
(5.28), ahj <Nh~j when h > j. Hence, for h>j+t,
hht]j< Z--- Z Nh-hNh-J2 ...Nj<-i~j
j\ jt-1
j < jt _ ! < ••• < J2 < Ji < ^
So that
b^<Nh-j. (5.32)
Clearly Bk~ 1 is the null matrix. Thus, on writing J = k\I and
D = Jk ~ 2 - Jk *" 3B + . . . + (- l)fc *■ 2Bk ~ 2
one obtains
AD = (B + J)D = Jk ~ x + (- l)fc" 2Bk ~ x = (k\)k ~ H.
Hence, by (5.31),
xD = {k\)k~'p,
so that
(k !)*" lpj = (k !)*" 2t, + "l (- l)'(fc!)"" 2 "' Y t^J ■
r = 1 h= j + t
Thus, by (5.32),
W^M Z 11^11^ "J-
h = j
Therefore, by (5.29) and (5.30),
\\(k\foLj(x -y)\\ < *£ II 7^)-^) II ^"7+1 (2<j^k). (5.33)
fc = j - i
Suppose that for some j with 2 < j ^ k there are a, g with
(a, g) = 1, q ^ NJ and la,- — a/g| ^q~2. Let
L = min(<7, JV). (5.34)

68 Vinogradov's methods
Then for each xe[1, L] the number of ye[l, L] for which
||(/e!)fca;(x-y)|| ^N'~j
is bounded by the number of ye[l, L] for which
\\(k\)ka(x - y)/q\\ ^N'~j + (k\)kLq- 2
and this is at most R where
R = ((k\)kLq- x + 1)(2^1 ~j + 2{k\)kLq- l + 1). (5.35)
Hence there is a set M of integers xe[l, L] such that M = card
satisfies M > L/(K -f 1) and such that for each pair x, y with xe
yeJt, x =/= y,
\\(k\)kaj(x-y)\\>Nl-J.
By (5.33), for every such pair x, y there exists an h for which
; - 1 < h < fc - 1 and
lly*W-y*(y)ll>iV-*.
Now Lemma 5.3 can be applied with / replaced by k — 1, with
Af; = sNy, with (5,. > N~{ with T = {y(m): meM} and with
a(n) = £' e((*i + • • . + xj)afc + ^+...+ xs)j8)
•* 1 5 • • • » -*S
where the sum is restricted to the solutions of the
simultaneous equations
xh!+...+xj = nh (Kfc^fc-1)
with 1 < x, < 2N. Therefore, by (5.22), (5.27) and (5.6),
I Iff (m, j3)|2s <l Jifc" ^(2^)^- 1)/2.
Hence, by (5.23) and Holder's inequality,
/(a)2s < (R/L)(log2N)2sJ{k ~ "(IN)^*- 1)/2.
This with (5.34) and (5.35) gives the following theorem.
Theorem 5.2 Suppose that there exist j, a, q with 2 ^ jf ^ /c,
|a7- — a/q\ ^ q~ 2, (a, q)= I, q*^ Nj. Then
/(a) < (J{k - X)(2N)Nfc(fc " 1)/2(qN'j + JV " ' + 4 " x))1/2s log 2N.
Combining this with Theorem 5.1 gives

The minor arcs in Waring's problem 69
Theorem 5.3 On the hypothesis of Theorem 5.2,
/(a) < N(Nt1(qN~j + AT l + q~ l))lf^k " ^logIN
where
_1„ _//e-2x/
i=i(k-iy
In particular, if N <^q<^Nj~ l, then
f{<x)<Nl-a\og2N
where
°=mt*2(h)i{l-iik-l)i^[))- (5-36>
Moreover 4o7e2log/e ~ 1 as /e—► oo.
All but the last part follow at once. To prove the last part observe
that when k ^ 3 the maximum is attained for a value of / satisfying
/e-lV1
< 1 where X is the larger root of the trans-
/-A(logfc_2
cendental equation
eA=i(/c-l)2(A+l).
2'
Now it is readily seen that A ~ 2 log /c and
A little calculation shows that Theorem 5.3 gives stronger results
than Lemma 2.4 when k ^ 12.
5.3 The minor arcs in Waring's problem
Let f(cc) be given by (1.6). Then
2s
|/(a)|2sda
0
is the number of solutions of
x\ + . . . + xk = y\ + . . . + yk
with 1 ^ xj9 yj ^ N. Hence
fi
o
|/(a)|2sda < Nkik-1)/2JS(N). (5.37)

70 Vinogradov's methods
Let N, P, 9W, °U be as in § 4.4 and let m = Jll \ sJft. Let a em. Choose a, q
so that (a, q) = l,q ^ n/Pand |a — a/g| ^ Pq~ 1n~ 1 (Lemma 2.1). Then
1 ^ a ^ q, and since a lies in no major arc Wl(q, a) it follows that q> P.
Hence, by Lemma 2.4 and Theorem 5.3
/(a)^N1_<T0 + £
where o0 = max (cr, 2x ~k) and cr is given by (5.36). Thus, by
Theorem 5.1 and (5.37) with s replaced by /c/, there is a positive
number S such that whenever
one has
/»
|/(a)|2sda^N2s"fc-^.
Combined with (4.33) and Theorems 4.4 and 4.6 this yields
Theorem 5.4 Let g0 = max(cr, 21 ~ k) and let s0 denote the least integer
such that
s0>mm(—(l--\ +2klY (5.38)
Then the asymptotic formula (2.27) holds whenever s > s0. Also
s0 ~ 4/c2log k as k —► oo.
Again, a modicum of computation shows that s0 < 2k + 1 when
/c ^ 11.
5.4 An upper bound for G(k)
The investigation here of Waring's problem has so far concentrated
on an asymptotic formula for the number of solutions of
However, Hardy and Littlewood (1925) observed that the required
size of s could be reduced by restricting the range of some of the
variables Xj. This technique was later greatly exploited by
Vinogradov and Davenport.
Vinogradov has shown that G(k)^ /e(log/e)(C + 0(1)) as fc-»oo,

An upper bound for G(k) 71
and over a period of nearly thirty years has reduced the permissible
value of C to 2. There is a fairly simple argument that shows that C
can be taken to be 3 and which motivates many of the underlying
ideas. For the reduction from 3 to 2, see Chapter 7, and for Wooley's
further reduction to 1 see Chapter 12.
Let Z be large and write
1 — 6^' ^j + 1 — 2^j •>
and let Qz(m) denote the number of solutions of
x\ + . . . + xk = m
with Zj < Xj ^ 2Zj. Then SmQz(m)2 is the number of solutions of
x\ + . .. + xk = y\ + . . . + ykt with Zj < xj, y} ^ 2Zj. (5.39)
Since
lYfc — vy\ ^ Iy — v \k7k~ l
and
\x\ + ... + xk-yk2-...-yk\<2kZk2+0(Zk3)<kZ\-1
it follows that (5.39) can only have a solution with x1=y1. By
repeating this argument it follows that x2 = y2, *3 = y^ and so on.
Thus
X Qz(m)2 <Zx...Zt^h Qz(m)) (Zx . . . Zt)
m \ m /
Moreover Zx .. . Zr > Zfc"fc(1 " 1/k)t and Qz(m) = 0 when
m>3"fcZfc + 0(Zfc"1).
- l
Thus
iQzimf « (iQzOnAV*^1" ^ (5.40)
and
Qz(m) = 0 when m>\Zk. (5.41)
The above argument also shows that Qz(m) is 0 or 1, and gives a seiJt
of natural numbers m not exceeding Zk for which m is the sum off /cth
powers and
card.^r>Zk"k(1"1/w.
Thus, for a comparatively small value off, say Ck log /c, the cardinality

72 Vinogradov's methods
of Ji can be made relatively close to Zfc. This construction, a slight
modification of that due to Hardy and Littlewood, is used in two
different ways on the minor arcs. Firstly, in an analogous manner to
Hua's lemma (Lemma 2.5) in order to save almost Nk9 and secondly
(and this is Vinogradov's contribution) to save a further small amount
on the minor arcs in a similar, but more efficient, way to Weyl's
inequality (Lemma 2.4).
Let
H(a) = X QN(m)e{(xm). (5.42)
m
Then, by Parseval's identity and (5.40),
|//(a)|2da <£ //(0)2AT k + fc(1 " 1/k)t. (5.43)
o
The following is due essentially to Vinogradov (1947).
Lemma 5.4 Let
V(*)= I I bye(zPky)
X/2 < p < X y^Y
where the b are arbitrary complex numbers. Suppose that a = a/q + /?
with \P\<±q-1X~k,q^ 2Xk, (a, q)=l, that Yp Xk, and that when
q^X one has \fi\P q~'X1 ~kY~ \ Then
/ V/2
v(ol)<uy1 + E I \by\
Note that the argument described below can be modified readily so
that the interval of summation (0, Y] for y can be replaced by an
arbitrary interval of length Y.
X bye(ccpky)
y^Y
(5.44)
Proof By Cauchy's inequality
V(a)2 « X X
X/2<p^X
When (/i, q) = 1, the number J of solutions of the congruence
xk = h(modq)
satisfies J <^ q£. Hence there is an L ^ q£ such that the primes p with
\X < p ^ X can be divided into L classes .^,..., SPl so that for two

An upper bound for G(k) 73
distinct primes px, p2 in a given class &j9mp\ = pk2(modq) if and only if
Pi = p2{modq).
Consider two such primes pl9p2. By the hypothesis
Mrf ~V\)\\ > \\a{p\-p\)iq\\ -h~ lX~kXk
provided that p1 # p2(modq). When q > X, the elements of & -} are
distinct modulo q. Hence, for pe&lj9 the ccpk are spaced at least \q~ 1
apart modulo 1. Therefore, by the one dimensional case of Lemma 5.3
(the large sieve inequality),
2
X bye(ccpky)
X/2 < p^ X
pe&j
y
v< Y
^1 IM2 (5.45)
y<Y
and the lemma follows easily from (5.44).
When q ^ X the argument may be refined as follows. Suppose that
p1 = p2{modq) but Pi=hp2. Then, by the hypothesis,
mp\ -p\)\\ = wm-m = m\p\-p\\
>q~1Y~1\Pi-p2l
Now \p1 — p2\ >z q, and so, combined with the argument above, this
shows that the ccpk are spaced > Y~ x apart modulo 1. Hence, by
Lemma 5.3, one obtains (5.45) once more.
Let X = N1/2, y = Xfcand
m*) = I I Qx(y)e(xpky).
X/2 < p^ X y
Adopt the notation of § 5.3 and suppose that a em. Choose a, q so that
(a, q)= 1, q^2Xk, |a —a/q\ ^½- lX~k. Then l^a^q and since
a is not in a major arc, when q^N one has
\oL-a/q\>q~1N1-k>q~1X1~kY~K Hence, by Lemma 5.4,
(5.40) and (5.41),
M/(a)^M/(0)(NM1-1/fc),-1+£)1/4.
Therefore, by (5.43) and (1.6),
f((xfkH(a)2W((x)e(-(xn)d(x < H(0)2W(0)n
m

74
Vinogradov's methods
where
*=i-7ll-
4/c 4
lY
Thus if t is chosen so that
r>(log5fc)/ -log 1-
k)Y
(5.46)
then it follows that there is a positive number 3 such that
m
Now
f(arkH(a)zW((x)e(-(xn)d(x < H(0)zW(0)n
H(ol)2W(ol) = ^ Q*(m)e((xm)
3-d
(5.47)
(5.48)
m
where
Q*(m) = z z z Z 2NK)e,vK)e*(>').
mt m2 X/2< p<X y
m\ + m2 + p y = ™
By (5.41), Q*(m) = 0 when m > \n. Hence, by Theorems 4.4 and 4.6,
f{0LfkH(a)2W{0L)e(- <xn)da
<m
= Z6*(m)
m
/(a) e( — (n — m)a)da
an
§>«3Ifi*(m).
m
Therefore, by (5.47) and (5.48),
f{<xTkH(<xyW{<x)e{- an)doL > n3H(0)2W{0) > 0.
o
On the other hand, the left side is the number of solutions of
x\ + . . . + x\k + y\ + . . . + yk + z\ + . . .
+ zk + pk(w\ + . . . + wk) = n
with the Xp yj? zJ? w,, p restricted in various ways. Hence
G(k)^4k + 3t.
The optimal choice of t in (5.46) occurs with t ~ /c log k. Thus
G(/c)^/c(log/c)(3 + o(l)) as /e->oo. (5.49)

Wooley's refinement of Vinogradov's mean value theorem 75
5.5 Wooley's refinement of Vinogradov's mean value theorem
It is evident that the proof of Theorem 5.1 makes use of differences of
the form xk — yk in which x = y (modp). A number of significant
advances have been made in recent years in which efficient differences
of the form (xk — yk)/pk with x = y (modpk) have played a crucial
role. Thus the objective here is to introduce differences of this and
related kinds. Especially important will be the modified forward
difference operator A? given by
Af(f(x);h,m) = m-'(/(* + hmk) -/(*)).
Definition 5.1 Let d and k be integers with 0 ^ d ^ k. Let A and X be
positive real numbers. Then it is said that the k-tuple of polynomials
*V =(T1(x),...,Tk(x))6ZW
is of type (d,A,X) when
(a) mt has degree i — dfor i ^ d and is identically zero for i < d, and
(b) the coefficient ci ofx1 ~ d in ^(x) is non-zero and bounded by AXd
for d ^ i ^ k.
There are three useful consequences immediate from the definition.
(i) Let A..eZ (1 <; < i ^ k) and put
*i = ^+Z VF,(l<i<fc).
j= i
Then <b is also of type (d, A,X).
(dm (z S\
(ii) Let J(¥, z) denote the Jacobian det lV j) . Then
V VZj /d+ 1 ^i,j^k
J(V) = det((i - d^z1-"- ^^ liiJik
/ k \ k — 1 k
= (k-d)\( n c« n n <*,-*,)•
\i = d+ 1 /r = d+ls = r+l
(iii) Let h and m be integers with 1 ^ hmk ^ BX and define the
polynomials Gt by
0.(z) = A*V¥.(z)\Km) (1 ^ i ^ k).

76 Vinogradov's methods
Then the system 0 is of type (d + l,kAB,X).
A generalization is required of Lemma 5.1.
Lemma 5.5 Suppose that the system *F is of type (d,A,X) and let
A(p, /r,*P ) denote the number of solutions in n e (Z/pkZ)k of the system of
congruences
k
X *,(«,) = hj (mod pJ) (d + 1 < j < k)
i = 1
with (J(*P,«),/?) = 1. Then
A{p,h^)^ (fc - d)!p"(k'd)
where fi(k,d) = \k{k -1)+ \d{d + 1).
Proof Write the congruence as
X ¥/!.,) = ft,- - I *>,) (mod p>).
i = d + 1 i = 1
Thus the number in question does not exceed
pkd max B(g)
g
where B(g) denotes the number of solutions of
X »P/n,) = ^-(mod p*)
i = d + 1
with nd + l5... ,nke(Z/pkZ)k~d. The condition on the Jacobian
implies that (cd + 1... ck9 p) = 1. Thus, by elementary row operations on
the matrix
\¥j(ni))d+ 1 ^ i,j^k
it follows that there is a w such that B(g) — C(w) where C(w) is the
number of solutions of
k
]T n\~d = Wj (modp7)
i = d + 1
with nd + l5..., nk as before. Referring again to the condition on the

Wooley's refinement of Vinogradov's mean value theorem 11
Jacobian one sees that the nd+ l9...9nk are distinct modulo p and
p > k — d. Thus one can appeal to Lemma 5.1 with k replaced by
k — d and j by j — d and deduce that
A{p,«,<P) < pkdmax C(w) < (k - d)\pkd + *(fc"d){k~d~l)
and this completes the proof of the lemma.
Of course, the underlying theme in the proofs of Lemmas 5.1, 5.2
and 5.5 is that one is counting the number of non-singular solutions.
Also, there are various occasions in the proofs of Theorem 5.1, and
Theorem 5.5 below when the starting point of an argument or process
is a separation of singular from non-singular solutions.
The process of bounding JS(X) in terms of Js _ k(X/p) is now much
more complicated than in §5.1 and involves a number of intermediate
steps which relate the system
k s
X OP/z,) - «F/r,)) + X (4 - y$ = 0 (1 < j ^ k) (5.50)
q = 1 r = 1
to the system
X Wj(zq) - Vj(tq)) + pj t (4 " vl) = 0(1 <; < k) (5.51)
q = 1 r = 1
Take KS(X9 F,*P) to be the number of solutions of (5.50) subject to
zq9tq^X(l^q^k) (5.52)
and
xr9yr^ Y(l^r^s) (5.53)
and LS(X9 Y9 0,/),¾1) to be the number of solutions of (5.51) subject to
(5.52),
zq = tq (modpfc) (1 ^ q ^ k) (5.54)
and
ur,vr^ YX~e (1 ^r^s) (5.55)
Suppose that *P is of type (d, A, X). Then there is a positive number
C0{k,A) such that

78 Vinogradov's methods
logX
uniformly for all X ^ 2 and all z with zr ^ X and J(^P, z) =^ 0. Thus
there is a positive number C^/c, ,4,0) such that the number
co(J(*P, z), X0) of different prime factors p of J(*P, z) with p > X0 does
not exceed C^/c, A, 0) uniformly for all X > 0 and all z with zr ^ X
and JOP,z)^0. Let ^(Z) denote the set of [20^,4,0)1 + 1
smallest primes greater than Z. It is now possible to state the
relationship between Ks and Ls. Such relationships are sometimes
described as a fundamental lemma.
Lemma 5.6 Suppose that O^d^s, X6 < Y ^ X and *P is of type
(d, A, X). Then there is a positive number C2{k, A, 0), a prime p in ^(Xd)
and a system <I> of type (d,A,X) such that
Ks(x,r,VK2kQYx\/s(y)
+ C2(X, 4,6)(X2sd + "(fc'd " 1)dLs(X, Y, 0,p,*))
where p. is as in Lemma 5.5.
Proof In the proof of this lemma implicit constants may depend on fc,
4 and 0 and KS(P,Q,¥) is abbreviated to K. Let /^(A) denote the
number of solutions of the simultaneous equations
I *W + I x; = fc,(i^M/c)
¢=1 r = 1
with z and jc satisfying (5.52), (5.53) and zl5... ,zk distinct, and let
K2(/r) denote the corresponding number with zl5...,zk not all
distinct. Thus K = Y,h(RiW + ^2(^))2 and so
K ^ 2/x + 2/2
where /, = I>;(A)2.
As in §5.1 the treatment of I2 is straightforward. Let
F(a) = X «(*•*)
z ^ X

Wooley's refinement of Vinogradov's mean value theorem
79
and
/(a) = ^ ¢(0^ + --- + <*kxk).
x^Y
Then
/2<
2s
F(a)2k" 4F(2a)2/ (a)2s | da
^,
By Holder's inequality, the integral here does not exceed
1 - 2/fc
F(<x)Zkf(<x)2s | da
^,
^,
\2/fc
F(2a)fc/(a)2s | da j .
By estimating trivially the function F in the second integrand one
obtains
I2^[ky K'-^lX1
\2/k
|/(a)|2Ma ,
^,
Therefore, when I2 ^ 1^ one has K ^ 2 (J) X JS(X), and in any case
/cNfc
X^2fcl2) J.W + 4/!
(5.56)
Clearly /x counts the number of solutions of (5.50) with the
variables subject to (5.52) and (5.53) and with the zl5... ,zk distinct
and the t19...,tk distinct. By the definition of ^, for each such
solution there is at least one member p of 3?(Xd) such that
(J(*,z)J(*,0,p) = 1
(5.57)
Therefore I± ^Yjpe&(xe)Ii(p) wnere Ii(p) denotes the number of
solutions of (5.50) subject to (5.52), (5.53) and (5.57). Again by the
definition of ^, it follows that card(^) <^ 1 and thus there is a prime
pe£P such that
h < /i(p).
Let
f{ar,y)= £ e{oLxx +--+ otkxk)9
x^Y
x = > (mod p)

80
Vinogradov's methods
and
G(*) =
X *(<*■*)
zi =¾ A\ .. .,z. ^ X
(J(<V,z),p)= 1
where
^) = ^) + -- + ^.(¾).
Then, since !P is of type (d,,4,X),
/i(p> ^
G(a)\2 I ••• I | H (a, h) 12da
^,
/11 = 1
*d=1
where
H(a,h)= X /(a^i) ---/(^¾)
ae^(li)
and ^(/r) is the set of solutions /i of (5.11) with /e = d. By Cauchy's
inequality and the arithmetic mean-geometric mean inequality one
has, in the notation of Lemma 5.2 and by Lemma 5.2,
\H(a,h)\2^B(p,h) £ |/(a,fll).../(a,£is)|2
ae^(fc)
ae3B{h) i = 1
Hence ^ <^ p2s~d- 1 Yfa= i /3(a) where
Ua) =
G(a)y(a,a)2s|da
^,
Therefore, there is an a such that
i^pt'-'iM.
The quantity I3(a) is the number of solutions of the simultaneous
equations
£ (^z,) - v,(g)
4= 1
s
+ X ((P«, + ay - (P»r + aY) =0 (1 < j < /c), (5.58)
r = 1

Wooley's refinement of Vinogradov's mean value theorem 81
with z91 satisfying (5.52) and (5.57) and u and v satisfying
— a/p < w,, vr ^ (Y — a)/p. (5.59)
By the binomial theorem
t Q(pur + ay(-ay-j=(pur)\
and it follows that each solution of (5.58) satisfies
k s
X (¢,(2,,) - ¢,((,)) + pl £ K " v'r) (1 < »' < *), (5.60)
q = 1 r = 1
where
¢-(^) = i Q ^(z)( - ay -j-
Conversely each solution of (5.60) satisfies (5.58). Thus I3(a) is the
number of solutions of the system (5.60) with z, t satisfying (5.52) and
(5.57) and u and v satisfying (5.59). Moreover, the system <b is of type
(d,A,X) and by using row operations on J(¥,z) one finds that
J(*,z) = J(V,z).
Let
g(<x) = X e(<*iPu + ••• + ockpkuk). (5.61)
u^YX °
and
£(<*) = X e(a.r),
zi ^ A",...,z, ^ A"
(J(*,2),p)= 1
where
1/2) = ¢^) +---+0.(2^.
Then
/i «P
2s -d
|£(a)2(l + #(a))2s | da.
Also
2s ^ 1 , i „/^\ I 2s
1 +0(a)|2s« 1 + 10(a)

82
Vinogradov's methods
and by considering those solutions of the underlying diophantine
equation with ur = vr one has
- d\s
(YX~U)
E(a)2 | da ^
^,
E(a)zg(ays \ da
®,
Hence
h<P
2s- d
2s
E(aYg{aYs | da
(5.62)
^,
The integral in (5.62) is equal to the number of solutions of the
equation (5.60) with the variables satisfying (5.52),(5.55) and
(J(*,Z)J(*,0,P)=1.
The function Of(z) is independent of z when i ^ d. Thus x^z) is
independent of z when i ^ d. Moreover for each solution counted one
has
xt(z) = Ti(t) (mod//) (d + 1 < i < k).
Now classify these solutions according to the common residue classes
modulo pl of the xt(z) and xt(t).
Let srf(h) denote the set of z in Z/pkZ with
Xi(z) = ht (modpfc) (d+l^i^k)
and (J(<b,z),p) = 1, let
E(a, n) =
Z
e(a.r),
zi ^ X,.. .,zk ^ A"
zi =«i (modpk),. ..,zk = nk (modpk)
and let
,d+ 1
D(a) = I
I
*d+1 = i (.,,= 1
Z £(a>«)
nestf(h)
Then
2s-d
/i ^rs_dK
where
K =
D(a) | #(aKs | da
^,

Wooleys refinement of Vinogradov's mean value theorem
83
By Cauchy's inequality and the notation of Lemma 5.5,
pd+i pk /»
0(*K Z ••■ I A{p9h9<t>) Z |£(a,/t)20(a)2s|da
^,1=1 *k = i
nejrf(h) J <&k
therefore, by Lemma 5.5,
Ix ^p2s + M(k.d-l) £ ... £
/ii = l n. = 1 »/
£(a,/i)2^(a)2s | da
^,
The lemma now follows easily from this and (5.56).
The system of equations (5.51) is now transformed using efficient
differences.
Lemma 5.7 Suppose that 1 < X6 < Y < X and that the system ® is of
type (d9A9X). Let H = Y1 ~ke and pe3?(6). Then there is an h with
1 < h ^ H such that the system 0 given by
satisfies
®j(z;h,p) = Aj(<t>j(zy9h9p) (1 < i ^ k)
LS(X9 r,0,p,*) < 2kkkXkJs( YX ~d)
+ 2k + lHkJs{YX ~ e)*Ks(X9 YX ~ *,©)*
Moreover, there is a positive number B depending at most onk9A99 such
that the system 0 is of type (d + l9B,X).
It can be observed that provided that X > X0(k9A96)9 so that
p < 2X\ the system 0 will be of type (d + l9k2kA9X).
Proof The final part of the theorem is immediate from the definition
of &>(0) and (iii) above. Thus one can concentrate on the main
inequality.
For brevity write L for LS(X9 Y9 69 p,<I>), and it will be useful to put
F(*)= Z
Z e(a&)
x= 1 z^X
z~x (mod p)
Let 1^ denote the number of solutions of (5.51) subject to (5.52),
(5.54), (5.55) and zq ^ tq (1 ^ q < k)9 and let I2 denote the number

84
Vinogradov's methods
with z = t for some q with 1 ^ q ^ K. Then by the definition of L,
L = IX+L
By (5.61),
12 ^ fcX
2s.
F(af " 11 #(a) | 2sda,
^,
and by Holder's inequality the integral here does not exceed
^,
)i - i/fc / >•
\ i/fc
I 0(a) I 2sda
= L1-1/kJ(YX-6)1
Ik
Hence, if /2 ^ /l9 then L ^ (2k)kXkJs( YX ~9), and in any case
L ^ (2k)kXkJs(YX-e) + 2/x. (5.63)
For each solution of (5.51) counted by Ix and each q with 1 ^ q ^ n
one has zq = tq (modpk) and zq ^ tq. Therefore, for some hq with
1 ^ I hq I ^ H one has
'« = *« + ft«Pk-
Let /3(1/) denote the number of solutions of the system of equations
X ,,0/z,; fc,, p) + £ W - «i) = 0 (1 < j < fc) (5.64)
¢=1 r = 1
in z,«, v, A and satisfying (5.52),(5.55) and
1 ^ h ^ H (1 ^ q ^ fc)
Then
/i< I ■• I ^3(1)-
rii = ± 1 >7k = ±1
Let #(a) be as in (5.61) but with p = 1, and let
z ^ X

Wooley's refinement of Vinogradov's mean value theorem
85
Then
n z go/,«,joW)12'd«
= 1 /i< H /
If Jfk\<l =
where the summation is over f/e { - 1, + l}fc. By Holder's inequality
El Z I Gfo,a, fc) I = ( X |G(or,h)Nk
^H""1 X |G(a,/i)|fc.
Hence, by Schwarz's inequality,
_ i
/i sS2\fir~*
g(a) 12% X
^,
M^H
2s.
G(a,h)\Zk\ g(<x)| 2sda
<%,
Thus, for a suitable choice of /i, one has
I, ^ 2kHkJs{YX~dfKs{X, YX~\ef
and the desired conclusion follows from (5.63).
Some useful information is now extracted from Lemmas 5.6 and
5.7.
Suppose that X is any positive number such that A < 2s and
JS(X) < Xk (5.65)
uniformly for all X ^ 1. Such numbers X certainly exist and /I = 2s is
an example.
Let j satisfy 1 ^ j'^ k and choose 6j,6j_1,.. .,81 iteratively by
putting
dj = i/k
(5.66)
and for i = j — 1J — 2,..., 1
a . /1 k + (2s + n(k,i-l)-X)Bt+l
9t = mm ^-, ^
Consider the inequality
Js + k(X)<Xk + * + X>'
(5.67)
(5.68)

86 Vinogradov's methods
where
X = /1(1 - 9,) + k + (2s + \k{k - 1))9,. (5.69)
This is established by showing, more generally, that for all systems *P
of type (0, l,X) one has
KS(X, X,V )<Xk + x + Xx' (5.70)
Clearly (z,z2,... ,zfc) is of type (0,1,X).
Put ¥(0) = ¥. Then, given ¥(0 take 6 in Lemma 5.6 to be 6t + , and
choose Pi and <I>(I) to be any p and <I> which satisfy Lemma 5.6. Then in
Lemma 5.7 put 6 = 0t + , and ^ = <$>{i) and define ¥(i + X) to be the 0
of that lemma.
As a first step it is shown inductively for i = j — 1J — 2,..., 0 that
Ls(X,Xr,0r + 1,Pi,<I><'>MMf+1 (5.71)
where
Xr = Xl~°l °r. (5.72)
The bound (5.71) is immediate from Lemma 5.7 when 0i+ 1 = l//c,
and this is certainly so when i = j — 1 and provides the first step in the
inductive proof of (5.71). Moreover, in view of this observation, one
may always suppose in the inductive step that 6i+ , < l//c, that is, by
(5.67), that
2k29i + ! = k + (2s + n(k, i) - k)6t + 2(0^i^j- 2). (5.73)
Suppose now that (5.71) holds with i replaced by i + 1 for some i
with 0 < i <j — 2. Then, by Lemma 5.6,
K8(X9Xi+l919<i+l))<XkX$+1 + X2sdi + 2 + ^i)di + 2 + kXt+2.
Since X ^ 2s it follows that
Ks(X9Xi+l9Wii+1))<X2"i + 2 + ll{k>i)ei + 2 + '<x}+2.
Hence, by Lemma 5.7,
L8(x9xi9ei + uPi^{i)) < xkxki+ x(i + xv)
where 2v = k + (2s + n(k,i) - X)6i + 2 - 2k26i + x. By (5.73) v = 0
which gives (5.71) as required.

Wooley's refinement of Vinogradov's mean value theorem 87
Now take i = 0 in (5.71) and combine with the bound in Lemma
5.6. It follows that when <P is of type (0,1,X) we have (5.70) with
(5.69), and so (5.68) is established.
The next task is to establish good choices for the above parameters.
Given s, suppose that (5.65) holds for
A = As = 2s-^/c(/c + 1) + rj
with
/I < 2s, n ^ 0.
Then
0<*7<£fc(fc + l)^/c:
(5.74)
First of all take )=1, so that 0X = £. Then X = 2s + 2/c —
^k(k +1) + n(l - £) and k + /I = 2s + 2/c - £fc(fc + 1) + 77 - /c. By
(5.74), fc + A < ;/. Hence
</s + fc(*M*A* (5.75)
with /I* = 2s + 2/c - £fc(fc + 1) + 77* and rj* = rj(l - £).
Since Jk(X) ^ fc!X\ it follows that rjk = ^k(k — 1) is permissible.
Now a simple inductive argument establishes that
is permissible, which recovers Theorem 5.1.
Now consider j = 2. Then 62 = k anc* 01 = min(£,£ — jfe) =
i-2^. Hence K = 2s + 2/c - %k(k + 1) + ^(1 - ^ + ^) and
/c + /I = 2s + 2/c - ^/c(/c + 1) + n - /c. By (5.74), £ < fa - f/c2)2, so
that fc + /I ^ A'. Thus (5.75) holds with
"* = ^-^ + ^} (5-77)
By (5.76), ?/zfc < ^/c2. Therefore the choice rj(l + 1)k = ?/Zfc(l — ^) is
permissible and so, generally one may take
^k = lfc(fc-l)(l-^y \ (5.78)

88 Vinogradov's methods
Finally, consider the general situation. Suppose that 7(/ — 1) ^ 2n
and i ^j — 1. Then n- = l/k and it easily follows by induction on
i = j — 1J — 2,..., 1 that
fc + (fc2+ ^(,-1)-,,)/&
^ 2P ^
and
U: = r .
2k1
Suppose further that S is a parameter at our disposal with
0<(5< 1
and that
j(j-l)^2Sn.
Then 0t ^ ^ + t^ + x where t = fc2"^2"^. Hence 0t - \i ^
T(^i + 1 ~~ A*) where [i = fc2 + (i _^. Now either 6{ ^ /i or 6{ > fi in
which case Or>jj. for all r with i'^ r ^ 7. Thus 0f — jj. ^ tj " l(£ — /i)
and this holds trivially in the former case. Therefore
where a = 21 ~ ^/c ~ 2. Hence one may take r\* = (k2 — r])d1 + r\ — k
and then
„ ^^ (fc2 - >?)/c(l +«) + (>?- /c)(/c2 + (1- %)
n " fe2 + (1 - %
/ 2 - 2<S\ 2fe " V + fc3a - <5/c^
^(1 k~)+ k> + (l-S)r, '
Hence 77* ^ ^/(1 — 2~k2d) provided that n ^ ^Sk2 and a ^ ^Snk~ 2,
i.e., by the definition of a, one has 77 ^ ^Sk2 and 4(5 " 1 ^ 2J. In order
for a suitable; to exist it suffices that log(4/(5)/log 2+1^ y/2n. Thus,
whenever

Wooley's refinement of Vinogradov's mean value theorem 89
one has
2 - 2(5
rj* ^ r\ I 1 —
Choose lx so that
1 / 5V1"1 1 1 / 5Y1"2
2k{k-l\l-4k) <4*fc<2k(k-1)(1"4*J ^^
and let /2 be the smallest /2 ^ /x such that
1,,(,-4-)--^. (,8o,
Now define ?/Zfc by (5.78) when / ^ /x and by
1 / 2-2(5Y"Zl
^ = ^Wl-^J (5.81)
when /x < / ^ /2. To see that this is permissible one argues as follows.
Firstly the case l2 = h is immediate since there are no / with
lx < I ^ /2. Secondly, if /2 > /1? then when / = ^ + 1 one has
2Vlog2; ^4
and v\(X_ 1)k < |<5/c2 = »7 say. Thus »7* ^ ^(1 — 1\lb) = ilk and one
may proceed by induction on /.
When I > l2 one is allowed to take, by (5.77),
** = ^(1-^ + ¾^} (5-82)
One shows in this case that
mk^ymM-^) * (5-83)
where y is the smallest (positive) root of the equation

90 Vinogradov's methods
It is shown first by induction on I that for I > l2 one has
"» < ^41" Yk) exp{ kW^) J'
Clearly, when I > Z2,
One has y ^ 1. Thus the case Z = Z2 + 1 is immediate. On the
inductive hypothesis, by performing the summation in the
exponential term, when I > l2+ 1, it follows that
*a-i*<*w(l-^)' ' '«P(3)^3))
Thus for I > l2 + 1 one has
«lk < *,- im.^1 - Tk)<*P[ k2{2k_3) )
and so on the inductive hypothesis once more
< ^ 3Y~\ A^l'=Ui-Ar-'^
"* ^ ** y-Tk) exp (—kw^)—J
as required. The desired conclusion (5.83) now follows easily by
summing the series in the exponential. A modicum of computation
shows that for k ^ 2 one has y ^ y0 < 7/5 where y0 = exp (y0/3), and
so
y= 1 +0(/e-2log2(8/<5)).
The following theorem has now been established.
Theorem 5.5 For each pair of natural numbers /c, Z there exists a
positive number C(kJ) such that for every X > 0
J^(X)^C(k9l)X2lk-m+l)f2 + ^

Wooleys refinement of Vinogradov's mean value theorem 91
where
" = !*<*-^-Ji)'"1- (5-85>
Furthermore, suppose that k ^ 3, S = 1/log k and lx and l2 are given by
(5.79) and (5.80). Then the above holds with n = nlk = n^ where nlk
satisfies (5.85) when I ^ /l5
^ = 4^(1 -hk^"^1^1^^ (586)
and
7 /log(8togfc)Y / 3V-'2
'» = 101 log2 j I1 " 2*) (/ > ^ (587)
Finally,
4k 1
/, = y log(21og/c) - -log(logfc) + 0(1)
and
''-'■-'*-"(■- Sit)" ' -((2^'^,8,0^)1 + °">
= /c(log /c — ^ log log /c — log log log k) + 0(k).
The following two theorems can now be deduced from the above by
arguments concomitant to those used to establish Theorems 5.3 and
5.4.
Theorem 5.6 Theorem 5.3 holds with
so that 2ak2 log k ~ 1 as k —► oo.
Theorem 5.7 Let a0 = max(cr, 21 ~k) and let s0 denote the least integer
such that
s0 > min^o-1^ + 2kl).
i

92 Vinogradov's methods
Then the asymptotic formula (2.27) holds whenever s ^ s0. Also
s0 ~ 2k2 log k as k —► oo.
5.6 Exercises
1 Show that if if is a sequence of natural numbers lk such that
Yuk= l 1/4 diverges, then for every s and /c0 there exists a k1 such that if
A(X) is the number of natural numbers n with n^ X which can be
written in the form
«= I x''
with the xfc non-negative integers, then A(x) > X1 ~L (X > X0(e9 k0)).
2 (Freiman's hypothesis, 1949; Scourfield, 1960). Let 5£ denote a
sequence of natural numbers lk. Then show that it has the property
that to every k0 there corresponds a k1 such that every natural
number n can be written in the form
«= z x<-
k
k0< k^ki
with the xk non-negative integers if and only if Y,k= i V4 diverges.
3 Let s0 be as in Theorem 5.4. Show that for 2s ^ s0 one has
1 q
z - Z
Q<q^RQ a=1
(a, 9) = 1
"!.
2s
^(NkQ_1+K)iV2s-k.
4 Show that \i X ^ \ and either fc = 2 and h^O or h
(0,... ,0, hk) with fck # 0, then
Jkk\X, Y,h)<$X1+£.
5 Suppose that 0 < B < 1 < A,
1 A
/:[l,oo)->R:/(x) = e
(7 = max/(/),
z
- Bx
and let x0 denote the positive root of the transcendental equation
eBx° = A{\ + Bx0).

Exercises 93
Show that
where 11 — x0 | ^ 1 and xx lies between x0 and /. Deduce that when
B ( ( B2
0 = - —- 1 +
1 + Bx0 \ \log At
6 (Wooley, 1995a). Show that Theorem 5.3 holds with
f ok1 log k ~ 1 as k —► 00.

Davenporfs methods
6.1 Sets of sums of /cth powers
It was demonstrated in §5.4 that the upper bound for G{k) could be
radically reduced by first of all constructing a set Ji of natural
numbers m not exceeding Zh which are the sum of t /cth powers. The
construction yields card^ > zka, where a = 1 — (1 — l//c)r, and is a
slight simplification of one due to Hardy and Littlewood (1925). In
fact, in their construction Z} is as above for 7 = 1,. . ., t — 1, but they
take Zt = Zt_ 1. The argument proceeds as before until the (t — l)th
step, when (5.39) reduces to
For each given pair yt-l9yt9 the number of choices for
^t - i> xt
is <^ Z\. It follows that
X6zM2^z1...zr_1zr1 +
m
2
< lez("<) (z1...z,_1zj-T^
m
Moreover
z^..zt>zk-*-2^-^-\
Hence, by Cauchy's inequality,
y 1 > zk~ik~2)(1 _ i/k)t~2 _f
m
Qz(m) > 0
Let Nt(X) denote the number of natural numbers m not exceeding
X such that m is the sum of at most t /cth powers. Then this yields
Nt(X)>X«<-£(X>X0(t,e)) (6.1)
with
••^-HX1-^"2- (62)
Note that a2 = 2/fc.

Sets of sums ofkth powers 95
There have been a number of refinements of this argument, which
have been effective in giving improved upper bounds for G(k) when k
is relatively small. The following theorem is a generalization of one
due to Davenport and Erdos (1939).
Theorem 6.1 Let t> 3,6=1- 1/fc, ^ = 1,
/e2-flf"3 k2 - k - 1
A2-k2 + k_ke<-3> AJ-k2 + k_ke<-30J (3^7^),
and Q(m) denote the number of solutions of
x\ + . . . + x* = m with Zx-> < x} < 2ZA-\ (6.3)
Then
XQ(m)2«ezAl + --- + A< + £.
m
Corollary T/ie inequality (6.1) toWs wif/i a, = 1 — p where
k3-3k2 + k + 2 Qt_3
"-p + e-M-** • (64)
The corollary follows by using Cauchy's inequality in the same way
as above.
Proof of the theorem Let Ms denote the number of solutions of
x* + . . . + xks = y\ + . . . + yks (6.5)
with ZAj < xpVj < 2ZAj and xs =£ ys. Since M1 = 0,
t
ZQM2^ Z MsZx>+i+- + x< + ZXl + ~- + \ (6.6)
m s = 2
Also M2 is the number of solutions of
x\ - y\ = x\ - y\
with x2 =£ y2 and ZA-j < xj5 y^ < 2Zkj. For each given pair x2, y2
with x2^=y2 the number of possible choices for xl5 y1 is <^Z£.
Hence
M2^Z2A2 + £^ZAl + A2. (6.7)
For s > 3,
MS = M; + 2M; (6.8)

96 Davenport's methods
where M's is the number of solutions of (6.5) with the additional
constraint x1 = yl9 and M's' is the number with x1 > y1. Then
M's<Zk'Ls (6.9)
where Ls is the number of solutions of
x*+... + xJ = /2+ ... + }£ (6.10)
Given x2,...,xs, the number of y2,...,_ys is <^ l(c.f. §5.4). Thus
Now consider M'J. The number of choices for x2, y2 is < Z2Al. For
any such choice (6.5) becomes
x\ -fi + A+ t (*5 ~ y)) = 0 (6.12)
./=3
where A is fixed. Let h = x1 — yv Then x\— y\> hZk ~ 1. Also
s
A + X (*5 - ykj) < zkkl-
7=3
Hence 0 < h <£ Zkkl ~k + \ and (6.12) can be rewritten in the form
A+(y,+hf-y\<Zkk\ (6.13)
For a given h let y and y + j be two possible values of y\ for which
(6.13) holds. Then
(y + j + h)k - (y +j)k -(y + hf + yk < Zu\
whence hjZk ~2 < Zkk\ Thus the number of possible choices for y1 is
<i + Zk^~k + 2h~1. (6.14)
For given x1? y1, (6.12) becomes
^i+ t (xk-yk) = o (6.15)
j=3
where ^ is fixed. The number of choices for y3, ...,ys-i
is <^ ZA3 + • + As-1 ancj for any sucn choice the number of choices
for x3,. .., xs _ 1 is <^ 1 (observe that xk4 + .. . + xks< Z^6 and that
there are <^ 1 values of x% in an interval of length ZA'0, and so on).
Given y3,. . ., ys_ x, x3,.. ., xs_ 1? (6.15) becomes
A2 + xk-yk = 0
where A2 is fixed, and since xs =fc ys the number of choices for xs, ys
is <ZE. Therefore, by (6.14),

Sets of sums ofkth powers 97
Ms < Z2k2 X (1 + zkkl ~k + 2h~ X)ZA3 + ••• + A- i + £.
0 < h < ZkX> ~k + l
Thus, by (6.8), (6.9), (6.11),
, y2X2rykX2 - k + 1 , V^3 ~k + 2)7*3 + • • • + As - 1 + 2e
The theorem now follows from (6.6) on observing that for s = 3,. .., t,
(k + 1)X2 - k < ks and A2 + kX3 - k + 1 < As.
The following theorem is due to Davenport (1942a).
Theorem 6.2 Suppose that l^j^k — 2, 0 < v < 1, s/ is a set of
natural numbers a with a^Zv + k~1,S = card s/9 Q(m) is the number of
solutions of
xk +a = m
with Z < x <2Z and aestf, and T = £m2(w)2. Then
T<$ZS(l+Zv + £(Z-2+Z-v~j- 'S)2'').
Proof Let A7 be as in § 2.2 and write
jPj = {h:hj>0; h1 < Zv; h2,. . ., h}< Z).
Let Pj(h, m) denote the number of solutions of
Aj(xk;h)+a = m with Z<x<2Z, aesrf, (6.16)
and put
mj= I Ip,(M). (6.17)
Clearly
T<ZS + MV (6.18)
Also, by Cauchy's inequality,
MJkZ^'-'S^ ZPj(h,a)2.
The double sum is the number of solutions of
.k . l\ , „ _ A /~k
Aj(x\ ;h) + a1= Aj(xk2; h) + a2=a
with Z <x1?x2 <2Z,a1e.srf,a2estf,aejrf,heJ#:>j. Since the elements

98 Davenport's methods
of j?/ are distinct this is <^ Mj + Mj+V Hence
Mj^Zv + j-1S + (Zv + j-1SMj+l)1/2.
Thus, by induction on j,
M1«Zv + 1"21'-'S + Z(v+1)(1-2"-')-^"-'S1-2'-'M?;J1. (6.19)
By (6.17), Mj+ 1 is the number of solutions of
Aj+1(xk;h)+a1 =a
with Z < x < 2Z, AeJf j+ 1? a1es^/,aes/. By Exercise 2.1, when
7"^ /c — 2, for each pair al9 a the number of choices for x, /r is <^ Z£.
Thus Mj+1< S2Z£. The theorem now follows from (6.18) and (6.19)
Theorem 6.2 is usually applied iteratively to give lower bounds for
Nt(X) for successive values oft. More generally, let jrf denote a strictly
increasing sequence of natural numbers a with the property that
A(X) = card {a :aes/, a ^ X} (6.20)
satisfies
A{X)>X«-£ {X>X0{e)), (6.21)
where 0 < a < 1, and let N(jtf9 X) denote the number of different
numbers of the formx* -fa with xk + a < X andaestf. Let Z = \X1,k.
Then, in the notation of Theorem 6.2,
N{s/,X)> ^ 1.
m
Q(m) > 0
and by Cauchy's inequality
I 1 Zfilrn)2^ lew >Z2S:
m / m \ m
Q(m) > 0
where S = A(ZV + k x). Hence, by Theorem 6.2,
N(s/, X)> ZS(1 + Zv + £(Z~ 2 + Z~v~j~ 'S)2'Jy'.
Thus, by (6.21),
N(*f, X)>XP~E (X>X1(e)) (6.22)
where
/f=I(l+a(fc-l) + T)
/c

Sets of sums ofkth powers 99
and
t= max sup (min(va, 21 ~J' — v(l — a),
1 ^ j<k-2 0 <v < 1
0 + l)2-J'-(/c-l)a2-^-v(l -a)(l -2~j))).
When j' + 1 ^(/c — l)a the above supremum is non-positive, so the
maximum occurs for a value of j with j + l>(k— l)a. For such a
given value of j the supremum occurs when v is the lesser of the two
values given by
vcc = 21 ~j — v(l — a),
va = 0" + 1)2"j - (k - l)a2"j - v(l - a)(l - 2" j\
i.e. by
; + l-(/c-l)a
V = 2 \ V= : .
2J - 1 + a
Thus
. f j+i-tfc-pa
i = a max mm 2 J, :
izjzk-2 \ 2J - 1 + a
Consider the inequality
j + l-(k-iyx ;-(/c-l)a
2j - 1 + a 2j ~ 1 - 1 + a
This is equivalent to each of the following inequalities
2l-,-^ + l-(/c-l)«
^ 2-* - 1 + a '
1 + (/c - l)a > j + 21 " ■'"(l - a). (6.23)
The right-hand side of (6.23) is a strictly increasing function of j. Thus
if J is the largest value of; such that (6.23) holds, then
J+l -(fc-l)a
t = a 1 ,
2J - 1 + a
and if there is no such value of 7, i.e. if a < l//c, then t = a. This
establishes the following theorem.
Theorem 6.3 Suppose that -$4 satisfies (6.20) and (6.21). Let
H = [(/c — l)a], and J = H + \ when
2H((k-l)(x-H)>\-oc and H + 1 ^ k - 2

100 Davenport's methods
and J = H otherwise. Then N(,tf, X) satisfies (6.22) with
n 1/, ,, , J+l-(/c-l)a
p = - l+«fc-l) + « ,
/c\ 2 — 1 + a
when a ^ l//c and /J = 1/fc + a vv/ien a < 1/fc.
It is useful, in the case of fourth powers, to have a slight refinement
of this. The above argument is not materially altered if Q(m) is taken
to be the number of solutions of m = x4 + a with
Z < x < 2Z, x = r(mod 16), ae-rf, a ^ Zv + 3. Likewise the argument
that gives (6.1) with (6.2) is essentially unchanged when each Xj is
restricted to a given residue class modulo 16. Thus
Theorem 6.4 (Davenport, 1939c) Let N{th)(X) denote the number of
natural numbers n not exceeding X in the residue class h modulo 16 which
are the sum of t fourth powers. Then for t^l and 0 ^ h ^ min(t, 16),
N[h)(X) >Xa<-e (X> X0(e, t)) (6.24)
where
1 _3 + 13ar
2' *' + x ~ 12 + 4ar
a2=-, cct +1 = ^ A . (6.25)
/m particular,
™ — 12 /v — 331 „ _ 5539 //: 9£\
a3 — 28? a4.— 4.12? a5 — 6268- ^U.Z,U;
Davenport (1942a) has given an improvement upon the argument
of Theorem 6.2 which is particularly effective when k = 5 or 6. With
the same assumptions as in Theorem 6.2, let Q(m) denote the number
of solutions of
xk + pka = m (6.27)
with Z<x<2Z, a^Zv + k~\ \Zl "v < pk < Z1" v, plfx. Also, let
Q(m, p) denote the number of solutions of (6.27), for a given p, with
Z < x < 2Z, a^Zv + k~ 1, p\x. Then, by Cauchy's inequality,
T = ZQ(m):
m
satisfies
T<PYYQ(m,p):
m p

Sets of sums ofkth powers 101
where P is the number of primes p with \Zl ~v < pk < Z1 ~v. For a
given prime p and integer r with p\r the number of solutions of
xk = r(modpk) is 0 or (k, (j){pk)\ Thus the integers x with p\x can be
divided into q{p) = (/c, </>(pfc)) classes ^1?..., &qip) such that, if x and _y
are in a given class ^n then xk = yk (mod pk) if and only if
x = y (mod pfc). Let Qr(m, p) denote the number of solutions of (6.27)
with Z < x < 2Z, a ^ Zv + k ~ 1, xe$r. Then, by Cauchy's inequality,
/q(p) \2
KPZI I Qr(m, p)
m p \r = 1 /
^L Iie>,P)2
r = 1 p m
where Qr(m, p) is taken to be 0 when r > q(p). The triple sum here is
bounded by the number of solutions of
x\ -\-pkal =x\ + pka2
with x1= x2(modpk) and xl9 x2, a 1? a2, p' satisfying the same
conditions as before.
Let Aj be as above and write
jfj ={h:hi>0,h1< 2ZV; /i2,. . . , hj < Z}.
Let pj(h, m, p) denote the number of solutions of
p~kAj{xk; pkh^ h2,. . .,hj) + a = m
and put
mj = Z Z Z Pjik a> P)-
Then, as in the proof of Theorem 6.2,
T^PiPZS + MJ
and
Mj<$Zv + j~1PS + (Zv + j-1PSMj+1)1/2.
Thus, if it can be shown that
Mj+1<$S2Z\ (6.28)
then it follows that
T < P2ZS(i +ZV + £(Z" 2 + Z~v-j^1p- 'S)2"J) (6.29)
and the extra factor of P~ 1 in the innermost bracket gives an
improvement over Theorem 6.2.

102 Davenport's methods
It is probable that (6.28) holds whenever j ^ k — 3, but this seems
rather difficult to prove in general. However, it can be established for
certain values of j. Consider the central difference operator V,- which
can be defined in terms of A, by
V-(/(a); pi9 . . . , p.) = A;(/(a -^-...- #.); pi9. . . , j?,).
Then
V/a*;^,.. .,£,.)
01= + 1 0, = ± 1
= 11 ■••! , ,,*' n*-k+JWi-P/
/0 /i /, »o-'l ■* • -'j-
21/! 2pj
/o + /i + ... + /j = k
= P1...PJ L --L /^(2/, + 1)1...(2/.+ 1)1-
/0 + 2(/1 + ... + /j) = fc- j
If /c —7 is odd, then /0 > 1 in every term, and so
V,V; pi9. . . , j?,-) = ajBi . . . jByp,(a; j?1?. . . , j?,)
where
Pj(cc; Pl9..., Pj)
_ _ k\2lo+1-k + jalopill...p^
f "fr (/0 + 1)1(2/^1)1...(2/,+ 1)1
/o + 2(/i + ... +lj) = k - j- 1
If k —j = 2, then
Vy(afc; j»x j87) = j»!...£y||y(12a2 + #+•.+ #)•
The number Mj + x can now be reinterpreted as the number of
solutions of
p~ kVj +1(ctk; hxpk, h2,...,hj+1)+a1=a2
with a = x + \hxpk + . . . + ify + i- When k — j — 1 is odd and positive
P"fcVJ+ i(afc; /i1pfc, /i2,. .., hj +1)
= a/]1 . . . hj+lpj+l{(x\h1pk, h29 ...,hj+1)

Sets of sums ofkth powers 103
which is positive. Given ax,a2 the number of choices for a,
hl9. . ., hj+ 1? i.e. for x, /il9. . ., hj+ 1? is <^ Z£. If moreover /c — j — 1
> 3, then pJ+ x(a; /?l9. . ., j?j+ J is a polynomial in /^ of degree at
least 2. Thus given au a2, oc, h1,. . ., hj+ 1? the number of choices forp
is < 1. Hence in this case one has (6.28).
When k-j - 1=2,
p~kVJ+1(<*k;hy, h2,...,hj + J
2^'+1/c!
2*3!
= hi---^+i^fcTr(12a2 + P2kh? + hi+... +^2+i).
Given a1? a2, the number of choices for /i1?. .., hj + 1 is <^ X€. Then
given a1?a2, /il9. . ., hj+ 1? the number of choices for a, p, i.e. for x, p, is
again <^ Xs, since the number of solutions of 3u2 + v2 = mis < m£.
Thus, if 7 = k — 3, then (6.28) holds once more. This yields
Theorem 6.5 (Davenport, 1942a) Suppose that l^j^k— 4 and k —j
is even, or thatj = k — 3. Suppose further that 0 < v < 1 and s# is a set of
natural numbers a with a^Zv + k~ 1. Let Q(m) denote the number of
solutions of
xk + pka = m
with Z < x < 2Z, aesrf, \Zl "v < pk < Z1 " v, p/x, let T = £m6(m)2,
and let S = card j&. Then
T <P2ZS(l+Zv + E(Z-2 + Z-v-j-1p-1S)2~J)
where P is the number of primes p with 2Zl ~ v < pk < Z1 ~ v.
Corollary Suppose that (6.1) holds, that l^j^k — 4 and k—j is even,
or that j = k — 3. Then
Nt+1(X)>X«< + i-£ {X>X0(t + l,s))
with
and
aI + 1=-(l+at(fc- 1) + 1,)
t ; = a, mm 2 * J, : T
- l

104 Davenport's methods
This follows from Theorem 6.5 in the same way that Theorem 6.3
follows from Theorem 6.2.
Suppose that k = 5. Then (6.2) gives a2 = f, the above corollary
gives
16 +85a, 1 ,
a' + 1=ItT6T5^) when *<««<*
and Theorem 6.3 gives
7 + 33a,
a, _l 1 =
i + 1 c/"7 i - \ w"wi 5 ^ ^r
when 1^ a, < 1
5(7 + a,)
Hence
Theorem 6.6 (Davenport 1942a) W^/zen /c = 5, (6.1) /?o/ds wzf/?
n, —2 ~ _5 _569 _6913439 (> C)Q\25^)
oc2 — 5, oc3 — 9, oc4 — 845, as — 15161i5{^>v.yiz.jj).
Now suppose that /c = 6. Then (6.12) gives a2 = ^, and the corollary
to Theorem 6.5 gives
19
19 +120a, , ,
a,+ !=-—— when T^a, <42
r+1 6(19 +6a,) 3 ' 42
43 +246a, 19 2
<*/+!= 1 — when if <• a, < 4
r + 1 6(43 +6a,) 42 ' 3
and Theorem 6.3 gives
15 +81a,
a,x,=
t+l~//K , v ™A1WA 3 ^ ^r
when 4^ a, < 1.
6(15 +a,)
Hence
Theorem 6.7 (Davenport, 1942 a) When k = 6, (6.1) /zoWs with
„ —I ^ __59_ _ 1661 „ _ 5549 „ _ 575 117
^2 — 3' a3 — 126' a4 — 2886' a5 — 8379' a6 — 787 182'
o/ _ 24 040 980 990 984 981 / ^ p OAfi Q1A
a13 — 25 335 323 032 000 606V > V.!7HO yV).

G(4) = 16 105
6.2 G(4) = 16
It is useful to introduce here the generating function
/i(a)= Y, e(ccxk)
X < x< 2X
and the corresponding auxiliary functions
w(a) = X -x1//c~ ^(ax)
X* < x < (2X)t< k
and
W(ol, q,a) = q~1S(q, a)xv((x —a/q)
where S{q, a) satisfies (4.10). The following lemma is then
from Theorem 4.1 on taking n = \2X~\k and n = [X]k.
Lemma 6.1 Suppose that (a, q) = 1 and a =a/q + /J. Then
/7(a) - W(a, 4, a) ^ <?1/2 + £(1 + Xk|j5|) (6.33)
and z/ moreover |j8| ^(2kq)~ ^2^)1 ~fc, f/zen
/7(a) - W (a, ¢, a) ^ <?1/2 + £. (6.34)
One reason for this kind of choice for h(cc) is that it fits more readily
into the area of ideas used in § 6.1. Another reason is exemplified by
the next lemma which shows that h(a/q + /J) decays like || j8|| " 1 as ||/J||
grows, rather than like || j8|| ~ 1/k as for f(a/q + /J) (cf. Lemma 4.6). This
is not usually of vital importance, but can often result in a reduction of
technical difficulties.
Lemma 6.2 Suppose that |j8| ^ \. Then
W(P)<x(i+xk\p\ri.
This can be shown in the same way as Lemma 2.8. As an immediate
consequence of this and Theorem 4.2 one has
Lemma 6.3 Suppose that (q,a)= 1. Then
W(a/q + P, q, a) < Xq~ 1/fc(l + XkW\\y \
The following theorem is due to Davenport (1939c) and is still the
best that is known for fourth powers. Exercise 2.2 gives G(4) ^ 16.
(6.30)
(6.31)
(6.32)
immediate

106 Davenport's methods
Theorem 6.8 Suppose that n^O or — 1 (mod 16) and n is sufficiently
large. Then n is the sum of fourteen fourth powers.
Corollary G(4)= 16.
Proof of Theorem 6.8 Choose h1,h2,j so that
hi+h2+j = n(modl6), 0^/1^4, 0^/i2^4, 1^^6.
Let
243
V = 1567' *-"1/4' (6J5)
and let srf(h) denote the set of natural numbers a such that a < X3 + v,
a = h(mod 16) and a is the sum of four fourth powers. Further, let
ae^(/ir)
Then, by Theorem 6.4,
3972
Kr(0) >*"-', /^=^ (6.36)
By (6.30) (with k = 4),
n
|/i(a)K»|2da = XGM2
0 m
where Q(m) is the number of solutions of
x4 +a — m
with X < x <2X and aesrf(hr). Hence, by Theorem 6.2 with k = 4,
7=2,
\h(a)Vr(oc)\2 doc <^ AT, (0)(1 + Xv + E(X~ 2 + X~ v" 3 K,(0))1/4).
o
Hence, by (6.36) and Cauchy's inequality,
IMa^K^^K^^Ida^^K^O)^^)^-^, y = ffff. (6.37)
o

G(4) = 16 107
Define the major arcs Wl(q, a) by taking P = (2X)/(2k) = X/k and
aR(g, a) = {0L:\0L- a/q\ < Pq~ ln~ l}.
Let 90¾ denote the union of all the $R(g, a) with 1 ^a ^ q ^ P and
(a, ^)=1. Then the Wl(q, a) are disjoint and lie in ^ =
(Pn~\ 1+Pn"1].
Let m = JU \SIR. By Weyfs inequality (Lemma 2.4) and the argument
used to prove Theorem 2.1,
h(oL)<$X1/8 + £ (aem).
Hence, by (6.35) and (6.37),
|fi(a)6^(^^(aJIda ^ n1/2 "^(0)^(0) (6.38)
m
where S is a suitable positive constant.
In a similar manner to the proof of Theorem 4.4, one has, for
1 ^ m ^ n,
h(oc)6e( - am)da = /(w)S(w) + 0(n1/2 ~ d) (6.39)
where
I(m)= ^ ••• I 4-6(x1...x6)-^
X4<x!^(2X)4 X4<x6<(2*)4
x i + ... + x6 = m
and <3(ra) is the singular series defined in Theorem 4.3. It is
readily verified, for instance by considering the with
XA < Xj < 2X4, that
I(m)>n1/2 when \n < m ^ n. (6.40)
By Lemma 2.15, when s = 6 and p > 2, one has M*(py) > 0.
Moreover, when s = 6, p = 2 and m = j (mod 16) with 1 ^ )^ 6, it is
trivial from the definition of M* in §2.6 that M*(2y) > 0. Hence, by
Theorem 4.5,
<3(ra) > 1 when m =j (mod 16).
If m = n — a1 —a2 with ares#(hr), then m satisfies fn < m ^ n and
m =7 (mod 16). Hence, by (6.39) and (6.40),
h(oc)6VMV2^)e(- ^n)d^ = J(n)+ 0(n1/2 ~dV\(0)V2(0))

108 Davenport's methods
where J(n)> ^^(0)^(0). Hence, by (6.38)
6
hioifV^a^i^ei- an)doi
o
R(n) =
satisfies
R(n)>n1/2V1(0)V2(0)>0.
Hence n is the sum of fourteen fourth powers as required.
6.3 Davenport's bounds for G(5) and G(6)
Theorem 6.9 (Davenport, 1942b) G(5)^23, G(6)^ 36.
The proof of this is similar to but simpler than that of Theorem 6.8.
On this occasion it suffices to adopt the notation of §4.4, so that
(4.29),..., (4.32) hold.
Let r = 7, t = 8 when k = 5, and r = 10, t = 13 when k = 6. Further,
let jrf denote the set of natural numbers a not exceeding j$n for which a
is the sum of t /cth powers and write
V(ol) = Y, e(ota).
ass/
By Theorems 6.6 and 6.7,
i
|K(a)|2da< K(0)2tt_"
o
where \i = 0.912 53 when k = 5, and \i = 0.948 91 when k = 6.
Let m = ^\9M. Then, by Weyl's inequality (Lemma 2.4),
|/(a)'K(a)2|da < nr/k~ ' ~dV(0)2 (6.41)
m
where S is a suitable fixed positive number.
By Theorem 4.4, when 1 ^ m ^ n,
I*
f((x)re( - am)da = Cmr/k ~ ^(m) + 0(nr/k ~ ' ~d) (6.42)
Jan
where C is a positive number depending only on k and r.
By Lemma 2.15 with s = r, k = 5 or 6 and n replaced by m, one has
M*(py) > 0. Hence, by Theorem 4.5, S(ra) > 1. It now follows easily

Exercises 109
from (6.41) and (6.42) that
n
f(a)rV((x)2e(- (xn)d(x |> nr/k ~ ^(0)2 > 0,
0
and therefore that G(k) ^ r + It when k = 5 or 6.
6.4 Exercises
1 Show that for X > X0 one has
N19(X)>X0'9668 when /c = 7,
N28(X)>X0-9838 when /c = 8.
Deduce that G(7) ^ 53*, G(8) ^ 73.
2 (Davenport, 1939a) Let Q(m) denote the number of
solutions of m = x3 + y3 + z3 with Z < x ^ 2Z, Z4/5 < y ^ 2Z4/5,
Z4/5 < z ^ 2Z4/5. Show that
X6(m)2^Z13/5 + £.
m
Deduce that (i) G(3) ^ 8, and (ii) almost every natural number is the
sum of four positive cubes.
3 (Davenport, 1950) Show that when k = 3 one has
N3(X)>X^5*-* (X>X0(e)).
4 (Vaughan, 1985) Let Q(m) denote the number of solutions of
m = x3 + y3 + z3 with Z < x ^ 2Z,y ^ Z5/6,z < Z5/6. Show that
r1
m JO /i =¾ H
where H = Z1/2,
A(°o = X e(a(* + ^)3 ~ a*3)>
Z < x ^ 2Z
g(cc) = £ ^(ay3).
y ^ Z5/6
*Note that the claim G(7) ^ 52 of Sambasiva Rao (1941) is based on an arithmetical
error.

110 Davenport's methods
Show also that whenever (a, q) = 1 and | a — a/q | ^ q ~ 2 one has
Z !//,(«) 12 ^ # z2 + 7<? + HZl + £ + <?z£-
Deduce that
Z6(m)2^Z8/3+£
m
and
JV3(X)>x8/9-£(X>*0(£))
when /c = 3.
5 (Vaughan, 1989a). Show that when k = 3 one has
N3(X)>X19>21-*(X>X0(e)).

7
Vinogradov's upper bound for G(k)
7.1 Some remarks on Vinogradov's mean value theorem
For the purposes of this chapter, the notation of Chapter 5 is
assumed.
By (5.3), J(k)(X, 0, h) is the number of solutions of
t(4-yl) = hj (l^j^k) with 0<Xr,yr^X. (7.1)
r = 1
This system of equations is not soluble when \hj\ > sXj for some j.
Hence, by (5.4),
£./ifc)(*, 0, h)< Xkik+ l)/2Js(X). (7.2)
h
On the other hand, the left side of (7.2) counts all the solutions of
(7.1) with h considered as an additional variable. Thus
Js(X)^X2s-k{k+i^2.
Recall that JS(X) is the number of solutions of
t(4~yi) = 0 (\^j<k) with 0<xr,>v^*- (7.3)
r= 1
Obviously the number Ts(x) of'trivial' solutions obtained by taking
the yr to be a permutation of the xr satisfies
\_XY^Ts(X)^s\Xs.
Thus
Js(X)p max(A'2s-*(fc+l)/2, Xs) (7.4)
which shows, incidentally, that (7.3) has 'non-trivial' solutions
whenever s>\k(k + \) and X is sufficiently large. For further
comments see §29 of Hua (1956) and Vaughan & Wooley (1995b,
1997).
It can be conjectured that, when k ^ 3, as X —► oo
JS(X) ~ Cs?fc max (X2s~k{k+ 1)/2, Xs). (7.5)
While this probability lies very deep, at any rate it is possible to

112 Vinogradov's upper bound for G(k)
establish it when s ^ k + 1 and when s is sufficiently large. The latter
is done by adapting the Hardy-Littlewood method to the /e-
dimensional unit hypercube tf/k. The minor arcs are dealt with by
applying Theorems 5.1 and 5.3 which can be thought of as the
analogues of Hua's lemma and Weyl's inequality respectively. For the
major arcs it is necessary to develop an asymptotic approximation for
the generating function
/(«)= X e(ai* + ... + afc:>cfc) (7.6)
x< X
and to estimate the corresponding auxiliary functions
Hfi) =
e(Piy + ... + Pkyk)dy,m (7.7)
0
S(q9a) = S(q9al9 . . . 9ak) = £ e^a^x + . . . +akxk)/q). (7.8)
x= 1
7.2 Preliminary estimates
Much of the material in this section is due to Hua (1940a, 1952,1965).
It is convenient here to recall that the polynomial congruence
0(x) = b0 + ^ix + • • • + bkxk = 0 (mod/?)
is said to have a root of multiplicity m at x0 when (t>(x) =
(x — x0)m(t)1(x)-\- p(t>2(x) with 0i(x) and (j>2(x) polynomials such
that pl<j)^x0\
Theorem 7.1 Suppose that (q, k) = 1. Then
S{q,a)^ql-llk + \
Proof In a similar manner to the proof of Lemma 2.10, when
(q, r) = (qr, al9. . . 9ak)= 1 one has
S(qr, al9 . . . , ak) — S(q, al9 ra2, . . . , rk ~ 1ak)
x S(r,al9qa2,- . . ,qk~ lak).
Thus it suffices to treat the case when q is a power of a prime.
Suppose that p\{al 9a2,...,ak) and px is the highest power of p dividing
(al92a2,...9kak). Let xl9...9xr denote the distinct roots of the
congruence

Preliminary estimates 113
p~ l(a1 + 2a2x + ... + kakxk ~ x) = O(modp)
and suppose that their respective multiplicities are m1?.. ., mr. Note
that r ^ k — 1. Further let m = m1 + . . . + mr. Then it suffices to show
that for I — 1, 2,. ..
\S(pl, a,,..., ak)\ ^ /c2max(l, w)p'"l/k. (7.9)
Since m ^ k — 1 this easily gives the theorem.
The case I = 1. The argument in this case is due to Mordell (1932)
and gives more, namely that
\S(p,a1,...,ak)\^kp1-1/k. (7.10)
It can be assumed without loss of generality that p\ak and p > k.
Consider
T= £ ... £ |S(p, Zl>..., zj2*. (7.11)
Z! = 1 Zk = 1
Then, by multiplying out the summand, applying (7.8) and inverting
the order of summation, one obtains
T = pkM (7.12)
where M is the number of solutions of the simultaneous congruences
x{ + . . . + x{ ee y[ + . . . + y{{modp) (1 < j^ k) (7.13)
with 1 ^ x{ ^ p, 1 ^ y• ^ p. In a similar manner to the proof of
Lemma 5.1, it follows that if x^..., xfc, y1?..., >'fc satisfy (7.13), then,
for each x, Y\i(x ~ xi) — Y\t(x ~ y;)(m°d pi Thus the x1?..., xfc are a
permutation of the yl9. . . , yk. Therefore M ^ k\pk, and so, by (7.12),
T^k\p2k. (7.14)
When p\u, ux + v runs through a complete set of residues modulo p
as x does. Let
bj = bj{u, v)= Y, ai[ • JhV"-'.
Then
\S(p, «!,..., ak)| = \S(p, bl9. . ., bk)|. (7.15)
Moreover frfc = akuk and frfc _ 1 = uk ~ 1 (vkak +ak_ x). Thus, as u varies,
bk takes on (p — l)(/c, p — 1)" 1 distinct values modulo p, and for a
given u, as i> varies, bfc _ x takes on p distinct values modulo p. Hence,
by (7.15) and (7.11),

114 Vinogradov's upper bound for G(k)
P(P ~ 1)
(K P -1)
\S(p,au...,ak)\2k^T.
Therefore, by (7.14),
\S(p9 al9...9 ak)\lk ^ k\2kp2k ~ 2 ^ k2kp2k ~ 2,
which gives (7.10) as required.
The case I > 1. This is proved by induction on /. Obviously px < k.
Thus, when 2 ^ / ^ 2t + 1, (7.9) is trivial. Hence it can be assumed that
/ ^ 2t + 2.
For brevity write <\>(x) = a1x-\-. . .-\- akxk. Recalling that
are the distinct solutions of p~x<t>'{x) = 0(modp) one obtains
where
S{pl9al9...9ak)=T0 + £ 7} (7.16)
j= i
7} = X e{<t>{x)p-1) (7.17)
X = 1
x = x}(mo6p)
and
7o= E ^ e(4>(pl ~* ~ 'z + y)p-l). (7.18)
y = 1 z = 1
Since I > 2t + 2 one has
0(p'-T-1z + y)^0(y) + ^-T-1z0'(y)(mod^).
Hence the innermost sum in (7.18) is zero. Thus it remains to estimate
the contribution to (7.16) from the 7} with j ^=0.
When r = 0, i.e. m = 0, there is nothing to prove. Suppose that
m > 1. When / < /c, the trivial estimate |7}| ^ p' ~ * in (7.16) gives
\S{pl9al9...9ak)\^kpl-1^kpf-lf\
Thus it can be supposed that I > k.
Consider the polynomial in x,
4>{px + xj) — (f)(Xj) = b^x + ... + bkxk
where
Let pp denote the highest power of p dividing (bl9 b2,. . ., bk). Clearly

Preliminary estimates 115
p ^ 1. If p > /e, then
and so p|afc, jp|afc_ 1?.. ., p\a1 contradicting (p, k) = 1. Thus
p<fc</. (7.19)
Let c{ = bip~ p and
ij/{x) = p~ p{<t>(px + x_y) — <t>{Xj)) = cxx + ... + ckxk.
Then, by (7.17),
\Tj\ =p»-1\S(pl-p,c1,...,ck)\. (7.20)
Since p~ x(j)'{x) = 0(mod p) has a root of multiplicity m} at xy one can
write p~x(j)'(x) in the form
p~ z(t>'(x) = (x- x -r^!(x) + p02(x)
with p/0! (x7) and deg 02 < m7. Now let p* denote the highest power of
p dividing (c1, 2c2,..., /ecfc). Then
p-axlj'{x) = pl-a-p<t>'{px + xj)
+ P02(P* +*/))•
The coefficients of this polynomial are all integers and at least one is
coprime with p. Since deg02 < rrij the coefficient of xmj is
so that g + p ^ 1 + t + ra^ Hence, if J > mj? then the coefficient of xd is
a multiple of p. Hence
p-ar(x)^p1-a-p + t(pmjxm^1(xj) + pcl>2(px + xj))(modp).
Therefore the degree of p~a\jj\x\ modulo p, is at most rrij and so the
number of solutions of the congruence
p->'(x) = 0(modp),
counting multiple solutions multiply, is at most my
Therefore, on the inductive hypothesis, (7.9), with / replaced by
/ — p, aj by Cp m by m-} one obtains, via (7.19) and (7.20),
\Tj\ ^ k2mjpp ~ V " P)(1 " 1/fc) < k2mjPl" //fc.
The desired conclusion, (7.9), now follows from (7.16) on summing
over all j > 1.

116 Vinogradov's upper bound for G(k)
The following theorem gives the asymptotic expansion for/on the
major arcs.
Theorem 7.2 Let (Xj = ajq-j + jS,- (j = 1,. . ., k) and suppose that
Q — \A\•> • • •><?*] and Aj = a^qqj 1. Then
f(x) = q-1S(q,A)I(P) + A
where
A<q(l + \PX\X + \P2\X2 + . . . + \Pk\Xk).
Proof By Lemma 2.6 with
cx = e{{Axx + ... + Akxk)q~ x)
and
F(y)=e{Piy + ...+pkyk)
and the observation
I cx= t e((Aiy + ... + Akyk)q-') £ 1
x<y y=l x^y
x = >>(modqf)
= ^-^(^) + 0(^),
one obtains
/(<*) = 9-^(4 ,4)( F(X)X-
where
AWl +
F'(y)ydy) + A
o /
^+...+ ^/-1^
0
Integration by parts gives the theorem at once.
Theorem 7.3 The auxiliary function I(P) satisfies
KPXXii + w^ +.. . + \pt\xkyllk-
Proof It can be assumed that
for then the general case follows by a change of variable. It can further
be supposed that
1^1 + ... + 10^1

Preliminary estimates 117
for otherwise the result is trivial. Let
r, = (|j8,| + ... + IM1/k,
Pi(a) = Pi + 2^2a + .-. + kfik(xk ~ 1
and
j2/ = {a:0<a^l, Ma)! ^ ^}.
Then «s/ can be dissected into <^ 1 intervals on each of which p[ (a)
does not change sign. Let £8 be a typical interval of this kind. Then
integration by parts gives
eOV + ... + j?fcafc)da^ Y\\
Thus it remains to show that
^{aiO^a^l.Ma)!^}
satisfies
measO^H *7 '• (7.21)
The argument now proceeds by iteratively constructing sets
^1? ^2> ^2> ^3> • • • as follows. If ^! is empty, then there is nothing
more to prove. Thus it can be supposed that there is an ax such that
(Xa^l and M«i)l < yi- Now M«i)l ^ \Pi\ ~ kY\ so that
if ijSJ > 2kY\, then ^J < fljBJ + 7¾1^ < ((1 + 1/(2^))1^1)^,
whence
Hence in this case (7.21) is trivial. Thus it can be supposed that, for a
suitable number C1 depending at most on /c, one has
and |pi(a)| < ClY2 for every ae(^1. Therefore it suffices to show that
meas^M Yi 1
where
^1 = {a :0 ^ a ^ 1, |a - aj > Y2 \ ^(a)! < Cx Y2}.
Let
Pi(a)-Pi(ai)
P2(a) =
a — cc1
Then Q)l<^(€2 where
*2 = {a:0<a<l,|p2(a)|<2C1y|}.

118 Vinogradov's upper bound for G(k)
Proceeding in this way, at thejth step one obtains a constant Cj _ 1?
a polynomial p -(a) of degree k—j and considers
Vj = {oL:0^oi^l9\pj(oL)\<2Cj.lYi}.
If Wj is not empty, then there is an a, such that \Pj(olj)\ < 2Cj _ 1 Yj.
Defining at each step
Pj+M = P-iia)-pjiaj)
a-a,-
it follows that
pM)= lV«*
/i = 0
with
and
Thus
k-j
Yh — Zj fi+ 1 aj - 1
i = h
ti» = (h+i)ph+1.
y\P=JPj + 0(\pJ+l\ + ... + \pk\)
and so it can be supposed that there is a number C ■ depending at most
on /c such that
\Pj\^CjYkJ+l (7.22)
and \Pj{(x)\ < CjYjj+ 1 for every ae^-. Let
^.= {a :0<a< 1, |a - a;| > 7;^, |p.(a)| < C;^ + J.
Then, by (7.22), one desires to show that meas(^) <^ Yj~+V
The process may stop because, for somej ^ k — 2, <&. is empty or the
inequality (7.22) is violated. Otherwise for j ^k — 2 one has
<&. a <€. + x and the process continues until one reaches Q)k_ v Now
y(ri) = kpk.
Thus
®*- i c {« ■ I?? " "fc" 'AT 1+«l< Q- ilftl" 1/k)
so that
meas(^fc_ x)< Y^1
as required.

An asymptotic formula for JS(X) 119
7.3 An asymptotic formula for JS(X)
Theorem 7.4 There are a positive constant C1 and positive numbers 3 (k)
and C2(k, s) such that whenever s > /c2(31og/c + log log k + Cx) one has
Js(X) = C2(k,s)X2s-k(k+1)/2+0(X2s-k{k+1)/2-d{k)).
Note that by using below Theorem 5.5 in place of Theorem 5.1 the
lower bound for s can be replaced by
5/c2 29/c2 Ik2
~Tlog/c + "3(rloglog/c + "T-log log log/c + Ck2.
Proof Let X denote a large real number, let
X = h 2l=Xl/2' Qj = Xi~k (2^j^k) (7.23)
and let %% denote the cartesian product of the intervals (Q J \ 1 +(2/1].
When qx < ^1/2, ^ < X* (2 <; < /c), and 1 ^ a} ^ q} with (qp aj) - 1,
let Wl(q, a) denote the cartesian product of the intervals
{ccilcc-aj/q^q^Q;1}.
The major arcs Wl(q, a) are pairwise disjoint and contained in ^*.
Let 3R denote their union. Then the minor arcs m are given by
m = ^\«W.
By Lemma 2.1, for each ae^f, there exist q, a such that (q^ a}) = 1,
la,- — aj/qj\ < q~ 1QJ 1 and q-3 < (¾. Let n denote the set of ae^/jf for
which in addition q} > Xk for some j with l^j^k. Then, by
Theorem 5.3 with / = [4/c log /c], it follows that there is a positive
constant C3 such that
/(aHX1"" (aen) with p~ 1 = C3k3 log/c. (7.24)
Now let $1 denote the set of ae^f for which there exist q, a such that
(^., aj) = 1, |a,- - ^/^| ^ ^ * Q" \ ^ < Ql9 a, ^ XA (2 <; ^ fc). Thus
wkjSR = ^1 (although nn^l may not be empty) and sIRc:^R. Let
Pj = ccj - aj/qp q = [<?!, . . . , qk\ Aj = qajqp so that
(q,Al9..>,Ak)=l. (7.25)
By Theorem 7.2 and (7.23),

120
Vinogradov's upper bound for G(k)
f{*)-q-lS{q,A)I{ft)
<qx... qk(l + X1'V ' + X"a2 ' + . • . + Xxq~ l)
<X'-\ (7.26)
If aem n, so that a<£9W, then q^ qx> \X1/2. Hence, by Theorem 7.1
and (7.7),
q~ 'S(q, A)I(P) <Xq*-llk<$Xl-k + \
Therefore (7.24) holds with n replaced by m. Let m = [C3] + 1 and
rj = |/c2(l - 1/fc)'. Then, by Theorem 5.1,
f((t)\2tk + 2mfc2da <^ Y2tk + 2mfc2 ~ k{k + 1)/2 + n ~ 2mfc2p
m
Moreover, for t > 3/c log k + k log log /c one has
rj-2mk2p<k2( 1--) - — -<0
\ k/ k log /c
and so when s^tk + mk2 there is a positive number S = S(k) such that
|/(a)|2sda^A'2s-fc(fc+1)/2-'5.
m
It remains, therefore, to treat the major arcs Wl. For ae5R((j, a),
(7.26) holds. Define K(a) = K(a, ¢, a) when ae$R (¢, a), K(a) = 0 when
aem. Then
2s i/( — \2s\a~ ^-j v2 — A
|/(a)2s- J/(a)2s|da«^
<m
2s- 2^
(|/(a)|2s-2 + |F(a)|2s-2)da.
4t\
(7.27)
By Theorem 5.1, if s — 1 > kl with / > 3/c log/c, then
|/(a)|2s" 2da <^ X2s~ 2 ~kik + 1)/2 +"
1/,2/1 1 /J,\/
with 77 = ^kz(l — l//c) < l/(2/c) = A. Hence there is a positive number
(5 = S(k) such that
X
2 - A
/(a)|2s"2da^X2s-fc(fc+1)/2-<5.
(7.28)
*
Let <xey)l(q, a). Then, by (7.25) and Theorems 7.1 and 7.3,
V(ol) < XqE ~ 1/k(\ + \fix\X + ... + \Pk\Xky 1/fc.

An asymptotic formula for JS(X)
121
Hence
it
\V(x)\2tdx<X2tWZ
>i/h
where
00
00
w= Z •• • Z 4i•• -<ik[.<i»---^]
<2i = 1 Qk = 1
2r(c- 1/fc)
and
fc pOO
j= i Jo
When t > 2/e2 one has
(l+PjXj)-2t/k2dPj.
00
00
w< Z • • • Z «i • • • iMi -ik) <°°
^1 = 1 ^k = i
and
z^ n x_j=x-fc(fc+i)/2.
Therefore, for s — 1 > 2/c2,
X
2 - A
|K(a)|2s-2da<^X2s-fc(fc+1)/2
— /
<&,
This with (7.27) and (7.28) shows that
l/(«)l2sd« =
<m
K(a)|2sda + 0(X2s-k{k+1)l2-*)
an
when s satisfies the hypothesis of the theorem with C1 suitably chosen.
It follows in a straightforward manner from Theorems 7.1 and 7.3,
that
K(a)|2sda = SJX2s~kik+ 1)/2 + 0(X2s~k{k+ 1)/2~3)
an
wjiere
oo
oo <ji
Ik
s= Z •
4i = i
•• Z Z •
qk = 1 «i = l
(«i.«i) = 1
•• z
flk = 1
(ak,qk) = 1
\q lS(q,A^
• , A)\2s
and
J =
r\
eiPiO. + . . . + Pkak)d(x
o
2 s
dp

122 Vinogradov's upper bound for G(k)
Note that q~ 1S(q,A1,..., Ak) = (q1 .. .qk)~ 1S(q1 ...qk9al9.. .,ak).
Also S < oo, J < oo and so the theorem follows with C2(/c, s) = <3J.
The positivity of C2(/c, s) is a consequence of (7.4).
For a more precise analysis, and several applications, of the above
theorem, see Hua (1965).
7.4 Vinogradov's upper bound for G(k)
It can now be shown, as an application of Theorem 7.4, that
G(k)
limsup — r^2. In many respects the proof builds on the ideas
k-oo /clog/C
of § 5.4.
Let n denote a large natural number, and write
JV = [n1/k].
Let K denote a natural number with
2K<k
and put
k.=^)^,^=^2. (7,29)
Now let Q(m) denote the number of solutions of the equation
(U1 + xj* + . . . + (t/z + xt)k = m
with Xj < 1^-, where / is a parameter to be determined suitably in terms
of /c at a later stage.
Consider
W(«) = I I e(mMapfcm).
X/2 <p< X m
By Holder's inequality, for any natural number r,
W{oi)2r < X2r ~ l £
X/2<p< X
Y,Q(m)e((xpkm)
m
where
= X2'~l X ZQ1(h)e(°LPkh)
X/2<p^X h
Qi(h)= E Q(ml)...Q{m2r)

Vinogradov's upper bound for G(k) 123
and the summation is over m1?..., m2r with
m1 -\-. . . -\- mr — mr+1— ... — m2r = h.
Hence, in the notation of §§4.4 and 5.3, and by Lemma 5.4,
W{z)2r<X2r-l(xU\ + EY.QM2) (aem). (7.30)
The sum X/,2i W2 is the number of solutions of
Z Lj(xj) = 0 (7.31)
with XjSll, V,.]4r and
Lj(y) = (Uj + yl)k + ... + (Uj + y2r)k
-(Uj + y2r + ,)k - ■ . • - (Uj + 3^)". (7.32)
The estimation of Q1(h) depends on a lemma, the proof of which
requires Theorem 7.4.
Lemma 7.1 Suppose that r > CK2\ogK where C is a suitably chosen
constant. Then the number Rj of different y in [1, Vj\*r for which Lj(y)
lies in a given interval of length jjf~K~1/2 satisfies
Rj^VJrUfK .
Proof For brevity, drop the parameter). By the binomial theorem,
W-lfyu'-iMM
where
MAy) = y\ +••• + r'2, - y2r +! -... - \V
Since je[l, J/]4' and J/ = [C/12] one has
£ (^U'-'M^yXU'-^^V2^1
i = 2K + \ \l J
Hence it suffices to show that the number K* of different y in [1, V~\
for which
lies in a given interval of length l/*~2X satisfies
4r

124 Vinogradov's upper bound for G(k)
K*<^ y^-ijji-iK^ (7.33)
Consider the number R** of 2K-tuples of integers z1?. .., z2K with
zf <^ K1 and for which
2K
i = 1
lies in a given interval of length (/~2X. The interval can be written in
the form
where w and v are integers with 0 < v < Uk~2K. Then
z2K =u(modU), z2X_! = (^-¾)^-1 (modify
and so on. Thus z2K is determined modulo (7, z2X_! is determined
modulo U by z2K, and so on up to z2. Moreover zx is determined
uniquely by z2x,. .., z2 since 0 < v < (7*~2X. Hence, on recalling that
K2 > (7, one has
#** <^ (|/2X (7~ ^(K2*- * IT x). . . (V2U~ x)
= yK(2K + l)-lul-2K^ (7J4)
By Theorem 7.4, given zx,. . ., z2A:, the number of solutions of
/c)M/(j) = zi(1^/^2X) with yell, K]4'
is <^ jA-*<2* + 1\ This with (7.34) gives (7.33) and so the lemma.
Suppose henceforth that the hypothesis of the lemma holds and let
xl,...,xl be a typical solution of (7.31). By (7.32), for
jcj+1e[1, Vj+ J4** one has
LJ + l(xJ+l)<uk;lvJ+1<u}-K-1'2.
Hence, by (7.29), L1 {xj lies in an interval of length < lA~K~x'2. Thus,
by Lemma 7.1, there are <V\rU\K choices for xv Then given
xl9 L2(x2) lies in an interval of length Uk2~K ~1/2, and so on. Hence the
total number of choices for xl9 jc2, .. ., jcz is
<(¥,... Vlfr{Ul...UlTK.
By (7.29), {U^.Mifp U{k~ 1/2)(1 -*'>. Hence, by (7.29) and (7.30),
for aem,
WiaXXV, ... vl(X-1Uk + e-{k-1,2H1-*yn*r)<W(0)NE-p

Exercises 125
where
Now take
K-
Then, by (7.29),
rjl = expl
= Ci
/log
p =
log /c],
0-i
1
16r
/ = 3fc
-.);
8r <'-*'■
, r = 1 + [CK2
|«exp(-3[±
logK]
log/c])<
(7.35)
where the implicit constants are absolute. Thus, if k is sufficiently
large,
1
c{(\ogky
say, where Cx is a suitable constant. Thus, in the notation of §5.4
(but with W(cc) as above),
f(a)4kH((x)2W((x)e(-(xn)d(x
<£ H(0)2W(0)n3 +il ~ 1/kY -a,k.
Choosing t optimally so that (1 — 1/kf < a/k gives
t ~ k log k.
Then the contribution from the minor arcs is
<tH(0)2W(0)n3~s
where S = S(k) is a suitable positive number.
The major arcs can be treated as in § 5.4. It follows that
G{k) ^ It + 4/c + /.
This with (7.35) yields
Theorem 7.5 Ask-*ao9 G(k) ^ /c(log /c)(2 + o(l)).
7.5 Exercises
1 Show that, when s ^ /c, JS(X) = s\XS + 0(XS ~ x).
2 Show that, when k = 2, J3(X) ^ X3 log X, and that (7.5) is false.
More precisely (Rogovskaya, 1986) show that

126 Vinogradov's upper bound for G(k)
J3(X)~ CX3\ogX
as X -> oo.
3 Let Gx(/c) denote the least s such that almost every natural
number is the sum of s /cth powers. Show that
limsup — -^ 1.
fc-oo /clog/c
4 (Vaughan and Wooley, 1995b) Show that when k ^ 3
as X -> oo.

8
A ternary additive problem
8.1 A general conjecture
Suppose that fcl5 /c2,. . ., ks are s integers satisfying
s
2 ^ k1 ^ k2 < . .. ^ ks and £ fc/ * > 1. (8.1)
j= i
Then the arguments discussed above, particularly in Chapters 2 and
4, suggest that the equation
s
Z xk/ = n (8.2)
j= i
has a solution in natural numbers xl5.. . , xs whenever
(i) for each prime p and large k the equation (8.2) is soluble modulo
pk with pKXj for some 7;
(ii) n is sufficiently large.
There are some exceptions to this, see Exercise 5, but they all seem
to have the general property that for some i there is a polynomial
sequence of n for which n — xk* has certain multiplicative properties
arising from its polynomial factorisation which are at odds with the
multiplicative properties of Yj= u» i-vyJ- ^ simplified form of this
phenomenon occurs in Exercise 2. Even in these examples it should be
true that (8.2) holds for almost all n.
There has been a great deal of work on questions of this kind, much
of it rather inconclusive in nature because the treatment of the minor
arcs in the present state of knowledge generally requires £ kj 1 to be
appreciably larger than unity.
The smallest value of s for which (8.1) is satisfied is s = 3. Then the
only case which has been completely solved is that of kx = k2 = k3 = 2,
the classical theorem of Legendre on sums of three squares. However,
in all the remaining cases it has been shown that almost all numbers

128 A ternary additive problem
can be represented in the form (8.2). The cases with kl = k2 = 2 and
with /c! = 2, /c2 = /c3 = 3 are due to Davenport & Heilbronn (1937a, b),
the case k1 = 2, k2 = 3,/c2 = 4 is due to Roth (1949) and the case k1 =2,
k2 = 3, k3 = 5 is due to Vaughan (1980a).
The last case is the hardest, and the remainder of this chapter is
taken up with its elucidation. The method can be readily adapted to
the other cases.
8.2 Statement of the theorem
Let E(X) denote the number of natural numbers not exceeding X and
not being the sum of a square, a cube and a fifth power of natural
numbers.
Theorem 8.1 There is a positive number S such that E(X) < X1 ~(\
In general principle the argument is similar to that of § 3.2. An
important feature is that the major arcs can be taken to be longer and
more numerous than the presence of the cube and fifth power might
suggest. However, a large part of the major arcs is treated, in some
respects, more like minor arcs.
Another feature of the argument is that there is some difficulty
connected with the convergence of the singular series. This is
overcome by replacing the singular series by a finite product.
8.3 Definition of major and minor arcs
Let n denote a large natural number and write
Further, let R(m) = R(m, n) denote the number of representations of m
in the form
m = x\ + X3 + X5
with Pk < xk ^ 2Pk, and let
/(m) = ZZZ^2-1/2^2/3^4/5 (8.3)
)>2 )>3 )>5
where the variables of summation satisfy Pl<yk^(2Pk)k and
>;2 + ^3 + y5 = m-

Definition of major and minor arcs 129
Also define
Sk = Sk(q,a) = J e(ark/q\ (8.4)
r = 1
A(m,q)= X (l~3S2S3S5e(-am/q) (8.5)
a = 1
(a,q)= 1
and
<5(m.X)= ^ 4(™,g). (8.6)
The first part of the proof of Theorem 8.1 is the establishment of
Theorem 8.2 There is a positive constant 6 such that for every
sufficiently large n
R(m) = I(m)&(m, n1/2) + 0(n1/3° ~s)
for all but <^nl ~ d values of m with n < m ^ 2n.
Proof Let
fck = M«)= I e((tx\ (8.7)
Pk < x < 2Pk
S = \0~\ P = nl3/30 + 1\ <% = (P/n,l + P/ri]. (8.8)
Then
R(m) = h2(cc)h3(cc)h5((x)e(— am)da. (8.9)
When 1 ^ a ^ q < P and (a, q) = 1, define the major arc 2R(g, a) by
2R(g, a) = {a : |a -a/g| < Pg" xn" *} (8.10)
and take 9W to be the union of all the major arcs. As usual, it is easy to
show that the 9Jl(g, a) are disjoint, and the minor arcs m are taken to
by °u\m.
There is an important further subdivision of 90¾. Let <SJl1 denote the
subset of 9ft formed from those $R(q, a) with q > n1/12, and let
W{q, a) = {a : |a - a/q\ < n3d " 14/15}. (8.11)
Now define 9JJ2 to ^e tne union of the 5R(g, a)\9l (g, a) with
1 ^a ^ q < n1/12 and (a, g) = 1. Then, if one writes
n = muTO1uaR2 (8.12)

130
A ternary additive problem
and
R1(m) =
the aim is to show that
/i2(a)/i3(a)/i5(a)e(-am)da,
rt
Xl«i(m)|2«n16/15-3d
m
and
I I
h2(oc)h3(cc)h5((x)e(— ccm)doc
a ^ n1/! 2 a = 1 J
(a,q)= 1
91(<J,a)
= /(m)S(m, n1/12) + 0(l).
(8.13)
The first of these estimates will follow from Parseval's identity if it is
shown that
|/i2(a)/i3(a)/i5(a)|2da < n16/15 ~ 3d.
(8.14)
8.4 The treatment of n
The minor arcs m can be treated in a straightforward manner. The
integral
Jo
2l4
fctlda
2"5
is the number of solutions of the equation
u2 -v2 + x5 - y5 +z5 -t5 =0
with P2 < u,v^ IP2, P5 < x, y, z, t ^ 2P5. The solutions are of three
kinds
(i) u±v,
(ii) u = v,x£ >',
(iii) w = v, x = y, z = t.
Hence the total number of solutions is
<P* + E + P2P2+E + P,P2
21 5-
Therefore, by (8.3),
ri
o
\h22hi\da4n9'l0+e.
(8.15)

The treatment ofn
131
Similarly
\hi\da<n2/3+E.
(8.16)
o
By Weyl's inequality (Lemma 2.4), it follows that for each aem
h2(oi) < n1/2+E(P~ x + n~ 1/2)1/2 < nE(n/P)1/2.
Hence, by (8.16),
r
\h22h\\doi^n5l2, + ^p-\
m
Therefore, by Schwarz's inequality, (8.15) and (8.8),
\h22hih25\doL<n16/15-33.
(8.17)
in
Let
WfcOJHXO/fcJx1'*-1**/**) (8.18)
X
where the variable of summation satisfies Pkk < x ^ (2Pk)k, and define
Wk = Wk(ct, q, a) = q~ 'Sk{q, a)wk((x - a/q). (8.19)
For aesIR, define (/>fc, Afc by
<l>k = 0k(a) = Wk(a, ¢, a) (aesJW(g, a)), Afc = Afc (a) = hk - 0k.
(8.20)
The first step in the treatment of 93^ u9W2 is to replace /i2 by 02. By
Theorem 4.1, when ae$ft one has A2(a) <^ P1/2 +£. Also, in a similar
manner to the proof of (8.16),
|/i2/i2|da<^8/15.
o
Therefore, by (8.8),
lAjfi^lda^n.
(8.21)
an
The next step is to estimate
4>jhihj\d(x.
a«i
To this end it is first necessary to consider the corresponding integrals

132
A ternary additive problem
2j„4i n„A \J.2l4\
with the integrand replaced by l^^sl and 102^3
By (8.19) and (8.20),
\<t>lhl\doL
an
< I X <r2is2i2
q < P a = 1
(a,q) = 1
f 1/2
- 1/2
|w2(/?)2M/?+a/4)4|d&
and, by (8.18),
\n2(P)\2=lM)e(-Ph)
(8.22)
where
W) = 14(^)-1
- 1/2
(8.23)
x,y
with x — y = h, \n = Pi < x, y ^ (2P2)2 = n. Moreover, by (8.7),
\h5(0L)\4 = Yc(h)e(0ih)
h
where tu\ V 1
c(h) = 2, 1
with x5 — y5 + z5 — t5 = h and P5 < x, y, z,t^ 2P5. Therefore
\n2(P)2h5(P +a/q)*\dp = YHh)c(h)e(ah/q).
- 1/2 h
Hence, by (8.22),
I02^I^IM%WI t q-2\S2\2e(ah/q).
SR h q^ P a = 1
(«,«) = 1
It is trivial that \S2\, defined by (8.4), is independent of a and satisfies
|S2|2 < q. Therefore, by (3.14), for h ± 0,
X q-2\S2\2e{ah/q)<q-1 £ d.
a = 1
(a,q)= 1
d\(q,h)
Thus
an
|0i*$|da < b(0)c(0)P + I ft(/i)c(/i)£ I -
/if 0 d|/i r < P/d r
It is immediate from (8.23) that b(h) < 1. Also, in a similar manner to

The treatment ofn
133
the proof of (8.16), c(0) < n2/5+E. Moreover £fcc(fc) < n4/5. Therefore,
by (8.8),
\(t>22h*\da<n5/6 + 8d.
(8.24)
an
Various forms of Holder's inequality are applied in order to
estimate
\4>22h$\d0L
and consequently an estimate is required for
|0*| da.
2R
By (8.20) and Lemma 6.3,
an
|0f|da< £ q"1
f 1/2
n2(l+nj8)-4dj8,
o
so that
|02|da <^ n
1 +£
2R
Therefore, by Holder's inequality and (8.16),
\4>22A3hl\da < (n1 + -)1'V'3 + £)3'4sup |</»2A3|
an
an
By Lemmas 6.1 and 6.3, for aesIR(g, a) one has
02(a)A3(a)^M1/V-
Hence
|^A3/^|da^tt5/4 + 2£.
an
By Schwarz's inequality, (8.25) and (8.16),
|0203A3/i2|da < (n1 + £)1/2(n2/* + £)1/2sup |03A3|,
jan 2«
and by Lemmas 6.1 and 6.3, when ae$Ji(g, a),
03(a)A3(a) < n1/3q1/6+E< n1/3P1/6+E.
(8.25)
(8.26)

134
A ternary additive problem
Hence, by (8.8),
\(t>22(t>3A3h23\d(x<n5/*.
(8.27)
2R
By Lemmas 6.1 and 6.3, and (8.8),
|0i0lAl|da < X q1/3 +
"1/2 ^5/3
an
q < P
« n5/4
Therefore, by (8.26) and (8.27),
o (1+«/*)'
-(1 + nyS)2d^S
.2 7.41 ^.. ^ .,5/4 +£
I02^3l^a ^ ^
+
an
I02«|da.
(8.28)
2R
Now consider the integral on the right. By (8.20), (8.19), Lemma 6.2
and Theorem 4.2,
an
\^t\^< Z Z q~6\S22S
nn „7/3
O (l+«0)'
d/?
,4i
„1/12 <q ^ p a = 1
(a, 4)= 1
^ n5/4 J
where
J= Z *"(«). F(q)= z <r4is;
q < P a = 1
(a,q)= 1
By Theorem 4.2, F(^M <?" 1/3- Thus £*= 3F(pfc)^ p" '. Also, by
Lemmas 4.3 and 4.4, |S3(pz,a)| <^ p//2 when /=1 or 2. Hence
]>jj=! F(ph)<^p~x. Moreover, by Lemma 4.5, F is a multiplicative
function of q. Therefore, there is an absolute constant C such that
j^n (i+Q-1).
Hence, by (8.28) and elementary prime number theory,
\(f)22h*\d(x<n5/* + E.
$>?!
Therefore, by Schwarz's inequality and (8.24),
|0^^^|da^M16/15
- 3d
««!
Hence, by (8.21),
\h2h2h2\d(x<n16115
- id
(8.29)
2«!

The major arcs ${(q, a)
135
Now consider 9ft2- By Lemma 6.3,
* f 1/2
|^|da« £ 4"1
9W2 q ^ P
<n4/5+e-93
By Hua's lemma (Lemma 2.5),
rv
n2>8- 14/15 (1 + nfiY
dfi
|/i||da ^ n5/3 + £,
o
\h%\doL<^n
1 +£
0
Hence, by Holder's inequality,
\4>lhlhl\d0L « (M4/5 + £ " ^)l/2(„5/3 + ejl/4(wl + ,)1/4
^ ^16/15- 3(5
2R;
Therefore, by (8.21),
\h\h\hM*<nl6ll5-*d.
J«n2
This with (8.29) and (8.17) gives (8.14).
8.5 The major arcs ^R(q, a)
To complete the proof of Theorem 8.2 it remains to establish (8.13).
A simple calculation shows that if k = 2, 3 or 5, if q^n1/12, and
if |/?|^n3<3"14/15, then
q1'2 + £(1 + n|j8|) < (n/q)1/k(l + n\P\)~ l.
Hence, by Lemmas 6.1 and 6.3, when <xe9t(g, a) one has
hk(a), Wk(oL)^(n/q)1/k(l+n\oi-a/q\)
- l
and
h2((x)h3((x)h5((x) - W2(ol)W3((x)W5(ol)
< (n/q)5l6(\ + n\0L-a/q\y lqlll + i\
Thus
I I
q< „1/12 a= 1
(a,q) = 1
\h2h3h5-W2W3W5\dx< 1
W(q,a)

136 A ternary additive problem
Let <P(g, a) = {a : n3d ~ 14/15 < |a -a/q\ <£}. Then, by Lemma 6.3,
z z
WjWj^lda^l.
q ^ „1/12 a= i Jy(q,a)
(a,q) = 1
Therefore, by (8.19), (8.4), (8.5) and (8.6),
z z
h2h3h5e( — ccm)dcc
q < «1/12 a = 1 J<n(<j,a)
(a,q) = 1
= /^)6(^1,71^) + 0(1)
where
/i(m) =
n
0
u>2(/?)w3(/?)w5(/?M-/?m)ci/?.
By (8.18) and (8.3), /1(m) = /(m), which gives (8.13) as required.
8.6 The singular series
The principal difficulty is that Y?=AA(n,q)\ apparently diverges.
This is resolved by approximating to 3(m, rc1/12) by a finite Euler
product.
Theorem 8.3 For all except ^n1 s values of m with n < m < 2n one
has
3(m,n1/12)= n ( Z A(m, ph)) + 0(exp(-(logn)3)). (8.30)
p^ n \h = 0 /
It is possible that one could show under similar conditions that the
finite product can be replaced by the infinite product, perhaps by a
method allied to that of Miech (1968). However there are attendant
difficulties which the method described here avoids. For a further
discussion of this matter in the case kl = 2, k2 = k3 = 3, see
Davenport & Heilbronn (1937a).
By (8.4), (8.5) and Theorem 4.2,
A(m, 1)=1, A(m, qXq'1130. (8.31)
Thus each of the series on the right of (8.30) converges absolutely.
For the proof of Theorem 8.3, precise estimates are required for

The singular series 137
A(m, ph). To this end the following formulae for Sk(ph, a), valid when
p\a, are basic. They are consequences of Lemmas 4.3 and 4.4.
When p > 2
Cphl2 (2|fc),
S2(ph9a)=</a\ (8.32)
when p > 3
fp[2,,/31 (/7^1 (mod 3)),
S3(/Aa)=<0 (fc = l(mod3),p = 2(mod3)), (8.33)
[s3(p, a)p2(fc" 1)/3 (/7 = p = 1 (mod 3)),
and when p > 5
pI4fc/51 (/7 #1 (mod 5)),
S5(p\a)=<0 (/7=l(mod5),p^l(mod5)), (8.34)
S5(p, a)p4(fc ~ 1)/5 (/7 = p = 1 (mod 5)).
Also, when k = 3 or 5 and p = 1 (mod k),
Sk(p,fl)= Z x(a)T(z) (8.35)
where <s/ denotes the set of k — 1 non-principal characters # modulo p
with #fc = Xo- Moreover,
|i(z)|=p1/2 and |S2(p, 1)| = p1/2(p> 2). (8.36)
Lemma 8.1 Suppose that h > 1 and p > 5. T/rerc
^(m, pfc) = 0 wte/7 h>\ and ph~llm, (8.37)
|/l(m, ph)\^8p-[{h-l)i30]-\ (8.38)
/l(m,p)= X c(x)x(m) (8.39)
/e.s/(p)
vv/iere -o/(p) is a collection of non-principal characters modulo p,
ktol^P-1 and card s/(p) < 8. (8.40)
Proo/ By (8.5), (8.32), (8.33), (8.34) and (8.35),
p*
/l(m,p*) = X Mz) Z Z(flH-flp"hm) (8.41)

138 A ternary additive problem
where stf(ph) is a subset of the set of characters modulo p and the b(x)
are suitable complex numbers. When h > 1 and ph~ 1][m the innermost
sum is
p ph ~1
X x(x)e(-xp~ hm) X e(-yp{~hm) = 0.
x = 1 y = 1
This gives (8.37).
The proof of (8.38) is by division into eight different cases.
(i) Suppose that 2\h, h ¢- 1 (mod 3) and h ¢- 1 (mod 5). Then srf(ph)
consists solely of the principal character and, by (8.32), (8.34) and
(8.35), one has
\A(m,ph)\^p*
where X = \h + [2/i/3] + [4/?/5] — 2h. The number X is an integer and
does not exceed - fc/30 < - [(¾ - 1)/30] - 1/30. Hence (8.38).
In all the remaining cases (8.41) holds with all the elements of jtf(ph)
being non-principal characters modulo p. Thus when ph\m the
innermost sum is automatically 0 and (8.38) follows at once. Also, by
(8.37) it can be supposed that either h = 1 or h > 1, ph ~ l\m and ph\m.
In either case the innermost sum in (8.41) is
(ii) Suppose that 2|/i, h ^ 1 (mod 3) and h ^ 1 (mod 5). Then
>^(ph) consists solely of the quadratic character, and
\k(y)\ = phl2 + W^ + W5! - 3h
Hence, by (8.42),
\A(m, ph)\ = px with X = ±h + [2/i/3] + [4/i/5] -2h-\.
The exponent X is an integer which does not exceed — h/30 — \. Thus
(8.38).
(iii) Suppose that 2|/i, h = 1 (mod 3) and h =k 1 (mod 5). When
p ^ 1 (mod 3), (8.33) gives A{m, ph) = 0. Hence it may be assumed that
p=l(mod 3). Then card sg(ph) = 2, and 1^1(^,^)1^2^ with X =
\h + 2(h — 1)/3 +\ + [4/i/5] —2h—\. The number / is an integer
not exceeding — h/30 — f. Therefore (8.38) holds.
(iv) Suppose that 2|/i, /7 ^ 1 (mod 3) and /7 = 1 (mod 5). The case
p ^ 1 (mod 5) is again trivial, so it may be assumed that p = 1 (mod 5).

The singular series 139
Then \A{m, ph)\ ^ 4pA with a = \h + [2/7/3] + 4(/7 -1)/5- 2/7, and the
argument is completed as before.
(v) Suppose that 2\h and h = 1 (mod 15). The case /? ^ 1 (mod 15)
is trivial, and when p = 1 (mod 15) one obtains |^4(m, ph)\ ^ 8/?A + 1/2
with X = \h + 2(h — 1)/3 + 4(/7 — 1)/5 — 2h. This is again an integer
which does not exceed — h/30 — ff < — [_{h — 1)/30] — 1. Hence the
exponent a + j satisfies a + \ ^ — [(/i — 1)/30] — §.
(vi) Suppose that h = 1 (mod 10) and /7 ^= 1 (mod 3). When
p ^ 1 (mod 5), (8.38) is immediate from (8.34), so it may be assumed
that /7=1 (mod 5). Then \A(m, ph)\^4px+1/2 with X=\{h-1)
+ [2/i/3] + 4(/] — 1)/5- 2/7. As in the previous case the exponent does
not exceed - [(/7 - 1)/30] -f.
(vii) Suppose that h=l (mod 6) and h ^ 1 (mod 5). This can be
dealt with in a similar way to case (vi).
(viii) Suppose, finally, that h = \ (mod 30). Then A(m, ph) = 0
for p ^ 1 (mod 15). This leaves the possibility p = 1 (mod 15).
Then \A(m,ph)\ ^ 8// with A =-^+ 2(fc - 1)/3+i + 4(fc - 1)/5
+ i — 2/7 — 2 . Again /I is an integer not exceeding — h/30 — f — \ =
-(/7-1)/30-1.
The proof of the lemma is now completed by establishing (8.39)
with (8.40). When /7=1, (8.32), (8.33), (8.34) and (8.35) give (8.41) with
b(x) = 0 unless p = 1 (mod 15). Thus, when p ^ 1 (mod 15) one obtains
(8.39) with (8.40) trivially satisfied.
When p=\ (mod 15), (8.41) holds with srf{p) consisting of the
characters x of the form x = XiX^Xs* where Xk denotes a non-principal
character of order k. Thus all the elements of s#(p) are non-principal
and cardj^(p) = 8. Moreover,
Hx) = s2(P, iHx3Mx5)p~3
Hence, by (8.41) and (8.42),
A(m,p)= ^ si(P> 1)?(X3)T(X5)P~3X(- lMx)z(m).
X e si (p)
If a character x belongs to s#(p) then so does X-
Also, by (8.36), one has
152(^1)1(^)1(^)/7-^(-1)1(^=/7-1.
This gives (8.39) with (8.40), as required.

140
A ternary additive problem
Let the set & consist of 1 and those natural numbers q such that if
p\q, then p^n and either p2\q or p^5. Let # denote the set of
squarefree numbers all of whose prime factors p satisfy 5 < p ^ n.
Finally, let Q) denote the set of natural numbers with no prime factor
exceeding n. Then each q in Q) can be written uniquely in the form
q = rs with re&, se%> and (r, s) = 1.
The next stage of the argument is to estimate
Z A(m,q)
(8.43)
U <q ^ V
qeS>
where
(7 = n1/12, K = exp((logn)1+2<3). (8.44)
By (8.5) and Lemma 4.5, A(m, q) is a multiplicative function of q.
Therefore
Z A (m, q)
U <q<V
qs°J
< X E|/t(m,rM(m,s)|
r > n80d se<#
rs3
+ Z \A(m9r)\
r < n80<5
Z A (m> s)
U/r < s ^ V/r
(8.45)
The first double sum is
<
w-2^Xr1/40|>l(m,r)|Yxi>lKs)l
(8.46)
The first sum here is
OC
00
]1 1 + I phlM\A(m, p")\ Ml 1 + E Ph/4°\A(m, ph)\
5<p<n\ /i = 2 //p<5\ /i=l
and, by (8.31), (8.37) and (8.38), this is
where the product is over all pairs p, r with pf||m. This is
<n\
By (8.38), the second sum in (8.46) does not exceed
n (1+8/^-1)^.
p < n

The singular series
141
Hence, by (8.31) and (8.45),
X A(m,q)<n~d + F(m)
where
qe<Z
F(m)= ^
r < n80<5
(8.47)
Z A(m9s)
U/r < s ^ V/r
(s,r)=l,.setf
(8.48)
By (8.39) and (8.40) and the multiplicative property of A(m, s),
A(m, s) = X* c(X)x(m) (seV) (8.49)
x
modi
where £* denotes a sum over primitive characters,
|c(z)l<s_1 (8.50)
and, for X > 0,
X* Mz)|"<8ro(s)5-A. (8.51)
mods
Lemma 8.2 Let I denote a natural number. Then, for arbitrary complex
numbers b(x),
N
I
vX= 1
I I.*b(x)x(x)
«<G x
mod 4
Wl £*|fc(Z)|
\Q^Q X
3/2\ 2/3
21/(21- 1)
(2/- 1)/2/
mod 4
where
B = (N1/2 + g1/z)N1/6(log(Nze))(Z4 " 1)/(6Z)
arcd t/ie implied constant is absolute.
Proof By the case I = 1 of Lemma 5.3 (the large sieve inequality), for
arbitrary complex numbers cl9. .., cN,
q
I I
q<Q a=1
(«,«) = 1
JV
X cxe(ax/q)
X = 1
JV
^ (N + Q2) X |c;
(8.52)
X = 1
By the theory of Gauss sums (see § 20 of Hasse (1964) or §9 of
Davenport (1966)) for a primitive character x modulo q,
q
*(X)-1 Z X(y)e(yx/q) = x(x)
y= i

142
A ternary additive problem
where \t(x)\ = <?• Therefore
N q N
Z c^(x) = t(X)_1 Z X(y) Z cxe(yx/q).
y= 1 x = 1
Hence
Is
X = 1
JV
Z cxX(x)
x = 1
JV
Z xoo Z ^{yx/q)
y=1 x=1
^-T1 I
mod </ mod </
Hence, by the orthogonality of characters, and (8.52)
I Z:
q^Q X
N
modi/
Z CxX(*)
x= 1
JV
<(N + Q2) £ \c:
x= 1
Applying this to the /th power of Z, = i cxX(x) giyes
z z*
1^ Q *mod<,
/V
Z Cx*M
X = 1
«(N' + e2)ZK
where
d,= Z'
y ^, wxi ' • • ~x,
xi .. .x, = y
and Z' denotes that the summation is restricted to Xj with Xj ^ N.
Suppose that / > 2. Then, by two applications of Holder's inequality,
Wf^d^y)
2 - 2/A
r
2/A
C C
JCi . ..x,= >•
and
„ , £! \l -2/A / W \2(M
ZKI2<( Z ^(y)(2A-2,/(A-2,» ' ^ '-"
zi.
'3" i ~ — / I ^ i-x
y \y = i / \x = l
where dt(y) is the number of solutions of x1 .. . xt = y in xl5. . ., xz.
Hence, by Theorem 288 of Hardy, Littlewood & Polya (1951),
A/(A- 1)\(A- 1)/A
Z 1*HX)X(X)
N
Z
,x = 1
q^Q X
modi/
WZ Z*IM%)I
2//(2/ - 1)
(2/- 1)/(2/)
where
modi/
ba = (nz + e2)1/(2Z)( z ^/(y)(2A"2)/(A"2))
,(1 - 2/1)/(21)

The singular series
143
Let k = 3. Then the lemma follows provided that, for X > 1,
X ^bO^XdogXe)'4-1.
In fact, it is easily seen by induction on r that dr(xy) ^ dr(x)dr(y) and by
induction on s that
and
I ^(y)^X(log^r-1
y<X
£ dr(jf 3T ^ (log x*r-
Let Q0 = Ur 1 and Q, = nl/2, let fr(#) = c(#) when q, the modulus of #,
is in <#, (^, r) = 1 and C//r < g < V/r, and let fefa) = 0 otherwise. Then,
by (8.48) and (8.49),
F(m)= £
r < «803
I Z* HxMm)
<l X
mod ^
(8.53)
By Lemma 8.2, Holder's inequality, (8.50) and (8.51),
In
I
m - n + 1
Z Z* Wx)X(m)
Ql- 1 <q<Qi X
mod 4
< n(l log(2ne))('4- men q-wdx^ (1 + 8p- ip - 1)/(20
p < n
This is
<|M7/8(/log(2^))(Z4-1)/(6/)(log2M)(8z-4)//
or
«U(y-1/2r1/2(logn)4
according as / > 1 or / = 1. Hence, summing over / with Qx_ x ^ K
gives, via (8.44) and (8.53),
2n
X F(m) « n47/48.
m = n + 1
1_<5 values of m with n<m^2n one has
Hence, for all but <£ n
F(m) < n~d, and so, by (8.47), when m is not exceptional,
Z A{m,q)<n-*.
qe9
(8.54)

144 A ternary additive problem
The proof of Theorem 8.3 is completed by examining
Z A(m,q).
q> V
qe&
Let X = l/(logtt). Then
X \A(m,q)\<: X (q/V)x\A(nu q)\
q > V qe9
qeC/
= V~XY\ (l+ t phX\A(m,ph)\).
Hence, by (8.31), (8.38) and (8.44),
Z \A(m,q)\
q > I
qe'S
< exp ( - (log n)2i) ft ( 1 +240 J p<30A ~ Uk + 3(U" '
5<p<n\ /c = 0
<^ exp (- (log n)s).
Therefore, by (8.54), (8.44) and (8.6), for all but < n1 ~d values of m
with n < m ^ 2n one has
fl ( I A(m,p")))-'Z(m,ni'12)<cxp(-{\ogn)i),
vp < n \h = 0
as required.
8.7 Completion of the proof of Theorem 8.1
By Theorems 8.2 and 8.3, for all but ^n1'3 values of m with
n < m^2n one has
R(m)
= /("■)(( 11 (Z ^(m,^)n + 0(exp(-(lQg^))J+0(n1/30^).
Consider I(m), given by (8.3), when n<m^2n. For y3, y5 satisfying
\m — \n < y3,y5 < \m — \n one has \n<y3,y5<n and \n<
m — y3 — y5 < n. Hence 7(ra)^>n1/3°. It is trivial that I(m)<£n1/3°.

Completion of the proof of Theorem 8.1 145
Thus it suffices to show that
00
n I A(m, ph))> (log nyc. (8.55)
p ^ n \h = 0 /
By (8.38), there is a constant C such that
oc
11 (Y.A(m,^))> ft (l-Cp-^Xlogn)
C < p^ n \h = 0 J C < p< n
-C
Therefore it is only necessary to show that for each prime p one has
OC
X A(m,ph)>p~6. (8.56)
/i = 0
It is easily deduced from (8.5) (cf. Lemma 2.12) that
p2' £ A(m,ph) = M{m,p') (8.57)
/i = 0
where M(ra, pr) is the number of solutions of
x2 + y3 + z5 = m(mod p') (8.58)
with l^x, >\ z^p1. Let y(2) = 3, y(p)=l (p > 2). When p/a the
congruence x2 =a{mod pr) has a solution for each f ^ y(p) whenever it
has one for t = y(p). Thus if it can be shown that (8.58) is soluble when
t = y(p) with p/x, then p2t' 2Hp) different solutions can be produced in
the general case t ^ y(p) by taking any y', z with y' = y(mod py(p)),
z' = z(modpy(p)). Thus
M(m, pr)>p2'~2y(p)
which, by (8.57), yields (8.56).
It is trivial that (8.58) is soluble with 2\x when p = 2 and
t = y(p) = 3. It remains to establish the corresponding result when
p > 2.
The number of cubic or zero residues modulo p is at least
(p — 1)/(3, p — 1)+ 1. Hence the conclusion will follow by the pigeon
hole principle if it can be shown that the number N of residues modulo
p of the form x2 or x2 + 1 with 1 ^ x ^ p — 1 satisfies

146 A ternary additive problem
This is readily done starting from the formula
8.8 Exercises
1 Show that almost every natural number is of the form p + xk.
2 (Babaev, 1958) Show that card {n:n # p + x\n ^ X) p X1/k.
3 Let R(n) denote the number of solutions of
x2 + y3 + z6 = n
with x > 0, y > 0, z > 0. Show that
(ii) r(f)r(f)r® = 0.73...,
(iii) x2 + y3 + z6 = n(mod ^) is always soluble with (x, q) = 1.
4 Obtain an asymptotic formula for the number of representations
of a number as a sum of two squares, two cubes and two fifth powers.
5 (modified version of Jagy & Kaplansky, to appear) Suppose that
p = 3 (mod4), p > 3, v = x2 + y2 = (18p)3 — z9 and u = 18p — z3.
Show that z = 3 (mod 4), (2p,u) = 1, (u,v/u)\ 35, u = 3 (mod 4) and 3
divides u to an even power. Prove that there is a prime number q = 3
(mod4) and an odd natural number s such that qs \\ u and q\v/u.
Deduce that x2 + y2 + z9 = (18p)3 is insoluble.

9
Homogeneous equations and
Birch's theorem
9.1 Introduction
Let F(x1,. . . , xs) be a homogeneous form of degree k ^ 2 with integer
coefficients. A natural question is to ask whether the equation
F(xl9...,xs) = 0 (9.1)
has a non-trivial solution, i.e. a solution in integers Xj not all zero.
Obviously when k is even the equation may only have the trivial
solution. However, when k is odd there is more hope. Lewis (1957)
building on earlier work of Brauer (1945) showed that if s is
sufficiently large, then any cubic form in s variables with integer
coefficients has a non-trivial zero. Shortly afterwards this was
extended by Birch (1957) to forms of arbitrary odd degree. Indeed,
Birch proved somewhat more than this. The object here is to give an
account of Birch's theorem. For references to later work on this and
related topics the interested reader should see Davenport's collected
works (Davenport, 1977).
The proof of Birch's theorem rests on a special case, namely on the
solubility of the additive homogeneous equation
^+... + ^ = 0, (9.2)
and this can be treated by an application of the Hardy-Littlewood
method.
9.2 Additive homogeneous equations
The methods of Chapters 2, 4 and 5 are readily adapted to give the
following theorem, and so the proof is only given in outline.
Theorem 9.1 Let k > 2 and s0 be as in Theorem 5.4, and suppose that
s > min(s0, 2k + 1) and s > 4/c2 — k + 1. Suppose further that when k

148 Homogeneous equations and Birch's theorem
is even not all of the integers c1, . . . , cs are of the same sign. Then the
equation (9.2) has a non-trivial solution in integers x,, . . . , xs.
Throughout this section, implicit constants may depend on
c c
*-" 1J • • • J ^s"
If there is ay' such that Cj = 0, then the conclusion is trivial. Thus it
may be assumed that, for every j, Cj =£ 0. Also, when k is odd, it can be
assumed (if necessary by replacing x1 by — xj that not all the cj are of
the same sign. Let R(N) denote the number of solutions of (9.2) with
1 ^ Xj^ N. Then the methods developed in Chapters 2, 4 and 5 give
R(N)=&J(N) + 0(Ns-k-d)
where
OC
3 = n T(p), T(p) = £ S(ph),
p /i = 0
q s
s(<l)= X II (q~lS(q,aCj)),
a = 1 j = 1
(a,q)= 1
and
J(N)= E ... I nm,...™,)1"-1.
mi = 1 ms= 1
c\ml + . . + c\ms = 0
They further show that there is a number C, depending at most on
cx,. . ., cs, such that
n t(p)>$
p>C
and that
J(N)>Ns~k.
Now it suffices to show that T(p) > 0 and, again, this will follow if it is
shown that MF{q\ the number of solutions of
F(x1? . . . , xs) = cxx* + . . . + csxks = 0(mod q)
with 1 < Xj ^ q, satisfies, for t sufficiently large,
MF(Pt)>C(p)PHs~1) (9.3)
for some positive number C(p) depending only on c1?. . ., cs and p.
In order to treat MF it is necessary to transform the variables so as

Additive homogeneous equations 149
to obtain a new form H in which an appreciable number of the
coefficients are coprime with p. Choose iy so that plj || Cj and choose
hj, lj so that Xj = hjk + \- and 0 ^ \] < k. Then
F(x1? . . ., xs) = G(phixu ..., phsxs)
where
G(xx, . . . , xs) = d1pllx\ +... + dsplsxks
with dj = CjPjXj. Now let h = max hy Then
F(p*-*%,..., p*-fc-xs) = pfckG(x1,...,xs)
and, for t > h,
pt - h + hi pt - h + hs
MF(P')> I ... Z 1
X! = 1 Xs = 1
phkG(x,, ..,xs) = 0(modp')
>MC(p'"*k)n p
j=i
hk- h + h
j
whence
Mpip^^Mcip1'^). (9.4)
The form G can be rewritten as
G = G(0) + pG(1) + . .. + ^-^-^
where
fc
Go-> = GU)(XU-)) = £ ^X*.
I = 1
', = J
Clearly there exist i and r with r ^ s/k and G(0 containing at least r
variables. Consider the form
H(xl9 . . . , xs) = ( £ pJGiS)(px{S)) + X pJGiJ\xiJ)) V \
Now
Mc(pr)^Mw(^-') (9.5)
and H has the shape
H = H{0) + pH(1) + . . . + pk~ lH(k- l)
with H{0) containing at least r variables, where r ^ s/k, and all its
coefficients relatively prime to p. It can be assumed, if necessary by

150 Homogeneous equations and Birch's theorem
relabelling the variables, that
H(0) = /j<°>(Xl,. . . , xr) = dxx\ + ... + drxk.
By (9.5) and (9.4), to prove (9.3) it now suffices to show that there is a
positive number C1{p) such that for t sufficiently large
MH(pt)>Cl(p)p^~l\ (9.6)
Let t denote the highest power of p dividing k and write y = z + 1
when p > 2 or t = 0, and y = t + 2 when p = 2 and t ^ 1. Then, as in
§26, (9.6) will follow on showing that, for each m,
d^x\ +...+ drxkr = m(mod py) (9.7)
is soluble in x1,. . . , xr with p\xv
Let K = py~x~ l(k, px(p — 1)). Then the number of /cth power
residues modulo py is (p(py)/K. Hence, by Lemma 2.14, the set M ■ of
residues m modulo py which can be written in the form
dxx\ + ... +djXkj (pJlXi)
satisfies card^> min{pyJ<j)(py)/K). Thus, ifr>4k, i.e. s > 4/c2 —/c,
then (9.7) has a solution of the desired kind, and this completes the
proof of Theorem 9.1.
Suppose that cl9. . . , cs are integers such that for every q the
congruence
c^ + . . . + csxks = 0(mod q)
has a solution with (xj9 q) = 1 for some/ Then, following Davenport
& Lewis (1963) c1?. . . , cs are said to satisfy the congruence condition.
They define T*(/c) to be the least s such that every set of s integers
c1?. . . , cs satisfies the congruence condition. They further define
G*(/c) to be the least number t such that whenever s ^ t the equation
cxx\ + . . . + csxks =0
has a non-trivial solution in integers when cx,..., cs are not all of the
same sign when k is even and satisfy the congruence condition.
The argument above gives T*(/c)^ 4/c2 — k + 1 and G*(/c)^
min(s0, 2k + 1). Davenport and Lewis show (i) that T*(/c)^/c2 + 1,
(ii) that r*(fe) = k2 + 1 when k + 1 is prime, and (iii) that G*(fe) ^
k2 + 1 when/c > 18and/c ^ 6. Vaughan (1977b, 1989a) has removed
the gap in (iii) by using the methods of Chapters 5, 6, 7 and 12.

Birch's theorem 151
For small values of/c, T*(/c) is known. (See Bierstedt (1963), Bovey
(1974), Dodson (1967), Norton (1966).) Also, following earlier work of
Norton (1966) and Chowla & Shimura (1963), Tietavainen (1971) has
shown that
r*(2fe+l) 2
lim sup —— = .
k->oc /clog/c log 2
9.3 Birch's theorem
Theorem 9.2 (Birch, 1957) Let j, I denote natural numbers and let
/c1? . . . , kj be odd natural numbers. Then there exists a
number T-(fcl9 ..., fc-, I) with the following property. Let
Fx(x), ..., Fj(x) denote forms of degrees /c1? . . . , kj respectively in
x = (xl9 . . . , xs) with rational coefficients. Then, whenever
^4^,...,/^,/)
there is an l-dimensional vector space V in Qs such that for every xeV
Fl(x) = ... = Fj(x) = 0.
The first step in the proof is to establish the case when j = 1, F x is
additive and k > 3.
Lemma 9.1 There is a number <£(/c, I), defined for natural numbers k, I
with k odd and k ^ 3, such that, if s ^ 0(/c, /), then for each form
cxx\ + . . . + csxks with cx, . . ., cs rational, there is an l-dimensional
vector space V in Qs such that for every xeV
cxx\+... + csxks=0. (9.8)
Proof By Theorem 9.1 there are t = t(k) and yx, . . ., yt not all zero
such that
cxy\ +. .. + ctyk = 0.
Similarly for
and so on. Hence, when s ^ It, the point
(uxy1, . . . , u^yt, u2yt+ !,..., u2y2v • • • ■> ui);it-> v), . . . , 0)
satisfies (9.8) for all ux,. . . , ut.

152 Homogeneous equations and Birch's theorem
Proof of Theorem 9.2 Let A: = max/c,-, so that k is an odd positive
integer. The proof is by induction through odd values of k. The result
for k = \ is straightforward. For k ^ 3 the principal step is to show
that if the result holds for systems of forms with max k( ^ k — 2, then it
holds for a single form of degree k. The conclusion is then easily
extended to a system of forms of degree at most k.
For a form
F(x) = F(x1,. . . ,.\\s)= ^ <'.-, <vx<, •••*.■,
11 '\
of (odd) degree /c, consider
<i <\
...o/0)r+---+"»+>.vri,>
= I uj,---»jk i <■„ >jV ■ ■ ■ c■
ji ./k «1 «k
0 < ./,. < H + 1
Now define e{1) = (1, 0, 0, . . .), e(2) = (0, 1, 0, . . .), and so on, and take
u0 = u, j(0) = j, y n = e(1), v(2) = e{1\ .... Then a further regrouping
of terms gives
F(vy + ule{l) + . . . + un + xe{n+ 1])
k
= Z vh Z Uh--uh-kF(y'^h,--*h-h) (99)
h = 0 j\....Jk-h
1 < j,. < fl + 1
where
F(y;hJ1 Jk-h)
is a form of degree h in j = (yx,. . . , vs). The total number of such
forms with h odd, 1 ^ h < /c — 2 and 1 ^yr ^ n + 1 does not exceed
k(n + l)fc. Hence, on the inductive hypothesis, and provided that
s > %(n+1)fc(fc-2,...,/c-2,1),
one finds that the corresponding simultaneous equations
F(y*Aj1,...Jk-h) = 0
have a non-trivial solution z{0) in Qs.
If z(0\ e( l \ . . . , e(" + l > are linearly dependent over Q, then omitting

Birch's theorem 153
one of the e{j) gives a linearly independent set of n + 1 points in Qs.
Thus, in any case, by taking one of the uj to be zero in (9.9) and, if
necessary, relabelling, one obtains z(0), z(l\ ..., z(n) that are linearly
independent and such that
k - 1
F(vz{0) + ulzil) + ... + unz(n))=cvk+ ^ vhGh(u) + G0(u) (9.10)
/i = 2
h even
where Gh(u) is a form of degree k — h in u = (wp . . . , w„).
The linear independence of z{0K..., z{n) ensures that, when
x = vzi0) + u1zn) + ... + unz{n\
non-trivial choices for (i\ ux,. . . , un) give non-trivial values for x.
Consider the system of forms
G„(m) = 0, /?even, 2^/?^/c-l. (9.11)
The degree, k — h, is odd in each case. Hence, a further application
of the inductive hypothesis shows that when n ^ 4/fc(/c — 2,. . .,
k — 2, m), i.e.
.s^.s0(/c, m),
the system (9.11) is soluble for every member u of an m-dimensional
vector space U in Q". Let w(1>, . . . , w(m> denote m linearly independent
points in U and consider
u = wxu{l) + . . . + wmu{m\
The linear independence again ensures that non-trivial w in Qw give
rise to non-trivial u in Q". Hence, by (9.10), for non-trivial
(r, u'j,. . . , wwi) there are non-trivial jc = (x„ . . . , xs) such that
F(jc) = c^ + H(w)
where H is a form in w = (w1}..., wm) of degree /c, i.e. F represents
ci;fc + //(w).
Continued repetition of this argument shows that if s^s{(k, /),
then F represents a diagonal form
cxv\ + . . . + ctvkt
with t = 0(/c, /). Lemma 9.1 now gives the case j = 1, k1 = k of the
theorem.
To complete the inductive argument, it remains to establish the
general case of j simultaneous equations Fx = . . . = Fj = 0 with

154 Homogeneous equations and Birch's theorem
max fc, = fc. This is done by subinduction on j. The case 7 = 1 has
just been dealt with. Suppose j > 1. Without loss of generality
it can be supposed that fc ■ = k. By the case 7 = 1, given m, if s >
*¥i(kp m), then there is an m-dimensional vector space U in Qs such
that Fj(x) = 0 for every jc in U. The points of U can be represented by
where Jt(1),. . . , x{m) are linearly independent points of Q\ For these
points the forms F1?. . . , F}_ x become forms in y = (y1, . . . , ym). If
max kt < k — 2
1 < 1 < j - 1
then one uses the main inductive hypothesis. If
max fc,- = fc
1 < i < j - 1
then one uses instead the subinductive hypothesis. In either case,
provided that m ^ 4^ _ 1(kl,. . . , fc;_ 1? /), there is an /-dimensional
vector space V in Qm on which each Ft vanishes. This completes
the proof of the theorem.
9.4 Exercises
1 Adapt the methods of Chapter 7 to show that
limsup — -^ 2.
fc^oc fclogfc
2 Adapt the methods of Chapter 6 to show that G*(3)^8,
G*(4) ^ 14, G*(5) ^ 23, G*(6) < 36.
3 Show that r*(2) = 5, r*(3) = 7, r*(4)=17, and that
r*(fc)^min(50,2fc + l).

10
A theorem of Roth
10.1 Introduction
van der Waerden (1927) proved that given natural numbers /, r there
exists an n0(l, r) such that if n > n0(l, r) and {1, 2,. . . , n} is partitioned
into r sets, then at least one set contains / terms in arithmetic
progression.
For an arbitrary set ja/ of natural numbers, let
A(n) = A(n,^/)= £ I D(n) = D(n, rf) = -A(n) (10.1)
a < ii H
ae,<J
and write d and 5 for the lower and upper asymptotic densities of a/,
d = d(«s/) = liminfD(n) and 5 = 3(,s/) = limsupD(n) (10.2)
respectively. When d = d\etd = d(s#) denote their common value, the
asymptotic density of stf. Erdos & Turan (1936), in discussing the
nature of the known proofs of van der Waerden's theorem,
conjectured that every set sJ with d(s^) > 0 contains arbitrarily long
arithmetic progressions. An equivalent assertion is that if there is an
/ such that s4 contains no arithmetic progression of / terms, then
d(*t) = 0.
The first non-trivial case is 1 = 3. The initial breakthrough was
made by Roth (1952, 1953, 1954) in establishing this case by an
ingenious adaptation of the Hardy-Littlewood method.
By a different method, Szemeredi (1969) proved the conjecture for
/ = 4, and Roth (1972) has given an alternative proof by an approach
related to that of his earlier method.
In 1975, Szemeredi established the general case. Unfortunately
Szemeredi's proof uses van der Waerden's theorem. More
recently Furstenberg (1977) has given a proof of Szemeredi's theorem
based on ideas from ergodic theory. Although this does not use van
der Waerden's theorem it apparently has a similar structure and so
still does not yield the sought after insight.

156 A Theorem of Roth
Ideas stemming from the attacks on this problem have enabled
Furstenberg (1977) and Sarkozy (1978a, b) to establish that if
d{srf) > 0, then the set of numbers of the forma —a' withaeja/, a'eja/
contains infinitely many perfect squares.
In this chapter, Roth's theorem is established using his version of
the Hardy-Littlewood method, and a proof of the Sarkozy-
Furstenberg theorem is developed along the lines of Furstenberg but
avoiding the ergodic theory.
Throughout this chapter implicit constants are absolute.
10.2 Roth's theorem
Let Mil)(n) denote the largest number of elements which can be taken
from {1, 2,. . . , n} with no / of them in progression. Let
Then Szemeredi's theorem is the assertion lim„^ ^ fi{l)(n) = 0, and this
obviously implies the Erdos-Turan conjecture. As the following
lemma shows, it is quite easy to prove that the limit exists. Its value is
another matter.
Lemma 10.1 For each integer /, lim„ _ a fi{l)(n) exists. Also, for m > n
one has fi{l)(m) < 2fiil){n).
Proof It is a trivial consequence of the definition of M(Z) that
M{l)(m + n) ^ M(Z)(m) + M(l)(n).
Hence
Mil){m)^
m
n
M{l){n) + Mil)(m-n
m
n
YYl
^-M{l)(n) + n
n
Therefore fi{l)(m) ^ fi{l)(n) + n/m, so that
lim sup fiil)(m) < fi{l){n)
m -* oc
whence
lim sup fiil){m) < lim inf fi{l)(n).
m -* oo n -*■ oc
Also, when m > n, M{l)(m) < (m/n + l)M{l)(n) ^ 2M{l)(n)m/n.

Roth's theorem 157
The following theorem not only shows that when / = 3 the limit is 0,
but gives a bound for the size of M{3)(n).
Theorem 10.1 (Roth) Letn>3. Then fi{3)(n) < (loglogn)" K
It is henceforward supposed that I = 3, and for convenience the
superscript (/) is dropped.
Choose M cz {1, 2,. . . , n) so that card^ = M(n) and no three
elements of Ji are in progression. Let
m € ,Al
Then
a
M(n)
/(a)2/(-2a)da (10.3)
o
since the right-hand side is the number of solutions of m1 + m2 = 2m3
with m-eJi and, by the construction of Ji, such solutions can only
occur when m1 —m2 — ra3.
Let k denote the characteristic function of Ji, so that
/(a) = 5>(xMa*). (10.4)
Suppose that
and consider
m < n, (10.5)
v(oL) — fi(m) ]T e(ax) (10.6)
X = 1
and
Then
E(a) = i;(a)-/(a).
£(a) = X c(*)e(ax) (10.7)
X = 1
with
c(x) = fi(m) - k(x). (10.8)
The idea of the proof is that, if M(n) is close to n, then
2
/(a)2/(-2a)da
0

158 A Theorem of Roth
ought to be closer to M(n)2 than to M(n)(d. (10.3)). To show this, one
first of all uses the disorderly arithmetical structure of Ji to replace f
by v with a relatively small error. It is a fairly general principle,
observable from the applications of the method in previous chapters,
that sums of the form
Z e(*x)
x ^ n
X€£j/
tend to have large peaks at a/q when the elements of srf are regularly
distributed in residue classes modulo q. Note that v(ol) has its peaks at
the integers.
Let
m - 1
F(a)= X ^(az). (10.9)
= = o
Lemma 10.2 Let q be a natural number with q < n/m, and for
y = 1, 2, . . . , n — mq let
m - 1
<r(y) = o(y;m,q) = £ c(y + X(l)- (10.10)
x = 0
Then
a(y)>Q (y =1,2,...,n-mq) (10.11)
and
n — mq
F((xq)E((x)= £ (T(y)e(a(y + mq-q)) + R((x) (10.12)
y= i
where R(cc) satisfies
\R((x)\<2m2q. (10.13)
Proof By collecting together the terms in the product FE for which
x + zq = /7 + mq — q one obtains
n
F(aq)E(cc)= Yj e{<x(h + mq — q))
h — 1 + q — mq
m - 1
x ^ c(h + q(m — I — z)).
z= 0
l^/i + q(m- 1 - z) **n
The innermost sum is at most m in absolute value, and so the total

Roth's theorem 159
contribution from the terms with h ^ 0 and h> n — mq does not
exceed, in modulus, m(mq+(m—l)q) <2m2q. For the remaining
values of h one has 1 < h + q(m — 1 — z) ^ n for all z in the interval
[0, m - 1]. This gives (10.12) and (10.13).
By (10.8) and (10.10),
m - 1
<x(v) = M(m)- £ K(y + xq).
x = 0
Let
m - 1
r= Z *(}>+ *<?)•
x = 0
Then r is the number of elements of .# among y, y + q,. . .,
y + (m — i)q. Let these elements be y + xxq,. . ., y + xrq. Then no
three are in progression. Hence no three of x1?. . ., xr are in
progression. Likewise for l+xl5...,l+xr. Moreover 1+x^ra.
Hence r ^ M(ra), which gives (10.11).
Lemma 10.3 Suppose that 2m2 < n. Then, for every real number a,
|£(a)| < 2n(fi(m) — fi(n)) + 16m2.
Proof By Lemma 2.1, there exist a, q such that (a, q) = 1, 1 ^ q ^ 2m
and |a — a/q\ ^ l/(2qm). Then
F(oLq) = F(xq-a) = F{P)
where |j8| ^ l/(2m). Hence, by (10.9),
\F(*q)\ =
sin rnnfi
sinn/l
2m
n
Thus, by Lemma 10.2,
\m\E((x)\ ^-m\E((x)\
71
< \F(aq)E(oi)\
n — mq
< ^ o{y) + 2m2q
y= i
< mE(0) + 8m3.
Moreover, by (10.7) and (10.8)
•>
£(0)- £ ifi{m) - k(x)) = n(n(m) - n(n)).
x= 1

160 A Theorem of Roth
The lemma follows at once.
Proof of Theorem 10.1. Let
'•i
I
Then, by (10.4) and (10.6),
/(a)2r(-2a)da. (10.14)
o
/ = I I /*(»»)•
ae M be.At
2\a + b
Thus, if Mj is the number of odd elements of M and M2 the number of
even elements, so that M1 + M2 = M(n\ then
/ = n{m){M\ + M\) > ^(m)M(n)2. (10.15)
By (10.3) and (10.14),
|/(a)|2da.
|M(n)-/|^(max|£(a)|
o
Therefore, by Lemma 10.3 and Parseval's identity, when 2m2 < n one
has
\M(n) -/I** (2n(fji(m) - ft(n)) + 16m2)M(n).
Hence, by (10.15),
fi(m)fi(n) <: 4(fi(m) - fi(n)) + 34m2n~ x (2m2 < n). (10.16)
Letting n—> oo and then m—► oo shows that t = lim„_+ ^/^(^) satisfies
t2 ^ 0. To establish the quantitative version of this, let
X(x) = fi(23Xy
By Lemma 10.1, it suffices to show that A(2x) <^x~ 1.
By (10.16),
A(y)A(y + 1) ^ 4(A(y) - A(y + 1)) + 34 x 2" 3>.
Dividing by A(_y)A(_y + 1), summing over j/ = x, x + 1,. . . , 2x — 1 and
appealing to Lemma 10.1 gives one
x < 4A(2x)" x + 200xA(2x)" 22~ 3\
When /l(2x) > 1/x the second term on the right is < \x for x
sufficiently large, so that /l(2x) < 8/x, which gives the desired
conclusion.

A theorem of Furstenberg and Sarkozy
10.3 A theorem of Furstenberg and Sarkozy
161
Theorem 10.2 Let srf bea set of natural numbers with d(srf) > 0, and let
R(n) denote the number of solutions of
a—a' = x2
in a, a', x with aesrf, a'estf, a^n. Then
\imsupR(n)n~3/2>0.
n -»• oo
This theorem is somewhat stronger than Theorem 1.2 of
Furstenberg (1977). The approach of Sarkozy (1978) is different. He
adapts the methods of § 10.2 to show that ifa — a' = x2 has only trivial
solutions, then
A(n) < n(\og log n)2/3(logn)~ 1/3.
Let *y0 denote an infinite set of natural numbers such that
lim n~ M(n) = 3(j?/),
n -*■ oo
let
and let
Wln(q, a) = {a : |a - a/q\ ^ q~ ln~ 1/2],
(10.17)
/(a)- £ e(a.a).
a < n
aesf
It is necessary to show that f has fairly orderly behaviour on Wln(q, a).
For n ^ 4,
/(a)|2da^
Wln(q,a)
|/(a)|2da ^ n.
0
Hence
\f(a)\2n~lda
Wln(q,a)
is bounded uniformly in q, a, n. Therefore one may choose infinite sets
J^(q, a) of natural numbers such that
jV(1, 1) = ^"(1, 0) c.V09 ¥{q + 1, 1) c f(q, q - 1),

162
A Theorem of Roth
Jf(q, a') cz ^V(q, a) when l^a<a'^q — 1 and (a', q) = (a, g) = 1 and
I/Ml2"" Ma
p(q, a) = lim
n -»• oo
Wln(q,a)
ne^ (q,a)
exists. Thus, given Q, for all sufficiently large n with neJ^(Q, Q — \)
one has
Z Z Pte,a)<i+ Z Z
4 < Q a = 1
(«,«) = 1
q< Q a= 1
(a,q) = 1
|/(a)|2n- Ma
9Wn(4,fl)
and the 9W„(g, a) with 1 < a ^ q < Q and (a, g) = 1 pairwise disjoint.
Hence
Z Z p(q,a)<\ +
q^Q a=1
(a,q) = 1
n
0
l/taJpn-Ma < 2.
Therefore
00
Z Z Pfoa)
4=1 a = 1
converges.
10.4 The definition of major and minor arcs
Suppose that 0 < rj < 1 and choose Q = Q0(rj) so that
Z Z p(q,a)<r] and 2>
q = Q + 1 a = 1
(«,«) = 1
Now define
/c = (g!)2, P = k100
and henceforth suppose that
neJ^(P, P- 1).
Then, given
choose
n0 = n0(>/,X)^X2
1
(10.18)
(10.19)

The definition of major and minor arcs
163
so that when
n^n
o
and 1 ^ a ^ q ^ P with (a, q) = 1 the major arcs
SRn.xfo fl) = {« : |a - a/q\ < Xq~ ln~ l}
are pairwise disjoint,
|/(a)|2tt Ma < p(q, a) + f]P
(10.20)
9Wr,(4,fl)
and
4(n)>fdn. (10.21)
By (10.17), 9K„>X(^, a) c <3Rn(q, a) for n 3* n0. Hence, by (10.20),
|/(a)|2n_1da<p(^,a) + ?/P
(10.22)
**H,xto>a)
Let 5)¾ denote the union of the major arcs WlntX{q,a) with
1 ^ a ^ q < P and (a, g) = 1, and define the minor arcs m by
m = {Xn~\l + Xn-^\m.
Further define
N
g(fl= I e(Px2)
x= 1
(10.23)
where
N^(n/k)1/2.
(10.24)
Then, by (10.19),
R{n)^0t where
By Theorem 4.1, when (a, g) = 1
n
<,(/ca)|/(a)|2da.
(10.25)
o
a
g(y) = q~lS(q9 a)h[y--) + 0[ q9,16( 1 + N2
y
a
q
(10.26)
where
S(q,a)= £ e(ax2/<ll HP)
x= 1
/NV2
0
.U- 1/2
a" 1/2e(j?a)da. (10.27)
Also, by Theorem 4.2, S(q, a) < qm.

164
A Theorem of Roth
10.5 The contribution from the minor arcs
Suppose aem. Choose a, q so that (a, g)=l, g<N4/3, \kcc—a/q\
^q~ 1N~413. Let a1=a/(k,a\ q1= qk/(k,a). Then |a — a1/q1\
^q~ 1N~413 and (al,ql)=l. Since aem, either q1> P or
|a —a1/q1\> Xn~ 1q^ \ In the first case, by (10.26),
g(koc)<tq~1/2N + q9/16l 1 + N:
kcc —
a
1
and in the second |/ca — a/q\ > Xn~ 1q~ 1, so that, by (10.26), (10.27)
and Lemma 2.8,
1/2
g(ka)<q~ 1/2
kcc
a
1
+ N3/4
Hence, by (10.19), in either case
g(koi) < Nk~ 40 + n1/2X~ 1/2 + N3/4.
Therefore, by Parseval's identity,
g(ka)\f(a)\2doL < (Nk~ 40 + n1/2X~ 1/2 + N3/>. (10.28)
m
10.6 The contribution from the major arcs
Now suppose that ae^)ln x(q, a) where 1 ^a < q < P and (a, q) = 1
Let qx = q/(q, k\ ax =ak/(q, k). Then, by (10.26),
g(ka) = q1 ^(q^ajhlklct
a
q.
a
a
q
+ Olq91/16U+N2k
The error term here is majorized by P + N2kXn~ *. Hence
g(ka)\f((x)\2d(x = MX+ 0(Pn + N2kX) (10.29)
2R
where
*i= I I
q^P a= 1
(a,4) = 1
a
^^(^fli)/! fc a-=))|/(a)|2da.

Completion of the proof of Theorem 10.2
165
By (10.22) and (10.18) the terms here with q ^ Q + 1 contribute an
amount which in absolute value does not exceed
P q
£ X Nn(p(q,a) + r]P~2)<2r]Nn.
q = Q + 1 a= 1
ia,q)= 1
Also, when q ^ Q, by (10.19) one has g|/c, so that q1 = 1. Hence
>2 + OfoAfa) (10.30)
where
I I
hi ki a
q^Q a= 1 Jan„,x(q,a)
(a, 4)= 1
a
q.
||/(a)|2da.
It is easily shown that for every positive number Y
a~ 1/2cosada > 0.
o
Hence, by (10.27),
Re/iOS)HjT1/2
rN2\p\
k(x. 1/2cos27rada > 0.
(10.31)
0
Therefore, on discarding all the terms in 0t2 with the exception of that
with a = q = 1, one obtains
ri/4nn
Re@i>
Re/i(/ea)|/(a)|2da.
1/47TH
Also, when |a| ^ l/(4nn), one has
/(a) -/(0)= £ 27cix
so that
x= 1
X6 J2/
■a
e(Px)dp
o
|/(a)| > /(0)(1 - 27r|a|n) > ±/(0) = ^(n).
Therefore
ReM2^±A(n):
(•1/4-nn
Re/i(/ea)da.
(10.32)
o
10.7 Completion of the proof of Theorem 10.2
By (10.24) and (10.31), Re fc(fca) ^ \N whenever 4nn\(x\ ^ 1. Hence, by
(10.32) and (10.21),
32nn
250

166 A Theorem of Roth
Thus, by (10.25), (10.28) and (10.30),
R(n)>@ = Re@
= Re^2 + 0((Nk~40 + n1/2X~ 1/2 + N3/4)n +rjNn)
d2
> nN - C((Nk~ 40 + n1/2X~ 1/2 + N3/*)n + rjNn)
250
(10.33)
for a suitable constant C ^ 1.
The proof is completed by making suitable choices of the
parameters. Let
*y= 10" 4a2C_1.
This fixes Q = Q0(rj) and so k and P. Note that, by (10.18) and (10.19),
k>Q>l/ti.
Let
X = rj-2k
and suppose that n^n0(rj, X) with neJr(P, P— 1). Finally, let
N = [(n/fc)1/2], so that (10.24) holds. Now for n ^ n^),
C((Nk-*° + n1/2X" 1/2 + N3/4)n + yyNn)
< C(rjNn + rjn3/2k~ 1/2 + f/Nn + ^iVn)
< 5CrjNn
d2 Nn.
2000
Hence, by (10.33),
lim sup R(n)n~ 3/2 ^ 32/c" 1/2 > 0
n -* oo 3UU
as required.
10.8 Exercises
1 Prove the theorem of Sarkozy stated in § 10.3.
2 Show that if d(s/) > 0 and R(n) denotes the number of solutions
of a —a' = p — 1 with aestf, a'estf, a^n, p prime, then
limsupJR(n)(logn)n 2 > 0.
n -* oo

11
Diophantine inequalities
11.1 A theorem of Davenport and Heilbronn
All of the forms of the Hardy-Littlewood method described so far
have dealt with the solution of equations in integers. For instance, in
Chapter 9 it was shown that if s is large enough, then given integers
cl9 . . ., cs (or equivalently given that c1?. . ., cs are all in rational
ratio), not all of the same sign when k is even, the equation
cxx\+.. .+ csx* = 0
has a non-trivial solution in integers x1,.. ., xs. Now one can ask
what happens when the c1?. . ., cs are not in rational ratio. It is no
longer sensible to insist that the form represents 0, but one can ask
instead that it take arbitrarily small values.
In order to answer this question, Davenport & Heilbronn (1946)
introduced an important variant of the Hardy-Littlewood method.
This enabled them to establish the following theorem.
Theorem 11.1 Suppose that s ^ 2k + 1 and tlmt /^,..., Xsare non-zero
real numbers not all in rational ratio, and not all of the same sign wlien k is
even. Then for every positive number n there exist integers x1? . . . , xs, not
all zero, such that
1^+...+^1^. (11.1)
It suffices to prove the theorem when n = 1, for it can then be applied
with Aj replaced by Aj/rj. Moreover, when k is odd, replacing, if
necessary, x\ by ( — x^f enables one to assume in this case also that
not all the Aj are of the same sign.
By relabelling it can be supposed that kJX2 is irrational. If
kJX2 > 0, then consider any j for which ^Jkj < 0. Then, when a^a- is
rational, A2/A7 is irrational and negative. In any case, by further
relabelling it can be supposed that
a1/a2 is irrational and negative. (11.2)
In all of the forms of the Hardy-Littlewood method considered so

168 Diophantine inequalities
far, the line of attack has been via fourier transforms on the torus
T = M/Z. For the present problem it is more appropriate to work on
U. The obvious analogue of (1.8) is the identity.
<•*
/ ^sin27ca fl (|jB| < 1),
eioLp)— da = <
. - x ^ 10 (|/J| > I)-
However there are difficulties associated with this transform because
the integral does not converge absolutely. It is more convenient,
therefore, to use instead
/(/0 =
1 *- /sin7iax
e(aj8)K(a)da, K(ol)=[ . (11.3)
, ticl
oc \
A straightforward application of the Cauchy integral formula gives
/(/*) = max(l-101,0). (11.4)
Lei
/(«)= £ e(axk), fj(oi) = f{Aja). (11.5)
X = 1
Then for the method to be successful one requires a positive lower
bound for
R(N) =
oc
11 fj(a)\K(a)da, (11.6)
- oc \j = 1 /
for by (11.4) and (11.5) this is
N N
£ ••• L max(1-1^+...+^1,0)
Xi = 1 xs = 1
which can only be positive if there are x ^,. . . , xs for which (11.1) holds
with r\ = 1. Thus Theorem 11.1 follows from
Theorem 11.2 Suppose that s > 2k. Then there are arbitrarily large N
for which
R(N)>Ns~k.
Note that throughout this chapter implicit constants may depend
on /1? . . ., /s.
11.2 The definition of major and minor arcs
The form of the Hardy-Littlewood method used here is somewhat
simpler than that described hitherto. The most important

The treatment of the minor arcs
169
simplification arises from the fact that for suitable choices of N the
integrand has only one really big peak, that at the origin. This is
because the irrationality ofk1/A2 ensures that one of/^ ,/2 is relatively
small when a is not near the origin.
Let
v = 4 P = N«. (11.7)
Then R is divided into three regions. These consist of the sole major
arc
m = {a:\a\^PN-k}, (11.8)
the pair of minor arcs
m = {a:PAr*<|a|<P}. (11.9)
and the 'trivial' region
t = {a:|a|>P}. (11.10)
The trivial region can be dismissed quickly. By Hua's lemma
(Lemma 2.5),
rx +1
\2k
\fj(*)rd*<N
2k-k + E
so that, by Holder's inequality,
rx + l
2k
II /y(«)
j= 1
d(x<N
2k-k + E
(11.11)
Thus, by (11.3),
Jt
n m
K(oc)d(x
<
'00
n m
a 2da
00
<N
s — k + £
I (h + P)
/i = 0
- 2
Therefore
n /»
X(a)da ^ N
s-k-d
(11.12)
where here, and below, 3 is a fixed positive number depending at most
on /c, s, /l1?.. ., as.
11.3 The treatment of the minor arcs
It is on m that use is made of the irrationality of kjk2, and the
argument requires a specialization of N.

170
Diophantine inequalities
Lemma 11.1 Let a, q be any pair with (a, q) = 1 and
/lx a
k2 q
<q~2.
Further let N = q2. Then
supminfl/^a)!, |/2(a)|)^iV
l - d
aem
The existence of arbitrarily large q, and hence N, satisfying
Lemma 11.1 is ensured by Lemma 2.1 and the irrationality of kjk2.
Such N may occur rather infrequently, but at any rate there are
infinitely many of them. The lemma fails if no specialization of this
kind is made. For instance it can be shown that, for suitable kjk29
limsup ( — sup min (1/^ (a)|, |/2(a)l) I > 0.
N-+ oo
N
m
Proof of Lemma 11.1 Suppose that N ^ N0(/l9... ,/s), let aem and
Q = Nk~v/2 and choose qj9 ap in accordance with Lemma 2.1, so
that
(qj9 aj) = 1, qj < g, \kp - aj/qjl < l/(qjQl
The first step is to show that at least one of ql9 q2 is relatively large. If
a-} were to be 0, then one would have
\oc\^l/(qjQ\kj\)<Nv-k
and so a would lie in $ft, not m. Thus aj =fc 0. Also one has
^=^+ J
7' 1 + J
Hence
Qj QjQ Qj\ ajQ
kx Xxol q2a1f1 + 0X
with |0,.| ^ 1
k2 k2cc a2qxy ' a^Q
which, since AT, and so Q, is large, gives
1 +
6.
- i
«26,
1
2
/2
<
q2ax
a2qx
<2
/2
and therefore
k1 = q2p±
k2 a2qy
+ 0((2-1)

The treatment of the minor arcs
171
On hypothesis,
l2 q
with |0|<1.
Hence
a q2a1
- 1 _ „- 2
<^e~ +<? ^ =4 ,
9 «2^1
so that
If the left-hand side is non-zero, then |a2<hl ^ <?> anc* if it is zero, then
a/q = (q2a1)/(a2q1), which again implies that \a2qi\f>q- Since
a2 = X2VLq2 — 02Q~ 1 <^ q2P it follows that q1q2 f> qP~ *. Therefore,
by (11.7),
max(^1?^2)>N1/5. (11.13)
Now, by Weyl's inequality (Lemma 2.4), for j = 1, 2,
fj{*) < N
1 +£
1
1 , *1
21 ~fc
+ TT +
Hence, by (11.13),
minfl/^UAWIMW1-*
as required
For the remainder of the proof of Theorem 11.2 it will be assumed
that N is chosen in accordance with the specialization given in
Lemma 11.1. Let
m1 = {a :aem, 1/^ (a)| < |/2(a)|}, m2 = m \m1.
By (11.3), X(a) <^min(l, a"2). Also the argument that gives (11.11)
can be readily adapted to show that
rx + i
2*+ 1
n /A*)
i = 1
da<^N
2k-k +£
so that
m
2k + 1
i = i
X(a)da ^ JV
2k-k + £

172
Diophantine inequalities
Therefore, by Lemma 11.1, when; = 1 or 2,
2k + 1
n /•(«)
Jm
i = 1
X(a)da < N
2k+l-k-d + e
Thus, there is a positive number 3 such that
X(a)da^Ns-fc-^.
Jm
n m
j= i
(11.14)
11.4 The major arc
In view of (11.12) and (11.14) it remains only to show that, for N
sufficiently large,
[I fj(*))K(*)daL>N
an \j = l
s- k
(11.15)
Let a6¾)¾. By (11.7), (11.8), Lemma 2.7, and the remark after the proof
of that lemma, one has
/)(01) = Vj(a) + 0(N2V)
where
e(Xj<xpk)dp. (11.16)
Vj((X) =
o
Therefore
f1...f,-v1...v,= zt(fj-vJ)[Y\ft)(Y\vl
i< j / \i^ J
<$N
s- 1 + 2v
Hence, by (11.8),
rs — k — d
(11.17)
OR V/ = 1 J=l /
A change of variables in the integral in (11.16), together with the
observation that
-y1/k-1e(y)dy
o
is <^ 1 uniformly in X ^ 0, shows that
i;7-(a) <^ |a|~ 1/fc.

The major arc
173
Hence, by (11.7) and (11.8),
f] v(ol) )K((x)d(x ^
00
a"s/fcda
Nv-k
^ ^y(s-fc)(l-v/fc)
Thus
17 ^(a))X(a)da =
OR V/ = 1
'00
n Vj(ol) )K(a)da
- 00 \j = 1
+ 0(Ns-fc-*).
(11.18)
By (11.16),
'00
n Vj{(x) )K(a)da
- 00 \j = 1
■00
rN
da
— 00 ,/
/*JV
dp!..
0
e((X&+... + Aj?»K(a)dft
0
Since K(a) <^ min (1, a 2) and the integrand is continuous the order of
integration may be interchanged. Hence, by (11.3) and (11.4),
'00
f] Vj{ol) )K(a)da
-00 \j = 1
rN
rN
dp,
o
max(l-^/¾+... +A,^|,0)d/S,
0
rN*
= k
— s
rN*
dcc1
o
(a,... a,)1""1
0
x max(l — \^1oc1 +... + /lsas|, 0)das.
It is now that one requires the hypothesis that ^1/^2 < 0- Consider the
region
& = {(a2,. . . as): 3Nk ^ a2 ^ 23N\ 32Nk ^ a,. ^ 232Nk (3 ^ ; ^ s)}.
Then, for 3 sufficiently small, whenever (a2,..., cts)e& one has
232Nk < - {k20L2 + ...+ Asas)A" l < \Nk
and so every cc1 with lA^ +...+ AsaJ ^ \ satisfies 32Nk <cc1 < Nk.
Therefore
Too
n Vj(oL))K(a)da>(Nl-kY
-00 V/ = 1
da-. . . . da.
dax
j2/(a2,...,as)

174 Diophantine inequalities
where s#(ct2,. .. , as) denotes the interval with end points
(— (A2 a2 + ... + Xsas) + |)/l [" *. Obviously the volume of 0& is
^iV^-1. Hence
Too / s \
- ao\j= 1 J
This with (11.17) and (11.18) establishes (11.15), and thus completes
the proof of Theorem 11.2.
11.5 Exercises
1 (Davenport & Roth, 1955; Vaughan, 1974b) Obtain
Theorem 11.1 for any s^C/clog/c where C is a suitable constant.
2 Let /l1? /l2, /l3, /i, n denote real numbers with X} =/= 0, rj > 0, Xl/X2
irrational, and kjk2 < 0. Show that there are primes pl9 p2, p3 such
that
3 (Baker, 1967; Vaughan, 1914a) Modify the argument used to
answer 2 above so as to show that there are infinitely many triples of
primes pl9 p2, p3 such that
l^iPi + ^iPi + ^3^3 + lA < (log maxpj)~ \
j
4 (Baker)."^" Let F(N)-+0 as N-+oo. Prove that the statement 'for
every sufficiently large N there are primes p1, p2, p3 such that pj ^ N
and I X1p1 + X2p2 + /l3Jp31 < F(N)' can be false for suitable Al9 A2, A3
with Ai/^2 > 0 and A.JA.2 irrational.
tCommunicated in conversation in June 1973.

12
Woo ley's upper bound for G(k)
12.1 Smooth numbers
Many of the most recent developments in additive number theory
have come about through the multiplicative properties of suitable sets
of natural numbers. The progenitors of these methods can be found in
Lemma 5.4 and Theorem 6.5. Very effective use has been made of sets
of numbers of the form
s/(X, Y) = {n ^ X: p | n implies p ^ Y}, (12.1)
sometimes called smooth numbers when Y is relatively small by
comparison with X. Let
A(X, Y) = cardj^(X, Y) (12.2)
be their counting function, and define p(u): U —► U to be continuous
for all u 7^ 0, differentiate for all u ^ 0,1 and to satisfy
p(u) = 0 (m < 0), = 1 (0 < m ^ 1), (12.3)
up'(u) = - p(u - 1) (u > 1). (12.4)
It is useful to establish a connection between A and p, and some
basic properties of p.
Lemma 12.1 (i) p is positive and strictly decreasing on [1, oo).
(if) There is a real number B > 1 such that for all X ^ 1 and u > 0,
| A(X,X1/U) - Xp(u)\ ^ BuX{\og2Xyl.
Proof (i) By (12.3) and (12.4), when u ^ 1, up(u) = juu_1 p(v)dv.
Suppose that there exist u ^ 1 such that p(u) = 0, and let u0 be the
least such u. Then u0 > 1 and 0 = u0p(u0) = \uu°o_ x p(v)dv > 0.
(ii) It suffices to show that, for some number B> 1, for each
72 = 0,1,2,...,
\A(X,X1/U)-Xp(u)\ ^B^iloglX)-1 (n<u^n + l). (12.5)

176 Wooley's upper bound for G(k)
The case n = 0 is immediate from the observations
- A(X,X1/U) + Xp(u) = {X} < e/2 ^
log2JT
The inductive step of the proof uses a special case of the Buchstab
identity, namely
A(X,X^) = A(X,X^) - £ A(X/p,p) (12.6)
X1!" < p^ Xl<v
valid when 1 ^ v ^ u. This is easily established by sorting the
elements of
s/(X,Xllv)\s/(X,Xllu)
according to their largest prime factor. The proof of (12.5) for n > 1 is
by induction on n.
When 1 = v < u ^ 2, (12.6) becomes
A(X9X1'») = IX]- X [X/p-]
xl<u < p^X
and the case n = 1 of (12.5) follows from elementary prime number
theory.
The case n > 1 now follows from (12.5) with v = n and the
inductive hypothesis, again by elementary prime number theory,
together with some partial summation. This completes the proof of
the lemma.
It has already been seen in Chapters 5, 6 and 7 that an important
role is played in Waring's problem by the number of solutions of
auxiliary equations of the form
*i + --- + ** = yi + --- + ysfc (12.7)
with the variables lying in various subsets of the natural numbers.
Here the interest is in the number SS(P,R) of solutions of (12.7) with
xpyjes/(P9R)(l ^j^s). (12.8)
The aim ultimately is to bound Ss + 1 in terms of Ss. This will be done
via a differencing procedure. For simplicity of exposition, as well as

The fundamental lemma 111
for other applications, it is useful to deal with a more general
situation. Thus take ^(z, u) to be a polynomial with integer
coefficients in the variables z and u = (u1,..., ut) and of degree at least one
in z, and denote dx¥(z, u)/dz by *F'(z, u). Further, define SS(P, Q, R) =
SS(P,Q,K;*F) = SS(P,Q, £;¥;*) to be the number of solutions of
the equation
*F(z, uj + xf +--- + xsfc = 4/(w,v) + /1 +---+^ (12.9)
with
x.,y.G^(g,^)(l <7<s), (12.10)
1 ^ w,z ^ P and u,vef (12.11)
where <% is a subset of (Zn [l,CP])f with cardinality U and C a
number depending at most on k and the coefficients of *F.
12.2 The fundamental lemma
The first objective is to relate the equation (12.9) to the equation
xF(z,u) + mVi +••■ + xJ) = *F(w,u) + mk(/i +•■• + yks). (12.12)
Define T8(P,Q,R,M) = T8(P,Q,R,M;V) = T8(P,Q,R,M',*',«0 to
be the number of solutions of (12.12) with z, w, u as before, and
M < m ^ MR, Xj,yjE^/(Q/M,R), z = w (modmfc). (12.13)
The expression
(x¥(w,u)-x¥(z,u))m~k
is often referred to as an efficient difference.
Theorem 12.1 Suppose that 1 < M < Q ^ P. 77ien
SS(P,Q,KMSS(P,M,K) + JV
+ QMPESS _ x(P, g, K) + PEU(MR)2s ~ ' T8(P, g, K, M)
where N denotes the number of solutions of (12.9) subject to (12.10),
(12.11) and *F'(z, u) = *F'(w, v) = 0. TTie implicit constant may depend
on the coefficients and total degree ofH*.

178 Wooley's upper bound for G(k)
Before proceeding with the proof of Theorem 12.1 it is useful to
establish the following combinatorial lemma. For a given natural
number n let s0(n) = Hp\np denote the squarefree kernel of n.
Lemma 12.2 Suppose that r is a natural number and X ^ 1 is a real
number. Then for each positive number s,
card {n ^ X :s0(n) = s0(r)} <^ XE.
Proof Let pl9... ,pt denote the different prime factors of r in
increasing order. Then it is necessary to bound the number of choices of
natural numbers u1,...,ut such that w1logJp1+ ••• +
ut\ogpt ^ logX. Let v be an integer parameter at our disposal. The
number of choices for ul9...,uv is at most (logX/log2)v and the
number of choices for uv+1,..., ut is bounded by the number of
choices with uv + 1 + • • • + ut ^ M where M = [log Ar/log(i; + 1)],
and (cf Exercise 1.5.1) the number of such choices is
(-l)M-it-v)(-(t-V)-l\ = ( M \<2M
{ } \M-(t-v) J \t-v)^
A choice such as v — [1/e] gives the desired conclusion.
Proof of Theorem 12.1 The proof is divided into four different cases. It
is helpful in order to categorise the cases to introduce the notation
x@(X)y
to mean that there is some divisor d of x with d ^ X such that each
prime factor of x/d divides y.
Let S(1) denote the number of solutions of (12.9), (12.10), (12.11) for
which
min{;c;,y;} ^ M (12.14)
for at least one j and let S{2) denote the number for which
¥'(2,11) = 0 or *F'(w,v) = 0. (12.15)
Further let S{3) denote the number for which x} > M and y, > M for
every;, (12.15) does not hold and
Xj^iM^'iz^u) or ^(M^'^v) (12.16)

The fundamental lemma 179
holds for at least one j. Finally let 5(4) denote the number for which
Xj > M and y} > M for every 7, (12.15) does not hold and (12.16)
holds for no j. Then
Ss(P,Q,KK4max{S0)}.
(i) Suppose that S(1) is maximal, so that SS(P,Q,R) ^ 4S(1). Let
f(*9X)= X *(«**)
and
z, u
where the sum is over z and u satisfying (12.11). For brevity write/(a)
for/(a,Q). Then
1
s(1> <
\ F(a)2f (a,M)f (a)2*-^da.
0
Hence by Holder's inequality,
S8(P, 6, R) < S8(P, M, K)1/(2s)Ss(P, g, K)1 " 1/(2s),
which gives the theorem in the first case,
(ii) Suppose that S{2) is maximal and let
G(a)=Ze(aW(z9u))
z,u
where the summation is over z, u satisfying (12.11) and *F'(z,u) — 0.
Then
s{2)<
I G(a)F(a)/(a)2s | da
0
and so by Schwarz's inequality
SS(P,Q,R)<SS(P,Q,R)1/2N1/2
and the theorem follows once more.
(iii) Now suppose that S{3) is maximal. Given z and u satisfying
(12.11) and ^'(z.u) =^ 0, denote by #*(z,u) the set of integers x such

180
Wooley's upper bound for G(k)
that x ^ Q and x0(M)*F'(z,u). Let
h(ol) = X Z e(axk + aXJ/(z>u))>
Z,U X6^"(Z,U)
where the summation is over z and u satisfying (12.11) and
*F'(z,u)#0. Then
s(3U
H(a)F(a)/(a)2s_1|cia,
o
and so by Schwarz's inequality
SS(P,Q,R)<SS(P,Q,R)^2
1/2
H(a)2/(a)2s-2|da
o
Hence
S8(P,Q, R)<ZV(q,r)9
(12.17)
IS
where V(q, r) denotes the number of solutions of the equation
^(z, u) + dkxk + x\ + • • • + xks _ !
= ^(w, v) + ekyk + y\ + • • • + yk _ 1
with z, u, w, v satisfying (12.10) and (12.11) subject to
¥'(z, u) # 0, *F'(w, v) # 0, g | ¥'(z, u), r | ¥'(w, v),
1 ^ d, e^ M, x ^ Q/d, y ^ Q/e, s0(x) = q, s0(y) = r.
Let
Gq(a)=£'e(ay(z,u)),
(12.18)
z,u
where the summation is over z,u satisfying (12.11), ^'(z.xx) ^ 0 and
q | ¥'^,11). Then by (12.17), for some constants C1 and D (the total
degree of *F),
ss(p,g,km
/(«)2/(«)
2s - 2
da
0

The fundamental lemma
181
where
J(a)= £ G,(a) X I *(«****)•
q ^CiPD d^M x^Q/d
so(x) = q
By Cauchy's inequality
/(a)l2^
( Z 1^)12) z
\q**CiPD J q^CxPD
X ^ *(«****)
d^M x^Q/d
so(x) = q
By interchanging the order of summation
I
I I e(ocdkxk)
d*k M x^Q/d
so(x) = q
= I
9«C,PD
I I e(«d*x*)
x ^ Q d^ M
s0(x) = qd^Q/x
By Lemma 2.2 the latter expression is
<^ X P£M ^ Q/x < 2MP
q^CiP0 x^Q
so(x) = q
2e
Hence
S8(P,Q,R)<QMI*
( Z |G,(a)|2W
0 Kq^CiP0 J
)
2s - 2
da.
J U \q^t|
By considering the underlying diophantine equation the integral on
the right here is
<pbs8.1(p9q9r)
and the thereom now follows in the third case.
(iv) Finally suppose that S(4) is maximal. Then for a given solution
of (12.9) counted by 5(4), for every j,
Xj > M, yj > M, 4"(z, u) # 0, 4"(w, v) # 0
and neither
x^(M)4"(z,u) nor y^(M)4"(w, v).
(12.19)
Let m be the greatest divisor of Xj with the property that (m,
*F'(z,u)) = 1. If m ^ M, then xy®(M)^'(z,u) contradicting (12.19).

182 Wooley's upper bound for G(k)
Thus m > M, and since no prime divisor of Xj exceeds R there is a
divisor rrij of Xj with
M <rrij^ MR and (m^'^u)) = 1.
The same conclusion also holds for a suitable divisor lj of yjt
Therefore
SAP&RXVt,
where V1 is the number of solutions of the equation
*F(z,u) + £ m^ = ¥(w,v) + £ l)y)
j = i j = i
with z,w, u,v satisfying (12.11) and
XjS st(Q/M), yjes/(Q/M), M < mj9Z,- ^ MR (12.20)
and
(m,,^z,u)) = (Zy,¥'(w,v))=l.
Let
F(a,/i)=X'e(a¥(z,u)),
z,u
where the summation is over all z,u satisfying (12.11) and (h,
*F'(z,u)) = 1, and let
/}(a) =f(m/*,Q/M) (1 ^; ^ s),/s+.(a) =f( - lp,Q/M) (1 ^; ^ s).
Then
/• i
m,l
F(a,fl)F(-a,&) f] /,(a)da
o j= 1
where the summation is over m = m1?..., ras,l = /1?..., /s satisfying
(12.20), and a = m1 ... ms, b = 1^.. .ls.
By Holders inequality
^ < n <w™.

The fundamental lemma 183
where
', = 1
m,l »/
| F(a, a) |2 |//a) 12sda
0
and
^=1
m,l
|F(a,b)|2|/s + ,.(a)|2Ma.
o
,2s - 1
Clearly neither /. nor J7 exceeds (MK)2s V2 where K2 is the number
of solutions of
¥(z,u) + mk(x\ + ... + xk) = *F(w, v) + mk(/i + • • • + }>*) (12.21)
with z, w,u,v satisfying (12.11), M < m ^ MK and
xpyjEs/(Q/M,R) and 0F(z,u)4"(w, v),m) = 1.
Hence
Ss(P,Q,R)<(MR)2s-1K2.
For a given m, let ^(/i,u) denote the set of solutions of the
congruence
^(z, u) = /i (modrafc)
with (^'(z, u), m) = 1. Let p be a prime divisor of m and pa \\ mk. Then
by Theorem 107 of Hardy and Wright (1979) and repeated
application of Theorem 123, ibid., the number of solutions of ^(z^u) = h
(mod//7) with p\x¥'(z,u) is bounded by the absolute degree of *F.
Hence, by the Chinese remainder theorem,
card#(fc, u) <^ m£. (12.22)
Obviously in (12.21)
xF(z,u) = xF(w,v) (modmfc).
Hence each solution of (12.21) can be classified according to the
common residue class modulo mk of ¥^,11) and ^(u^v). Let
g(cc, x, u; m) = £ ^(avF(z, u)).
z = x(mod m )

184
Wooley's upper bound for G(k)
Then, by (12.21),
V2 < Z V(m),
M < m ^ MR
where
K(m) =
G(a,m)|/(mka,Q/M)|2sda
(12.23)
o
and
m*
G(a,m)= £
/i = l
X X #(a,x,u;m)
u xe^(/i,u)
Hence, by Cauchy's inequality and (12.22),
m"
G(cc,h)<m£U Z Z Z l^(a,x,u;m)
/l = 1 U X6^(/|,U)
m'
^ m£(7X Z l0(a,x,u;m)
U X = 1
The theorem now follows from this and (12.23) on considering the
underlying diophantine equation.
For the purposes at hand it is necessary to specialise the function *F.
First, define the forward difference operator
A,.(/(x);h;m)
by
A1(f(x);h;m) = m k(f(x + hmk)-f(x)),
(12.24)
A1(AJ.(/(x); h1,...,hJ;m1,..., m7); hj+1;mj+1) (12.25)
and for convenience put
A0(/(x);h;m)=/(x).
Then put
^(x; h; m) = A,.(xk; h; m). (12.26)
Also, suppose that the real numbers 0-, $•, M-, Hj9 Q- satisfy

The fundamental lemma 185
0 < fa ^ 1/fc (1 < i < /e), 0>, = 4>l + • • • + 0,-, (12.27)
Mj = P*J,Hj = PMr\Qj = P1 ~\Q0 = P. (12.28)
Now take <% = % ^ to be the set of h1,..., hp ml9...,mj with
1 ^ ht ^ Hi9 Mf < w£ < MfK. (12.29)
Henceforward 77 will be a positive real number, usually sufficiently
small in terms of E,k,(j)l,... ,(j)k, and the implicit constants in
expressions will possibly depend on rj and (j) 1,..., (j)k as well as k and e.
Theorem 12.2 Assume the above notation and put R = Pv. Then for
j = 0,..., k- 1
Ss(P,QPR^j)<
pE \A HiMiR) Wj + i*)21'' T*(p> Qj> *• mj+i> ^)-
Proof The inequality ^-(zjhjm) > 0 holds for all choices of the
variables under consideration. Thus in the notation of Theorem 12.1,
N = 0.
Write SS(P, Mj + 1? R; ^) as an integral via exponential sums. Then
by making a suitable trivial estimate
SS(P,MJ+i,R;T,.) <MJ+lSs_ t(P,Qj,R;^)
<QjMJ+lSs.l(P,Q],R;Vj).
Hence, by Theorem 12.1,
SS(P,Qj,R;Vj) « PQjMj + ,SS_ ,(P,QpR;Vj)
+ PE(f\ HiMtRj(Mj + ,R)2s~ lTS(P,QpR,Mj+l;4>,).
If
P'QjMj +1Ss.t(P,Qj,R;%) < cSs(P,Q},R;¥,),
for a suitably small positive constant c, then the proof is complete.
Otherwise one may suppose that
P'QjMj + XSS _ t(P, QpR;Vj) > cSs(P, QPR\^).

186 Wooley's upper bound for G(k)
On writing Ss_ X(P,QpR'^j) as an integral and applying Holder's
inequality one has
Ss- ,(P, QPR; Vj) < S0(P, Qj,R; Vj)llsSs(P, QJ3 R, T/ " 1,/s.
Hence
SS(P, QPR; Vj) < (P'QjMj + ^(P,QpR; ¥,)
<(P°QjMJ+Jp(f\ H.M.R
On the other hand, by considering only the solutions to (12.12) in
which xt = yt (i = 1,... , s) and z = w one has by Lemma 12.1 that
fl HtMfl)(Mj + ,R)2'~ lT,(P,QpR,Mj+1;4>,)
J= 1 /
> (f\ H^AiMj^^'-'pffl H1M1r\mj+1R(Qj/Mj+J.
This completes the proof of the theorem.
12.3 Successive efficient differences
The differencing process continues by relating
Ts(P,QpR,Mj + j^j+l)
to Ss(Qj+ l9R) and Ss(P,Qj+1,R;x¥j+x). There are many ways of
dealing with Ts, as is done, for example, in Vaughan (1989a,fc),
Vaughan and Wooley (1993, 1994, 1995) and Wooley (1992, 1995d).
The manner chosen is relatively simple, and suffices for the
conclusions here.
Lemma 12.3 For j = 0,..., k — 1 one has
TS(P,Qj,R,Mj+i;Vj) « PMj + iRytl H<mASs(Qj + l,R)
+ Ss(Qj+l,R)l'2Ss(P,QJ + 1,R;Vj+l)1/2.
Proof By definition TS(P, Qp R, Mj + 1) is the number of solutions of

A mean value theorem 187
(12.12) with 1 ^z,w ^ P,z = w (modmfc), Mj+1<m^
Mj+1R, xi,yiestf(Qj+ 1?R), and /ii?ra; satisfying (12.29) when
1 ^ i ^ j. Thus
Ts(P9Qj,R,Mj+1)^U0 + 2Ul9
where U0 and L^ are the number of solutions with z = w and z < w
respectively. Clearly
U0^PMj+1R(t\ H,M^j5s(ej+1,K).
Thus it remains to deal with Ul. Let w = z + hmk. Then
1 ^ h ^ Mj+ x and the equation (12.12) becomes
xi + m•• + xs = tj + i(z\ ni,..., hj + ± \m^,..., wij + i) -\-
y\ + '~ + yks.
Thus U1 is bounded by the number of solutions of this with
1 ^ z ^ P, xi9ytestf(Qj+ i,R), and hi,mi satisfying (12.29) when
1 ^*^7 + 1. Now it is a simple matter to use an integral of an
appropriate product of exponential sums to count the number of such
solutions. A simple application of Schwarz's inequality then shows
that
tf i < Ss(Qj +1,R)1/2SS(P,QJ + 19R; Vj + x)1/2
and this completes the proof of the lemma.
12.4 A mean value theorem
It is necessary to make a suitable (simplifying) choice of parameters in
the previous theorem, and it is convenient to suppose that there is a
number Xs with 0 ^ Xs ^ 2s such that for each e > 0 there is an rj > 0
such that when R = P*1 one has
Ss{P,R)<P*s + E. (12.30)
Observe in passing that this implies that (12.30) holds whenever
R ^ Pn. For convenience put
S, = /L — 2s + k.

188 Wooley's upper bound for G(k)
This gives a measure of how close ks is to the 'ideal' value ks = 2s — fc.
For j = 1,..., k let
1 /1 1 \fk-d\k-j A ^
*'- r^+(* - r^Xin • (1231)
It is now proved by induction on j = k — 1,..., 0 that
TS(P, Qj9 R,Mj+l)<Pl + <k- j»Mj + xR2«k " » I J] H£M£K 1Q^Y-
(12.32)
By writing 5S(P, Qfc, K; *Fk) as an integral of an appropriate product of
exponential sums and making a trivial estimate one has
Ss(P,Qk,R;Vk) < P2(l\ H,M£*Yss(Qk,K).
Moreover (pk = l/fc so that Hk = 1. Thus, by Lemma 12.3, the case
j = k — 1 of (12.32) is established.
Now suppose that; ^ fc — 2 and (12.32) holds with; replaced by
; + 1. By (12.27) and (12.31),
<&y+1 < (/+ 1)/*<1- 1/*.
Hence, by Theorem 12.2 and the inductive hypothesis,
Sa(P9Qj+l9R^j+1)<
pi + (fc - J)e{Mj + ^)2, r pj HiMiR\2R2s{k - j + 1)Q^
Hence, by Lemma 12.3,
where
i/2=fmj+1jr n^i^^ <2;+t
J = 1
and
U3 = p(k-J)zR^k-J)l f[ H.MiR
xP1/2M^ + 2(Hj+1Mj+1)(Qj+1Qj + 2)^^.

A mean value theorem 189
By (12.27) and (12.31),
1 +2s0J. + 2 + As(l -^ + 2-^-+1)
+ 2(1 - (k - 1)0,- + x) - 2(1 + ct>j + x) - Xa(l - 0>, + x)
= l + (k-8a)<l>j + 2-2k<l>j+1=0.
Thus
U3 < P{k ~ j)£R2s{k -J)( f] HtMiR ) PMj + ^W
si = 1
and (12.32) follows.
By Theorem 12.2 and (12.32) with; = 0,
SS(P, P, K;*F0) ^P1+(k + 1)£MlsR2{k + 1)eQ\
+ E
s
Thus the following lemma has been established for all s ^ 1. The case
s = 0 is trivial from the observation that then X0 = 0, S0 = /c, S1 =
k- l,Xx = 1.
Lemma 12.4 Suppose that there is a number Xs with 0 < ks ^ 2s such
that for each s > 0 there is an n > 0 such that when R = P*1 one has
SS(P, R) < PA*+ £.
Let Ss = ks — 2s + /c,
1 /1 L_Y—-v-1
k + ds \k k + dj\ 2k
5s+l = <5S(1 — 9) + kd — 1 and Xs + t = 2s + 2 — /c + <5S + l. Then for
each £ > 0 there is an n > 0 smc^ that when R = P* one has
Ss + l(P,R)<Px*+>
+ e
It is useful to state a simple corollary of the above which follows
easily by induction on s on noting that 0 ^ £. Many refinements are
possible.
Theorem 12.3 Let
A. = 2s - fc + fc ( 1--

190 Wooley's upper bound for G(k)
Then for each e > 0 there is an n > 0 such that when R = P*1 one has
Ss(P,R)<Px* + £.
The following theorem is the main estimate for SS(P,R).
Theorem 12.4 Suppose that k ^ 4 and let os denote the root of the
transcendental equation
as + log<xs=l-^ (12.33)
with 0 < gs ^ 1 and let
ks = 2s - k + kos. (12.34)
Then for each e > 0 t/iere is an n > 0 such that when R = P*1 one has
Ss(P,R)<P*s + £. (12.35)
Moreover
Xs ^ 2s - fc + /cexp( 1 -y). (12.36)
Proof This is by induction on s = 0,1, The case s = 0 is
immediate from the observation that o0 = 1.
Assume the theorem for a given s. Then the hypothesis of Lemma
12.4 holds with 3S = kos. Thus the conclusion holds with As + 1 =
2s + 2 - k + £* and £* = Ss - 1 + (k - Ss)0 with
1 /1 1 \A-a.Y~1
0 = ^ ^ + t-
fc + (5S \fc k + dj\ 2k
Therefore
2 2 (k-b
*±s. l -
k + 5s k + SA 2k
2-1
s ' s
1 -k'
k + S,
= kas-(2-2l~k) °s
1 +a.

Wolley's upper bound for G(k) 191
By definition of gs and gs + 1?
2 . G
= °s - °s + 1 - log
s + 1
fc G&
Let
ffs ~ ffs + i
/J = , V = /J(l + (7S).
cr.
s
The expression cr + log cr is a strictly increasing function of g. Hence
0<ju<l,ju<v< 2/i. Moreover
2
- = v - \i - log(l - \i)
and — fi — log(l — /i) is an increasing function of ft when 0 < ft < 1.
Therefore
and so
(2-21_fc)
V < : .
Thus
(5* <fc(Ts-(2-21_k) ffs
^ /ccr, — /cv
1 + <7a
0".
s
1 + <75
= K + 1
and the case s + 1 follows.
12.5 Wooley's upper bound for G(k)
Let
M«)= Z Z e(apV). (12.37)
iX < p ^ X xerf{X, Y)

192
Wooleys upper bound for G(k)
Lemma 12.5 Suppose that ol = a/q + /J with | /? | ^ \q ~ xX ~\
q ^ 2Xk, (a, q) = 1, and that when q ^ X one has | /? | > q ~ 1X1 ~ 2k.
Then for every e > 0 there is an n > 0 swc/i t/iat when Y = Xv one has
h{a)< X1{1 ~ a) + E
where
g = max
1 k ( 2u
— — exp 1 —
4w 4w
(12.38)
Moreover,
g = ( 1 + 0\—i
2/c(l +t)V \fc2logfc
where x is the positive root of the transcendental equation
(12.39)
ex = ke(l + t).
(12.40)
and
1
G ~
2/clog/c
as k —► oo
(12.41)
Proof Let weN. By Holder's inequality
/i(a)" <^ X
u - 1
■kx < P ^ x
Z cme(ccpkm)
m
where cm is the number of solutions of the equation
x\ + • • • + xk = m
with x-e j^(X, F). Following the proof of Lemma 5.4 gives
\2u ^ \r2u — 1 + k + e V I ^, I 2
/j(a)2" « X
I
m
m
and the lemma follows from Theorem 12.4.
Theorem 12.5 Suppose that k = /clog^ where g is given by (12.38).
Then
G(k) < A + 0(k)
and X = /clog/c + /cloglogk + 0(k).

Exercises 193
Proof Let
ke 2 1,/1
A = ^B = k^ = Bl0" \2AB
and choose b, c so that | b — ft | ^ \ and
c = [Aexp( - bBJ] + 1.
Let s = 4fc + 2b + c and put P = n1/fc, Q = (£)1/fc, X = Q1/2,
K — 0*, q arbitrarily small,
/(a) = ^ *(««*),
fl ^ P
0(a) = X *(<*"*)>
W6JJ/(Q,«)
/i(a) as in (12.37) and consider
/(a)4fc#(a)2[7i(a)ce( - an)da.
r(n) =
o
Now we may proceed much as in §5.4, but via Theorem 12.4 and
Lemma 12.5. It follows that G(k) ^ 4/c + 2b + c. Moreover Bb =
Bft + 0(B),
e-Bb = e-B^{+ 0(B)) =(1+ 0(B))
AB
and c = § + 0(1). Thus
2 AB 2 1
2b + c = -log— + - + 0(1) = /clog- + 0(/e)
B 2 B a
as required.
12.6 Exercises
1 Show that when u > 1 one has wp(w) = j"jj _ i p(u)di;, and that
r>+ 1)" 2 ^p(u)^T(u + l)"1.
2 Let Gx(/c) denote the least s such that almost every natural
number is the sum of s /cth powers. Show that

194 Wooley's upper bound for G(k)
y GX(k) 1
hm sup——-^-.
fc-^oo k\ogk 2
3 (Ford, to appear) Show that there is a positive number c such
that if I > ck2 log /c, then every sufficiently large number can be
written in the form
i 4
j = k+l
4 (Vaughan, 198%) Assume the notation of §§12.1 and 12.2. Let
f^XtKm) = ^(x,/*, m),
/2(x1?x2,/i,m) =/1(x2,fi,w) -/^,/i,m),
F((X) = Z Z Z e(a/i(x,fi,w)),
h^ H M <m^ MR x ^ P
|F(a)|2Ma,
o
',=
and suppose that for _/= 1,2 the number Rj(n) of solutions of
fj(\,h,m) = n with xe[l,P]J',h<H, M <m ^ MR satisfies
Rj(n) < (nP)e. Show that
Ij < P2i ~j + E{MRH)2i ~ ^.
Put TS(P,M,R) = rs(P,P,K,M;^0). Show that
T2(P,R,M) < PMRS2(P/M,R) + P>+£(MRHfS3(P/M,R)\
T2(P,R,M) < PMRS2(P/M,R) + P*+£(MRH)*S4(P/M,R)>,
T3(P,R,M) < PMRS3(P/M,R) + P± + £(MRH)*S4(P/M,R)*.
Deduce that if /c = 4, then
S3(P,,R)^PA + £
where /I = 3 + + 2/jt anc* tnat if & > 5, then
S3(P, K) <^ PA3 + £, S4(P, K) <^ PA4 + £
where A3 = 3 + 26/, A4 = (4 + (fc - 3)6/)/(1 - 6/) and 0 is the smallest
non-negative root of 63(2k + 32) + 62(l5k - 48) +
0(3/e2 - llfc + 22)-3.

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Index
T(k), 24
T*(k), 150, 151
r0(/c), 25
addition of sets of residues, 23
additive homogeneous equation, 147,
151, 167
additive homogeneous equation,
non-trivial solution, 148, 151, 167
algebraic numbers, ix, 64
algorithm, Euclid's, 20
Apostol, 22
approximation, diophantine, 4
arc, major, 5, 8, 14, 27, 30, 38, 51, 70,
107, 112, 116, 119, 120, 125, 128,
135, 162, 163, 164, 168, 169, 172
arc, minor, 5, 8, 14, 27, 69, 112, 118,
125, 127, 128, 130, 162, 163, 164,
168, 169
arithmetic progression, 155
asymptotic density, 155
asymptotic density, lower, 155
asymptotic density, upper, 155
auxiliary equation, 176
auxiliary function, 14, 105, 112, 116
Babaev, 146
Bachet, 1
Baker, 174
Bertrand's postulate, 62
Bierstedt, 151
binary Goldbach problem, 6, 33
biquadrate, 1
biquadrates, Waring's problem for,
105
Birch, 147, 151
Boklan, 24
Bovey, 151
Brauer,147
Buchstab identity, 176
Cauchy-Davenport-Chowla theorem,
23
central difference operator, 102
character, 46
Chowla, 23, 151
condition, congruence, 53, 150
congruence condition, 53, 150
congruence, polynomial, 112
congruences, simultaneous, 58, 59, 76,
113
congruences, system of, 76
conjecture, Erdos-Turan, 155, 156
conjecture, general, 127
cubes, 6
cubes, four positive, 109
cubic form, ix, 147
Davenport, ix, 6, 23, 38, 94, 95, 97,
100, 103, 104, 105, 108, 109, 128,
136, 141, 147, 150, 167, 174
density, asymptotic, 155
density, lower asymptotic, 155
density, upper asymptotic, 155
Dickson, 1
difference operator, central, 102
difference operator, forward, 10, 25,
184
difference operator, modified forward,
75
differences, efficient, 75, 83, 177, 186
diophantine approximation, 3, 9
diophantine inequality, 167
diophantine inequality, non-trivial
solution, 167
Diophantus, 1
Dirichet, 3, 9, 14
distribution modulo one, uniform, 5
divisors, sum of, 7
Dodson, 151
Ellison, 1
efficient differences, 75, 83, 177, 186
equation, additive homogeneous, 147,
151, 167
equation, additive homogeneous,
non-trivial solution, 148, 150, 167
equation, auxiliary, 176
equation, homogeneous, ix, 147
equation, homogeneous, non-trivial

230
solution, 147, 151
equation, homogeneous, trivial
solution, 147
equation, transcendental, 69, 92, 192
equations, simultaneous, ix, 57, 60, 61,
78, 80, 151, 152, 153
equations, system of, 112
Erdos, 95, 155, 156
Erdos-Turan conjecture, 155, 156
ergodic theory, 155, 156
Estermann, ix
Euclid's algorithm, 20
Euler, 1, 6, 22
Euler-Maclaurin summation formula,
41,44
Euler product, finite, 136
Exodus 24:12, xi
Fermat, 1
finite Euler product, 136
Ford, 194
form, cubic, ix, 147
form, general, 7
formula, Euler-Maclaurin summation,
41,44
formula, Fourier's inversion, 15
formula, Poisson summation, 41
formulae, Newton's, 59
forward difference operator, 10, 25, 184
forward difference operator, modified,
75
four positive cubes, 109
four square theorem, 1
Fourier's inversion formula, 15
fourth powers, 100, 105
Freiman's hypothesis, 92
function, auxiliary, 14, 105, 112, 116
function, generating, 14, 38, 105, 112
function, Mobius, 28
function, partition, 3
function, von Mangoldt, 28
fundamental lemma, 78, 177
Furstenberg, 155, 156, 161
g(k), I
G(k), 5, 8
G*(k), 150
Gi(fc), 126, 193
Gauss sum, 46, 141
general conjecture, 127
general form, 7
general inequality, 7
generalized Riemann hypothesis, 6
generating function, 14, 38, 105, 112
Index
Goldbach, 6, 27
Goldbach binary problem, 33
Goldbach ternary problem, 27
Hardy, ix, 1, 3, 4, 6, 21, 24, 25, 32, 70,
72, 94, 142, 183
Hardy-Littlewood method, ix, 3, 6. 8,
112, 144, 155, 167, 168
Hardy & Wright, ix, 1, 21, 32, 183
Hasse, 141
Heath-Brown, 24
Heilbronn, 38, 128, 136, 167
Hilbert, 1
homogeneous equation, ix, 147
homogeneous equation, additive, 147,
151, 167
homogeneous equation, additive,
non-trivial solution, 148, 150, 167
homogeneous equation, non-trivial
solution, 147, 151
homogeneous equation, trivial solution,
147
homogeneous form, 147, 151
homogeneous inequality, ix
Hua, 6, 8, 38, 111, 112, 122
Hua's lemma, 12, 14, 72, 112, 135,
169
Huxley, 64
hypothesis, Freiman's, 92
hypothesis, generalized Riemann, 6
identity, Buchstab, 176
identity, Vaughan's, 28
Imperial College, ix
inequality, diophantine, 167
inequality, diophantine, non-trivial
solution, 167
inequality, general, 7
inequality, homogeneous, ix
inequality, Weyl's, 5, 11, 14, 17, 27, 55,
107, 108, 112, 131, 171
integral, singular, 4, 18
inversion formula, Fourier's, 15
Jacobian, 75, 76, 77
Jagy, 146
Kaplansky, 146
Karatsuba, 58
kernel, squarefree, 178
Kubina, 2
Lagrange, 1
Landau, ix

Index
large sieve, 64, 73, 141
Legendre, 127
lemma, fundamental, 78, 177
lemma, Hua's, 12, 14, 72, 112, 135,
169
Lewis, ix, 147, 150
Linnik, 6, 58
Littlewood, ix, 1, 3, 4, 6, 8, 24, 25, 70,
72, 94, 142
lower asymptotic density, 155
Mahler, 2
major arc, 5, 8, 14, 27, 30, 38, 51, 70,
107, 112, 116, 119, 120, 125, 128,
135, 162, 163, 164, 168, 169, 172
Mangoldt, von, function, 28
mean value theorem, 187
mean value theorem, Vinogradov's, 57,
58, 62, 75, 111
mean value theorem, Vinogradov's,
non-trivial solution, 111
mean value theorem, Vinogradov's,
trivial solution, 111
Meditationes Algebraicae, 1
Miech, 136
minor arc, 5, 8, 14, 27, 69, 112, 118,
125, 127, 128, 130, 162, 163, 164,
168, 169
Mobius function, 28
modified forward difference operator,
75
Montgomery, 6
Mordell, 38, 113
multiplicative number theory, 6
Newton's formulae, 59
non-singular solutions, 77
non-trivial solution, additive
homogeneous equation, 148, 151,
167
non-trivial solution, homogeneous
equation, 147, 151
non-trivial solution, Vinogradov's
mean value theorem, 111
Norton, 151
number theory, multiplicative, 6
numbers, algebraic, ix, 64
numbers, smooth, 175
operator, central difference, 102
operator, forward difference, 10, 25,
184
operator, modified forward difference,
75
231
partititon function, 3
Pillai, 1
Poisson summation formula, 41
Polya, 142
polynomial congruence, 112
postulate, Bertrand's, 62
power residue, 20, 22, 46
powers, sums of, 94, 108, 126
primitive root, 46
problem, binary Goldbach, 33
problem, ternary Goldbach, 27
problem, Waring's 1, 4, 38, 69, 70, 176
problem, Waring's for biquadrates, 105
product, finite Euler, 136
progression, 156
progression, arithmetic, 155
Rademacher, ix
Ramanujan, 3
Ramanujan's sum, 7, 32
region, trivial, 169
residue, power, 20, 22, 46
residues, addition of sets of, 23
Rieger, 1
Riemann hypothesis, generalized, 6
Riemann zeta function, 57
Rogovskaya, 125
root, primitive, 46
Roth, 128, 155, 156, 157, 174
Sarkozy, 156, 161, 166
Schmidt, ix, 38
Scourfield, 92
series, singular, 4, 20, 33, 48, 107, 128,
136
Shimura, 151
Siegel, ix
sieve, large, 64, 73, 141
simultaneous congruences, 58, 59, 76,
113
simultaneous equations, ix, 57, 60, 61,
78, 80, 151, 152, 153
singular integral, 4, 18
singular series, 4, 20, 33, 48, 107, 128,
136
singular solutions, 77
smooth numbers, 175
solutions, non-singular, 77
solutions, singular, 77
space, vector, 151
square theorem, four, 1
square theorem, three, 127
squarefree kernel, 178
squares, 3, 4

232
Index
Stemmler, 2
sum, Gauss, 46, 141
sum of divisors, 7
summation formula, Euler-Maclaurin,
41,44
summation formula, Poisson, 41
sums of powers, 94, 108, 126
sums of three squares, 127
symbol, Vinogradov, xi
system of equations, 112
Szemeredi, 155, 156
Szemeredi's theorem, 156
ternary Goldbach problem, 27
three squares, 127
Tietavainen, 151
transcendental equation, 69, 92, 192
trivial region, 169
trivial solution, homogeneous equation,
147
trivial solution, Vinogradov's mean
value theorem, 111
Turan, 155, 156
uniform distribution, 5
upper asymptotic density, 155
Vaughan, 6, 24, 55, 109, 110, 126, 128,
150, 174, 186, 194
Vaughan's identity, 28
vector space, 151
Vinogradov, 5, 6, 22, 27, 57, 70, 111,
122
Vinogradov symbol, xi
Vinogradov's mean value theorem, 57,
58, 62, 75, 111
Vinogradov's mean value theorem,
non-trivial solution, 111
Vinogradov's mean value theorem,
trivial solution, 111
Waerden, van der, 155
Waring, 1
Waring's problem, 1, 4, 38, 69, 70,
176
Waring's problem for biquadrates,
105
Watson, 6
Weil, 38
Weyl, 5, 10
Weyl's inequality, 5, 11, 14, 17, 27, 55,
107, 108, 112, 131, 171
Wilson, 64
Wooley, 58, 71, 75, 93, 126, 175, 186,
191
Wright, ix, 1, 21, 32, 183
Wunderlich, 2

The Hardy-Littlewood method is a means of estimating the number of
integer solutions of equations and was first applied to Waring's problem
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