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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 The Pitt Building, Trumpington Street, Cambridge CB2 IRP, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © 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 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 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 \ 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{) = X e(4>(x)) x= 1
Auxiliary lemmas 11 where
)|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(*) = 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- Lemma 2.5 (Hua's lemma, 1938b) Suppose that 1 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 c014 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 The major arcs 15 Proof For Y> 0, X e(amk/q) = J e(arfcAz) I l = Yq-lS(q,a) + 0(q\ m0). 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 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) 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) 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) 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)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 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. 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 \ 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) = \(y)e(ccxy) in the identity Z A(i,y)+ Z Z M(*,y) X < y < n XX y>X xy < n Here A is von Mangoldt's function, and the identity follows by observing that tx = 0 (lThe ternary Goldbach problem 29 Then S3= £S(Y) Yes* where Y 30 Goldbach's problems Theorem 3.2 Suppose that A is a positive constant and B ^ 2 A + 10. Then |/(a)|3da^2(logn) - A m The treatment of the major arcs, although straightforward, requires an appeal to the theory of the distribution of primes in arithmetic progressions. Lemma 3.1 Let v(P)= t e(M. (3.6) m=l Then there is a positive constant C such that whenever 1 ^ a ^ q ^ P, (a, q) = 1, cteyjl(q, a) one has fW = ^v(a-a/q) + 0(nexp(- C(log<2)). ¢(4) Proof Let /x(«) = Z (log pWp\ Then fx(a/q)= t e(ar/q)9(X9q9r) + 0((logX)(logq)) r = 1 (r,q) = 1 where S(X,q9r)= X log/>- p^ x p = r(modq) By Theorem 53 and (40) of Estermann (1952) it follows that whenever x/n < X ^ n one has fx(a/q) = ~^~ t e{arlq)^0{n^V{-Cl{\ognY'2)). (3.7) (The ternary Goldbach problem 31 Hence, by (3.1), (3.6), (3.7) and Lemma 2.6 with X = n. F(m) = e(Pm), P = a — a/q, e(am/q) log m — fi(q)/(t>(q) when m is prime, cm = - v(q)/(q) otherwise, one has Kq) /(a) —-—v(cc —a/q) <^ (1 + n|a — a/q\)nexp{ — C\(log n)1/2). ((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 <
• (3.9) q^P «=i ((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 /%/({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,
(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>(qr2Z Hence siM= Itt^i E 4-™/«) (3-24) q=l 36 Goldbach's problems Theorem 3.7 Let A denote a positive constant. Then n Y, l#i(w) — raS^ra)!2 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\(k2). Then (k2)pl~2v is an integer, and hence, by (4.3), \S(p\ a, b)\ ^ pvN where N is the number of solutions of the congruence kayk ~1 + b = 0 (mod pv) (4.4) with 1 ^ y <: pl ~v. Recall that max (0, t) < //2 ^ v. Thus the congruence is insoluble unless 6^ z and 0 — x is a multiple of /c — 1. If (4.4) is insoluble, then (4.2) is immediate. In the contrary case let X = (0 — z)/(k — 1). Then N is the number of solutions of (kp ~ x)awk ~ 1 + (bp ~e) = 0 (mod pv ~e) with 1 ^ w ^ pl ~v" A. Note that A < 0 < / - v. When /-v-a^v-0, one has N <^ 1, so that |5(pz,a,/7)|^pv. When /-v-A>v-0, then N <$ pl + e" 2v" A, so that |S(pz, a, &)| < pl ~ vpe. In either case \S(pf,a,b)\
40 The major arcs in Waring's problem It remains to consider the case when I is odd and pl{\). 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
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
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 (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'(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)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 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 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 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) 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)\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) < ( 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)\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)\ 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) 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 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, & 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
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) = 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)
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 Y58 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 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)
) (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 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 N. When h — S < a ^ h the same conclusion follows by considering jS with 1 -£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. 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^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 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{ 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_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) 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)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) yq~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 \n. Hence, by Theorems 4.4 and 4.6, f{0LfkH(a)2W{0L)e(- «3Ifi*(m). m Therefore, by (5.47) and (5.48), f{ 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,(l76 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,«,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 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( (mod p)
80 Vinogradov's methods and G(*) = X *(<*■*) zi =¾ A\ .. .,z. ^ X (J( ^ 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 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

«) 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) 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(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,), 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\' (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 )(I) to be any p and 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,<'>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 i)ei + 2 + '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

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 + (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 QR:/(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 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\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 Zj. 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— 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 -$4 satisfies (6.20) and (6.21). Let H = [(/c — l)a], and J = H + \ when 2H((k-l)(x-H)>\-oc and H + 1 ^ k - 2
100 Davenport's methods and J = H otherwise. Then N(,tf, X) satisfies (6.22) with n 1/, ,, , J+l-(/c-l)a p = - l+«fc-l) + « , /c\ 2 — 1 + a when a ^ l//c and /J = 1/fc + a vv/ien a < 1/fc. It is useful, in the case of fourth powers, to have a slight refinement of this. The above argument is not materially altered if Q(m) is taken to be the number of solutions of m = x4 + a with Z < x < 2Z, x = r(mod 16), ae-rf, a ^ Zv + 3. Likewise the argument that gives (6.1) with (6.2) is essentially unchanged when each Xj is restricted to a given residue class modulo 16. Thus Theorem 6.4 (Davenport, 1939c) Let N{th)(X) denote the number of natural numbers n not exceeding X in the residue class h modulo 16 which are the sum of t fourth powers. Then for t^l and 0 ^ h ^ min(t, 16), N[h)(X) >Xa<-e (X> X0(e, t)) (6.24) where 1 _3 + 13ar 2' *' + x ~ 12 + 4ar a2=-, cct +1 = ^ A . (6.25) /m particular, ™ — 12 /v — 331 „ _ 5539 //: 9£\ a3 — 28? a4.— 4.12? a5 — 6268- ^U.Z,U; Davenport (1942a) has given an improvement upon the argument of Theorem 6.2 which is particularly effective when k = 5 or 6. With the same assumptions as in Theorem 6.2, let Q(m) denote the number of solutions of xk + pka = m (6.27) with ZSets of sums ofkth powers 101 where P is the number of primes p with \Zl ~v < pk < Z1 ~v. For a given prime p and integer r with p\r the number of solutions of xk = r(modpk) is 0 or (k, (j){pk)\ Thus the integers x with p\x can be divided into q{p) = (/c, (pfc)) classes ^1?..., &qip) such that, if x and _y are in a given class ^n then xk = yk (mod pk) if and only if x = y (mod pfc). Let Qr(m, p) denote the number of solutions of (6.27) with Z < x < 2Z, a ^ Zv + k ~ 1, xe$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
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) ^ 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)-^ X4n1/2 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 + 7x8/9-£(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 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 (\^jv^*- (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 plPreliminary 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'{x) = 0(modp) one obtains where S{pl9al9...9ak)=T0 + £ 7} (7.16) j= i 7} = X e{{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(px + x_y) — {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'{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\•> • • •>>(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 ^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:0118 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 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 = [,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) \X1/2. Hence, by Theorem 7.1 and (7.7), q~ 'S(q, A)I(P) 3/c log k + k log log /c one has rj-2mk2p 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 An asymptotic formula for JS(X) 121 Hence it \V(x)\2tdxi/h where 00 00 w= Z •• • Z 4i•• - 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« = 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 2KVinogradov'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 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). Hence, by (7.29) and (7.30), for aem, WiaXXV, ... vl(X-1Uk + e-{k-1,2H1-*yn*r)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 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;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 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(', (iii) w = v, x = y, z = t. Hence the total number of solutions is
The treatment ofn 131 Similarly \hi\dafc, Afc by 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), \lhl\doL an < I X 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*\da22h$\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 |134 A ternary additive problem Hence, by (8.8), \(t>22(t>3A3h23\d(xjj=! 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?! Therefore, by Schwarz's inequality and (8.24), |0^^^|da^M16/15 - 3d ««! Hence, by (8.21), \h2h2h2\d(xThe major arcs ${(q, a) 135 Now consider 9ft2- By Lemma 6.3, * f 1/2 |^|da« £ 4"1 9W2 q ^ P 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*136 A ternary additive problem Let
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)= 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 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 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 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
The singular series 141 Hence, by (8.31) and (8.45), X A(m,q) 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) «142 A ternary additive problem where \t(x)\ = 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)\(A- 1)/A Z 1*HX)X(X) N Z ,x = 1 q^Q X modi/ WZ Z*IM%)I 2//(2/ - 1) (2/- 1)/(2/) where modi/ ba = (nz + e2)1/(2Z)( z ^/(y)(2A"2)/(A"2)) ,(1 - 2/1)/(21)
The singular series 143 Let k = 3. Then the lemma follows provided that, for X > 1, X ^bO^XdogXe)'4-1. In fact, it is easily seen by induction on r that dr(xy) ^ dr(x)dr(y) and by induction on s that and I ^(y)^X(log^r-1 y 1 or / = 1. Hence, summing over / with Qx_ x ^ K gives, via (8.44) and (8.53), 2n X F(m) « n47/48. m = n + 1 1_<5 values of m with n144 A ternary additive problem The proof of Theorem 8.3 is completed by examining Z A(m,q). q> V qe& Let X = l/(logtt). Then X \A(m,q)\<: X (q/V)x\A(nu q)\ q > V qe9 qeC/ = V~XY\ (l+ t phX\A(m,ph)\). 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" ' 5n1/3°. 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($ 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{pyJ4k, 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)= ^ <'.-, .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, 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 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{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 + * £(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 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
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(ax2/164 A Theorem of Roth 10.5 The contribution from the minor arcs Suppose aem. Choose a, q so that (a, g)=l, g P or |a —a1/q1\> Xn~ 1q^ \ In the first case, by (10.26), g(koc) Xn~ 1q~ 1, so that, by (10.26), (10.27) and Lemma 2.8, 1/2 g(ka). (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}. (11.10) The trivial region can be dismissed quickly. By Hua's lemma (Lemma 2.5), rx +1 \2k \fj(*)rd*170 Diophantine inequalities Lemma 11.1 Let a, q be any pair with (a, q) = 1 and /lx a k2 q 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\)The treatment of the minor arcs 171 On hypothesis, l2 q with |0|<1. Hence a q2a1 - 1 _ „- 2 <^e~ + 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(XjThe 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 (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 (n176 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 1 is by induction on n. When 1 = v < u ^ 2, (12.6) becomes A(X9X1'») = IX]- X [X/p-] xl 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 <7178 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)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)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)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 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)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,.) ,). 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 HA mean value theorem 187 (12.12) with 1 ^z,w ^ P,z = w (modmfc), Mj+1 0 there is an rj > 0 such that when R = P*1 one has Ss{P,R)
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)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)j + 2-2kj+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)
+ 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)
0 t/iere is an n > 0 such that when R = P*1 one has Ss(P,R)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 0192 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 +£(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 on representations of integers hy sums of powers. This introduction to the method deals with its classical forms and outlines some of the more recent developments. Now in its second edition, it has been fully updated; extensive revisions have been made and a new chapter added to take account of major advances hy Vaughan and Wooley. The reader is expected to be familiar with elementary number theory and postgraduate students should find it of great use as an advanced textbook. It will also be indispensable to all lecturers and research workers interested in number theory and it is the standard reference on the Hardy-Lilllcwood method. This series is devoted lo thorough yet reasonably concise treatment of topics m any branch ofmalhcmalics. Typically, a Tract lakes up a single thread in a wide subject and follows its ramifications, thus throwing light on its various aspects Tracts arc expected to be rigorous, definitive, and of lasting value to mathematicians working in the relevant disciplines. Exercises can be included to illustrate techniques, summarize past work, and enhance the book's value as a seminar lexl All volumes arc properly edited and typeset and published, initially at least, m hardback Cambridge UNIVERSITY PRESS www-umfaridge.org ISBN 0-521-57347-5