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Analytic Methods for Diophantine Equations
and Diophantine Inequalities
HaroldDavenportwasoneofthetrulygreatmathematiciansofthetwen-
tiethcentury. BasedonlectureshegaveattheUniversityofMichiganin
theearly1960s,thisbookisconcernedwiththeuseofanalyticmethods
inthestudyofintegersolutionstoDiophantineequationsandDiophan-
tineinequalities. Itprovidesanexcellentintroductiontoatimelessarea
of number theory that is still as widely researchedtoday as it was when
the book originally appeared. The three main themes of the book are
Waring’s problem and the representation of integers by diagonal forms,
the solubility in integers of systems of forms in many variables, and the
solubility in integers of diagonal inequalities.
Forthesecondeditionofthebookacomprehensiveforewordhasbeen
added in which three leading experts describe the modern context and
recent developments. A complete list of references has also been added.
Analytic Methods for Diophantine Equations
and Diophantine Inequalities
Second edition
H. Davenport
This edition edited and prepared for publication by
T. D. Browning
OxfordUniversity
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521605830
© Estate of Harold Davenport 1963, 2005
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2005
isbn-13 978-0-511-08086-9 eBook (NetLibrary)
isbn-10 0-511-08086-7 eBook (NetLibrary)
isbn-13 978-0-521-60583-0 paperback
isbn-10 0-521-60583-0 paperback
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First edition first published by Campus Publishers 1963
Contents
Foreword page vii
Waring’s problem, by R. C. Vaughan vii
Forms in many variables, by D. R. Heath-Brown xi
Diophantine inequalities, by D. E. Freeman xv
Editorial preface xix
1 Introduction 1
2 Waring’s problem: history 3
3 Weyl’s inequality and Hua’s inequality 7
4 Waring’s problem: the asymptotic formula 15
5 Waring’s problem: the singular series 24
6 The singular series continued 33
7 The equation c1xk1 +···+csxks =N 39
8 The equation c1xk1 +···+csxks =0 45
9 Waring’s problem: the number G(k) 51
10 The equation c1xk1 +···+csxks =0 again 63
11 General homogeneous equations: Birch’s theorem 67
12 The geometry of numbers 75
13 Cubic forms 85
14 Cubic forms: bilinear equations 92
15 Cubic forms: minor arcs and major arcs 99
16 Cubic forms: the singular integral 104
17 Cubic forms: the singular series 107
18 Cubic forms: the p-adic problem 111
v
vi Contents
19 Homogeneous equations of higher degree 120
20 A Diophantine inequality 125
References 134
Index 139
Foreword
Waring’s problem: Chapters 1–10
WhenDavenportproducedtheselecturenotestherehadbeenverylittle
progress on Waring’s problem since important work by Davenport and
Vinogradov something like a quarter of a century earlier, and the main
interest was to report on the more recent work on forms as described
in the later chapters. Indeed there was a generally held view, with
regardtoWaring’sproblematleast,thattheyhadextractedeverything
that could be obtained reasonably by the Hardy–Littlewood method
and that the method was largely played out. Moreover, the material on
Waring’sproblemwasnotintended,ingeneral,tobestateoftheart,but
rather simply an introduction to the Hardy–Littlewood method, with a
minimumoffussbyamasterlyexpositor,whichcouldthenbedeveloped
asnecessaryforuseinthestudyoftherepresentationofzerobygeneral
integral forms, especially cubic forms, in the later chapters. There is
noaccountofDavenport’sownfundamentalworkonWaring’sproblem,
namely G(4) = 16 (Davenport [18]), G(5) ≤ 23, G(6) ≤ 36 (Davenport
[19]), nor of Vinogradov’s [94] G(k) ≤ 2klogk+o(klogk) for large k
or Davenport’s proof [17] that almost all natural numbers are the sum
of four positive cubes. Nor, on a more technical level, was any attempt
made to obtain more refined versions of Lemmas 4.2 and 9.2, estimates
forthegeneratingfunctionT(α)onthemajorarcs,suchasthosedueto
Davenport and Heilbronn [25] or Hua [50], although such refinements
can be very helpful in applications.
InthelasttwentyyearstherehasbeenagooddealofprogressonWar-
ing’sproblem. Methodsofgreatflexibility,inspiredbysomeoftheideas
stemmingfromtheresearchesofHardyandLittlewood,Davenport,and
Vinogradov have been developed which have permitted the retention of
vii
viii Foreword
manyofthewrinklesintroducedintheearliermethods. Thebeginnings
of a glimmer of some of these seminal ideas can be seen in Lemmas 9.4
and 9.5.
The asymptotic formula for the number of representations of a large
natural number n as the sum of at most s kth powers established in
Theorem 4.1 when s ≥ 2k +1 was state of the art for 3 ≤ k ≤ 10,
butforlargerk methodsduetoVinogradovweresuperior(seeTheorem
5.4 of Vaughan [86]). The current state of play is that the asymptotic
formula is known to hold when s ≥ 2k (k = 3,4,5) (Vaughan [82, 84]),
s ≥ 7.2k−3 (k = 6,7,8) (Bok(cid:1)lan [8], following Heath-B(cid:2)rown [43]), and
s ≥ s1(k) where s1(k) = k2 logk +loglogk +O(1) (k ≥ 9) (Ford
[32]). The discussion in the Note in Chapter 3 in the case k =3 is still
relevanttoday. Althoughtheasymptoticformulaforsumsofeightcubes
isnowestablishedtheclassicalconvexityboundwasnotimprovedinthe
exponent when 2 < m < 4. The core of the argument of Vaughan [82]
is extremely delicate and leads only to
(cid:3)
|T(α)|8dα(cid:3)P5(logP)−γ
m
for a positive constant γ and a suitable set of minor arcs m. However
Hooley [47] has shown under the (unproven) Riemann Hypothesis for a
certain Hasse–Weil L-function that
(cid:3)
|T(α)|6dα(cid:3)P3+ε
m
andthisinturnimpliestheasymptoticformulaforsumsofsevencubes.
Unfortunately it is not even known whether the L-function has an ana-
lytic continuation into the critical strip.
ForG(k)thebestresultsthatwecurrentlyhaveareG(3)≤7(Linnik
[57, 59]), G(4) = 16 Davenport [18], G(5) ≤ 17, G(7) ≤ 33, (Vaughan
and Wooley [89]), G(6) ≤ 21 (Vaughan and Wooley [88]), G(8) ≤
42 (Vaughan and Wooley [87]), G(9) ≤ 50, G(10) ≤ 59, G(11) ≤ 67,
G(12) ≤ 76, G(13) ≤ 84, G(14) ≤ 92, G(15) ≤ 100, G(16) ≤ 109,
G(17) ≤ 117, G(18) ≤ 125, G(19) ≤ 134, G(20(cid:1)) ≤ 142 (Vaughan and(cid:2)
Wooley[90]),andG(k)≤s2(k)wheres2(k)=k logk+loglogk+O(1)
(Wooley [98]) in general. Let G#(4) denote the smallest positive s such
that whenever 1 ≤ r ≤ s every sufficiently large n in the residue class
r modulo 16 is the sum of at most s fourth powers. Then, in fact,
Davenport showed that G#(4) ≤ 14 and we now can prove (Vaughan
[85]) that G#(4) ≤ 12. Linnik’s work on Waring’s problem for cubes
doesnotusetheHardy–Littlewoodmethod, butinsteadisbasedonthe
