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Analytic Methods for Diophantine Equations and Diophantine Inequalities

Harold Davenport
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Mathematical Library
[Reviewed by
Álvaro Lozano-Robledo
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In 1952, Harold Davenport published his book The Higher Arithmetic, which is one of the most beautiful introductions to number theory available to date and has become a very popular book among number theorists and the general public, since it was written with the intention of being accessible to anyone. Later on (1967), Davenport published another great example of mathematical writing: Multiplicative Number Theory, this time aimed at a more advanced audience. Davenport passed away shortly after, in 1969.

The book under review was first published as a set of lecture notes by the University of Michigan in 1962, where Davenport spent some time in the early 1960's. The notes have been restored, with minor corrections, by the editor of the book, T. D. Browning and TeXed back to life. The popularity of The Higher Arithmetic and the quality of Multiplicative Number Theory make the book under review valuable for sentimental reasons for those who learned and enjoyed the number theory of Davenport's excellent expositions. However, Analytic Methods stands on its own as yet another great example of Davenport's style and ability to present a number theory topic. Moreover, the new edition of his lecture notes include a foreword written by three experts (R. C. Vaughan, D. R. Heath-Brown, D. E. Freeman) where the recent discoveries and state of the art on the topics covered in the book are summarized, adding a great amount to the total value of the volume.

The first 10 chapters are dedicated to Waring's problem, to which Davenport himself made important contributions. The main theme here is the Hardy-Littlewood circle method (as Davenport remarks, Vinogradov's name should tag along whenever this method in mentioned, since the russian mathematician improved their approach with some technical simplifications). It all begins with a brief history of the problem and Weyl's and Hua's inequality. Then he moves on to asymptotic formulas, the singular series and the study of the function G(k).

Chapters 11-19 are dedicated to the study of forms in many variables. The motivating question here is the following: let F be a homogeneous polynomial in n variables of degree d and integer coefficients. When does F = 0 have non-trivial integer solutions? Is there N sufficiently large such that all forms F of degree d with more than N variables have a integral solution? In these chapters, the author describes Birch's theorem and concentrates on cubic forms. In the final chapter, number 20, we can find an exposition of the work of Davenport and Heilbronn. They adapted the Hardy-Littlewood-Vinogradov method to the study of diophantine inequalities. Finally, I would like to point out that the book contains a comprehensive bibliography on the subject. The authors of the current foreword have added many articles of interest, updating Davenport's own list.

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.

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 c1x1k + · · · + csxsk = N 39
8 The equation c1x1k + · · · + csxsk = 0 45
9 Waring’s problem: the number G(k) 51
10 The equation c1x1k + · · · + csxsk = 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
19 Homogeneous equations of higher degree 120
20 A Diophantine inequality 125
References 134
Index 139