Analytic Methods for Diophantine Equations and Diophantine Inequalities

Based on lectures given by Davenport at the University of Michigan in the early 1960s.

H. Davenport (Author), T. D. Browning (Prepared for publication by)

9780521605830, Cambridge University Press

Paperback, published 7 February 2005

160 pages
22.8 x 15 x 1.3 cm, 0.26 kg

"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...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."
MAA Reviews, Alvara Lozano-Robledo, Cornell University

Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities. It provides an excellent introduction to a timeless area of number theory that is still as widely researched today 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. For the second edition of the book a comprehensive foreword has been added in which three prominent authorities describe the modern context and recent developments. A thorough bibliography has also been added.

Preface
Foreword
1. Introduction
2. Waring's problem: history
3. Weyl's inequality and Hua's inequality
4. Waring's problem: the asymptotic formula
5. Waring's problem: the singular series
6. The singular series continued
7. The equation C1xk1 +…+ Csxks=N
8. The equation C1xk1 +…+ Csxks=0
9. Waring's problem: the number G (k)
10. The equation C1xk1 +…+ Csxks=0 again
11. General homoogeneous equations: Birch's theorem
12. The geometry of numbers
13. cubic forms
14. Cubic forms: bilinear equations
15. Cubic forms: minor arcs and major arcs
16. Cubic forms: the singular integral
17. Cubic forms: the singular series
18. Cubic forms: the p-adic problem
19. Homogeneous equations of higher degree
20. A Diophantine inequality
Bibliography
Index.

Subject Areas: Number theory [PBH]