Analytic methods for Diophantine equations and Diophantine inequalities

Includes index

Includes bibliography (p. 134-138) references.

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

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.