Van der Corput's Method of Exponential Sums (London Mathematical Society Lecture Note Series, Series Number 126)

This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums. These arise in many problems in analytic number theory. It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory. The authors show how the method can be applied to problems such as upper bounds for the Riemann-Zeta function. the Dirichlet divisor problem, the distribution of square free numbers, and the Piatetski-Shapiro prime number theorem. TABLE OF CONTENTS......Page 5 Acknowledgments......Page 7 1.1 Basic Definitions......Page 9 1.2 Historical Overview......Page 10 1.3 Two Dimensional Sums......Page 11 1.4 The method of Bombieri and Iwaniec......Page 12 1.5 Notation......Page 13 2.1 Estimates Using First and Second Derivatives......Page 14 2.2 Some Simple Inequalities......Page 16 2.3 The Weyl-van der Corput Inequality......Page 18 2.4 Iterating Weyl-Van der Corput......Page 21 2.5 Upper Bounds for the Riemann Zeta-function......Page 24 2.6 Notes......Page 28 3.1 Introduction......Page 29 3.2 Lemmas on Exponential Integrals......Page 30 3.3 Heuristic Arguments and Definitions......Page 37 3.4 Proof of the A-Process......Page 40 3.5 Proof of the B-Process......Page 43 3.6 Notes......Page 45 4.1 The Riemann Zeta-function......Page 46 4.2 Sums Involving 0......Page 47 4.3 The Dirichlet Divisor Problem......Page 48 4.4 The Circle Problem......Page 50 4.5 Gaps Between Squarefree Numbers......Page 52 4.6 The Piatetski-Shapiro Prime Number Theorem......Page 54 4.7 Notes......Page 61 5.1 Introduction......Page 62 5.2 Preliminary Lemmas......Page 63 5.3 The Algorithm......Page 65 5.4 Applications......Page 71 5.5 Notes......Page 77 6.1 Introduction......Page 78 6.2 Generalized Weyl-van der Corput Inequality......Page 83 6.3 Omega Conditions......Page 87 6.4 The AB Theorem......Page 91 7.1 Introduction......Page 94 7.2 Preliminaries......Page 95 7.3 The Airy-Hardy Integral......Page 98 7.4 Gauss Sums......Page 101 7.5 Lemmas on Rational Points......Page 104 7.6 Semicubical Powers of Integers......Page 109 7.7 Proof of the Theorem......Page 112 7.8 Notes......Page 118 Appendix......Page 119 Bibliography......Page 125 Index......Page 128