Van der Corputʼs method of exponential sums / S.W. Graham and G. Kolesnik.

Format:

Book

Author/Creator:

Graham, S. W., author.

Kolesnik, G., author.

Series:

London Mathematical Society lecture note series ; 126.

London Mathematical Society lecture note series ; 126

Language:

English

Subjects (All):

Exponential sums.

Physical Description:

1 online resource (119 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1991.

Language Note:

English

Summary:

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.

Contents:

Cover; Title; Copyright; Contents; 1 Introduction; 1.1 Basic Definitions; 1.2 Historical Overview; 1.3 Two Dimensional Sums; 1.4 The method of Bombieri and Iwaniec; 1.5 Notation; 2 The Simplest Van Der Corput Estimates; 2.1 Estimates Using First and Second Derivatives; 2.2 Some Simple Inequalities; 2.3 The Weyl-van der Corput Inequality; 2.4 Iterating Weyl-Van der Corput; 2.5 Upper Bounds for the Riemann Zeta-function; 2.6 Notes; 3 The Method of Exponent Pairs; 3.1 Introduction; 3.2 Lemmas on Exponential Integrals; 3.3 Heuristic Arguments and Definitions; 3.4 Proof of the A-Process

3.5 Proof of the B-Process3.6 Notes; 4 Applications of Exponent Pairs; 4.1 The Riemann Zeta-function; 4.2 Sums Involving ψ; 4.3 The Dirichlet Divisor Problem; 4.4 The Circle Problem; 4.5 Gaps Between Squarefree Numbers; 4.6 The Piatetski-Shapiro Prime Number Theorem; 4.7 Notes; 5 Computing Optimal Exponent Pairs; 5.1 Introduction; 5.2 Preliminary Lemmas; 5.3 The Algorithm; 5.4 Applications; 5.5 Notes; 6 Two Dimensional Exponential Sums; 6.1 Introduction; 6.2 Generalized Weyl-van der Corput Inequality; 6.3 Omega Conditions; 6.4 The AB Theorem; 7 New Exponent Pairs; 7.1 Introduction

7.2 Preliminaries7.3 The Airy-Hardy Integral; 7.4 Gauss Sums; 7.5 Lemmas on Rational Points; 7.6 Semicubical Powers of Integers; 7.7 Proof of the Theorem; 7.8 Notes; Appendix; Bibliography; Index

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88429-8

1-107-36629-1

1-107-37101-5

1-107-36138-9

1-107-36918-5

1-299-40408-1

1-107-36383-7

0-511-66197-5