Area, Lattice Points, and Exponential Sums

by M. N. Huxley

Overview

In analytic number theory many problems can be "reduced" to those involving the estimation of exponential sums in one or several variables. This book is a thorough treatment of the developments arising from the method for estimating the Riemann zeta function. Huxley and his coworkers have taken this method and vastly extended and improved it. The powerful techniques presented here go considerably beyond older methods for estimating exponential sums such as van de Corput's method. The potential for the method is far from being exhausted, and there is considerable motivation for other researchers to try to master this subject. However, anyone currently trying to learn all of this material has the formidable task of wading through numerous papers in the literature. This book simplifies that task by presenting all of the relevant literature and a good part of the background in one package. The book will find its biggest readership among mathematics graduate students and academics with a research interest in analytic theory; specifically exponential sum methods.

Product Details

ISBN-13:	9780198534662
Publisher:	Oxford University Press
Publication date:	08/22/1996
Series:	London Mathematical Society Monographs , #13
Pages:	512
Product dimensions:	6.50(w) x 9.50(h) x 1.27(d)

About the Author

University of Wales

Table of Contents

Introduction
PART I: Elementary Methods
1. The rational line
2. Polygons and area
3. Integer points close to a curve
4. Rational points close to a curve
PART II: The Bombieri-Iwaniec Method
5. Analytic methods
6. Mean value theorems
7. The simple exponential sum
8. Exponential sums with a difference
9. Exponential sums with a difference
10. Exponential sums with modular form coefficients
PART III: The First Spacing Problem: Integer Vectors
11. The ruled surface method
12. The Hardy Littlewood method
13. The first spacing problem for the double sum
PART IV: The Second Spacing Problem: Rational vectors
14. The first and second conditions
15. Consecutive minor arcs
16. The third and fourth conditions
PART V: Results and Applications
17. Exponential sum theorems
18. Lattice points and area
19. Further results
20. Sums with modular form coefficients
21. Applications to the Riemann zeta function
22. An application to number theory: prime integer points
PART IV: Related Work and Further Ideas
23. Related work
24. Further ideas
Introduction
Part I Elementary Methods
1. The rational line
2. Polygons and area
3. Integer points close to a curve
4. Rational points close to a curve
Part II The Bombieri-Iwaniec Method
5. Analytic methods
6 C Mean value theorems.
7. The simple exponential sum
8. Exponential sums with a difference
9. Exponential sums with a difference
10. Exponential sums with modular form coefficients
Part III The First Spacing Problem: Integer Vectors
11. The ruled surface method
12. The Hardy Littlewood method
13. The first spacing problem for the double sum
Part IV The Second Spacing Problem: Rational vectors
14. The first and second conditions
15. Consecutive minor arcs
16 C The third and fourth conditions.
Part V Results and Applications
17. Exponential sum theorems
18. Lattice points and area
19. Further results
20. Sums with modular form coefficients m 21. Applications to the Riemann zeta function
22. An application to number theory: prime integer points
Part IV Related Work and Further Ideas
23. Related work
24. Further ideas
References