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:	506
Product dimensions:	6.50(w) x 9.50(h) x 1.27(d)

About the Author

University of Wales

Table of Contents

IntroductionPART I: Elementary Methods1. The rational line2. Polygons and area3. Integer points close to a curve4. Rational points close to a curvePART II: The Bombieri-Iwaniec Method5. Analytic methods6. Mean value theorems7. The simple exponential sum8. Exponential sums with a difference9. Exponential sums with a difference10. Exponential sums with modular form coefficientsPART III: The First Spacing Problem: Integer Vectors11. The ruled surface method12. The Hardy Littlewood method13. The first spacing problem for the double sumPART IV: The Second Spacing Problem: Rational vectors14. The first and second conditions15. Consecutive minor arcs16. The third and fourth conditionsPART V: Results and Applications17. Exponential sum theorems18. Lattice points and area19. Further results20. Sums with modular form coefficients21. Applications to the Riemann zeta function22. An application to number theory: prime integer pointsPART IV: Related Work and Further Ideas23. Related work24. Further ideasIntroductionPart I Elementary Methods1. The rational line2. Polygons and area3. Integer points close to a curve4. Rational points close to a curvePart II The Bombieri-Iwaniec Method5. Analytic methods6 C Mean value theorems. 7. The simple exponential sum8. Exponential sums with a difference9. Exponential sums with a difference10. Exponential sums with modular form coefficientsPart III The First Spacing Problem: Integer Vectors11. The ruled surface method12. The Hardy Littlewood method13. The first spacing problem for the double sumPart IV The Second Spacing Problem: Rational vectors14. The first and second conditions15. Consecutive minor arcs16 C The third and fourth conditions. Part V Results and Applications17. Exponential sum theorems18. Lattice points and area19. Further results20. Sums with modular form coefficientsm 21. Applications to the Riemann zeta function22. An application to number theory: prime integer pointsPart IV Related Work and Further Ideas23. Related work24. Further ideasReferences