Area, Lattice Points, and Exponential Sums by M. N. HuxleyArea, Lattice Points, and Exponential Sums by M. N. Huxley

Area, Lattice Points, and Exponential Sums

byM. N. Huxley

Hardcover | January 1, 1995

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In analytic number theory a large number of problems can be "reduced" to problems involving the estimation of exponential sums in one or several variables. This book is a thorough treatment of the developments arising from the method developed by Bombieri and Iwaniec in 1986 for estimating theRiemann zeta function on the line s = 1/2. Huxley and his coworkers (mostly Huxley) 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 potentialfor 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 thattask by presenting all of the relevant literature and a good part of the background in one package.The audience for the book will be mathematics graduate students and faculties with a research interest in analytic theory; more specifically, those with an interest in exponential sum methods. The book is self-contained; any graduate student with a one semester course in analytic number theoryshould have a more than sufficient background.
M. N. Huxley is at University of Wales, Cardiff.
Title:Area, Lattice Points, and Exponential SumsFormat:HardcoverDimensions:506 pages, 9.21 × 6.14 × 1.26 inPublished:January 1, 1995Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0198534663

ISBN - 13:9780198534662

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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 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

Editorial Reviews

`This book [is] very detailed ... The book is very well written. It is an excellent and important work for all mathematicians who deal with exponential sums and lattice point theory. It is accessible to graduate students beginning research.'Zentrallblat fur Mathematik, vol. 861, 1997