Area, Lattice Points, and Exponential Sums
Oxford University Press (Verlag)
978-0-19-853466-2 (ISBN)
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 the Riemann 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 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 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 theory should have a more than sufficient background.
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 ; 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 ; 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
| Erscheint lt. Verlag | 13.6.1996 |
|---|---|
| Reihe/Serie | London Mathematical Society Monographs ; 13 |
| Zusatzinfo | line figures |
| Verlagsort | Oxford |
| Sprache | englisch |
| Maße | 161 x 241 mm |
| Gewicht | 885 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| ISBN-10 | 0-19-853466-3 / 0198534663 |
| ISBN-13 | 978-0-19-853466-2 / 9780198534662 |
| Zustand | Neuware |
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