Area, Lattice Points, and Exponential Sums

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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.

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Product Details

Table of Contents

Introduction 1
1 The rational line 7
2 Polygons and area 24
3 The integer points close to a curve 42
4 The rational points close to a curve 63
5 Analytic lemmas 87
6 Mean value results 126
7 The simple exponential sum 142
8 The exponential sum for the lattice point problem 167
9 Exponential sums with a difference 185
10 Exponential sums with modular form coefficients 197
11 The ruled surface method 235
12 The Hardy-Littlewood method 255
13 The First Spacing Problem for the double sum 272
14 The First and Second Conditions 287
15 Consecutive minor arcs 307
16 The Third and Fourth Conditions 329
17 Exponential sum theorems 349
18 Lattice points and area 372
19 Further results 406
20 Sums with modular form coefficients 423
21 Applications to the Riemann zeta function 438
22 An application to number theory: prime integer points 452
23 Related work 467
24 Further ideas 478
References 484
Index 491
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