**CBMS Regional Conference Series in Mathematics**

Volume: 84;
1994;
220 pp;
Softcover

MSC: Primary 11; 42;
Secondary 41

Print ISBN: 978-0-8218-0737-8

Product Code: CBMS/84

List Price: $38.00

Individual Price: $30.40

**Electronic ISBN: 978-1-4704-2444-2
Product Code: CBMS/84.E**

List Price: $38.00

Individual Price: $30.40

# Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis

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*Hugh L. Montgomery*

A co-publication of the AMS and CBMS

This book contains lectures presented by Hugh L. Montgomery at the NSF-CBMS Regional Conference held at Kansas State University in May 1990. The book focuses on important topics in analytic number theory that involve ideas from harmonic analysis. One valuable aspect of the book is that it collects material that was either unpublished or that had appeared only in the research literature. This book would be an excellent resource for harmonic analysts interested in moving into research in analytic number theory. In addition, it is suitable as a textbook in an advanced graduate topics course in number theory.

#### Table of Contents

# Table of Contents

## Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xi12 free
- Notation xiii14 free
- Chapter 1. Uniform Distribution 116 free
- Chapter 2. van der Corput Sets 1732
- Chapter 3. Exponential Sums I: The Methods of Weyl and van der Corput 3954
- Chapter 4. Exponential Sums II: Vinogradov's Method 6580
- Chapter 5. An Introduction to Turan's Method 85100
- Chapter 6. Irregularities of Distribution 109124
- Chapter 7. Mean and Large Values of Dirichlet Polynomials 125140
- Chapter 8. Distribution of Reduced Residue Classes in Short Intervals 151166
- Chapter 9. Zeros of L-functions 163178
- Chapter 10. Small Polynomials with Integral Coefficients 179194
- Appendix: Some Unsolved Problems 195210
- 1. Uniform Distribution 195210
- 2. van der Corput Sets 196211
- 3. Weyl Sums 196211
- 4. van der Corput's Method 197212
- 5. Turan's Method 197212
- 6. Irregularities of Distribution 198213
- 7. Mean and Large Values of Dirichlet Polynomials 198213
- 8. Reduced Residues in Short Intervals 200215
- 9. Zeros of L-Functions 201216
- 10. Small Polynomials with Integral Coefficients 201216
- 11. Character Sums 202217
- 12. Diophantine Approximation 202217
- 13. Metric Diophantine Approximation 204219
- 14. Algebraic Integers 205220
- 15. Trigonometric Polynomials 206221
- 16. Miscellaneous 207222
- References 210225

- Index 215230
- Back Cover Back Cover1237

#### Readership

Analytic number theorists and harmonic analysts.

#### Reviews

The book is a masterpiece of exposition and can be highly recommended to anybody interested in the connections of analysis and number theory.

-- Zentralblatt MATH

Many analysts, not necessarily specialists in harmonic analysis, will be able to find much to interest them here … The long lists of references provided for each chapter are one of the invaluable features of a very readable and stimulating book.

-- Bulletin of the LMS

These lectures give a lively, often original, look at a selection of topics from the area described by the title and, once one accepts the restriction of the previous sentence, there is nobody, anywhere, better equipped to tell this tale … this book deserves the attention of every analytic number theorist …adds considerably more to the literature than a number of books which could serve that function.

-- Mathematical Reviews