**Graduate Studies in Mathematics**

Volume: 203;
2019;
356 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-6285-7
Product Code: GSM/203.S**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

**Electronic ISBN: 978-1-4704-5420-3
Product Code: GSM/203.E**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

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#### Supplemental Materials

# The Distribution of Prime Numbers

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*Dimitris Koukoulopoulos*

Prime numbers have fascinated mathematicians
since the time of Euclid. This book presents some of our best tools to
capture the properties of these fundamental objects, beginning with
the most basic notions of asymptotic estimates and arriving at the
forefront of mathematical research. Detailed proofs of the recent
spectacular advances on small and large gaps between primes are made
accessible for the first time in textbook form. Some other highlights
include an introduction to probabilistic methods, a detailed study of
sieves, and elements of the theory of pretentious multiplicative
functions leading to a proof of Linnik's theorem.

Throughout, the emphasis has been placed on explaining the main
ideas rather than the most general results available. As a result,
several methods are presented in terms of concrete examples that
simplify technical details, and theorems are stated in a form that
facilitates the understanding of their proof at the cost of
sacrificing some generality. Each chapter concludes with numerous
exercises of various levels of difficulty aimed to exemplify the
material, as well as to expose the readers to more advanced topics and
point them to further reading sources.

#### Readership

Undergraduate and graduate students and researchers interested in distribution of prime numbers.

#### Reviews & Endorsements

It's clear that Koukoulopoulos had a marvelous time putting together this beautiful material and producing a very readable and pedagogically sound text (replete with good exercises). The book is well-paced and reads very well. The careful reader, with pencil and paper in hand, keen to do exercises galore and have fun doing so, will learn a lot of beautiful number theory and find out marvels about the secret life of the set of primes: they are elusive but not unyielding.

-- Michael Berg, Loyola Marymount University

The book under review is a really beautiful guide to the mysteries involving the distribution of prime numbers. The book is written in such a manner to introduce beginning graduate students as well as advanced undergraduate students to the related methods of analytic number theory. A very nice aspect of this work is that the author gives emphasis on demonstrating the main ideas involved, thus making the presentation and flow of the book much more natural and reader friendly.

-- Michael Th.Rassias (Zürich)

#### Table of Contents

# Table of Contents

## The Distribution of Prime Numbers

- Cover Cover11
- Title page i2
- Contents v6
- Preface vii8
- Notation xi12
- And then there were infinitely many 114
- Part 1 . First principles 720
- Part 2 . Methods of complex and harmonic analysis 5164
- Chapter 5. An explicit formula for counting primes 5265
- Chapter 6. The Riemann zeta function 6275
- Chapter 7. The Perron inversion formula 7083
- Chapter 8. The Prime Number Theorem 8497
- Chapter 9. Dirichlet characters 95108
- Chapter 10. Fourier analysis on finite abelian groups 100113
- Chapter 11. Dirichlet 𝐿-functions 110123
- Chapter 12. The Prime Number Theorem for arithmetic progressions 118131

- Part 3 . Multiplicative functions and the anatomy of integers 129142
- Part 4 . Sieve methods 173186
- Part 5 . Bilinear methods 233246
- Part 6 . Local aspects of the distribution of primes 299312
- Appendices 335348
- Bibliography 344357
- Index 354367
- Back Cover Back Cover1370