Softcover ISBN:  9781470462857 
Product Code:  GSM/203.S 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Electronic ISBN:  9781470454203 
EPUB ISBN:  9781470468309 
Product Code:  GSM/203.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 

Book DetailsGraduate Studies in MathematicsVolume: 203; 2019; 356 ppMSC: Primary 11;
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.ReadershipUndergraduate and graduate students and researchers interested in distribution of prime numbers.

Table of Contents

Chapters

And then there were infinitely many

First principles

Asymptotic estimates

Combinatorial ways to count primes

The Dirichlet convolution

Dirichlet series

Methods of complex and harmonic analysis

An explicit formula for counting primes

The Riemann zeta function

The Perron inversion formula

The Prime Number Theorem

Dirichlet characters

Fourier analysis on finite abelian groups

Dirichlet $L$functions

The Prime Number Theorem for arithmetic progressions

Multiplicative functions and the anatomy of integers

Primes and multiplicative functions

Evolution of sums of multiplicative functions

The distribution of multiplicative functions

Large deviations

Sieve methods

Twin primes

The axioms of sieve theory

The Fundamental Lemma of Sieve Theory

Applications of sieve methods

Selberg’s sieve

Sieving for zerofree regions

Bilinear methods

Vinogradov’s method

Ternary arithmetic progressions

Bilinear forms and the large sieve

The BombieriVinogradov theorem

The least prime in an arithmetic progression

Local aspects of the distribution of primes

Small gaps between primes

Large gaps between primes

Irregularities in the distribution of primes

Appendices

The RiemannStieltjes integral

The Fourier and the Mellin transforms

The method of moments


Additional Material

Reviews

...this is an excellent book introducing the reader to a wealth of modern techniques for studying prime numbers. There is a lot of new material here that has never appeared before in book form. The author took great care in explaining both the intuition behind this very technical subject and in providing the 'best' proofs, especially proofs that are short and understandable. The book will be an excellent introduction to anybody interested in primes at a research level (or rather, interested in quickly reaching this level).
Maksym Radziwi, University of Texas at Austin 
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 wellpaced 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)


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

Chapters

And then there were infinitely many

First principles

Asymptotic estimates

Combinatorial ways to count primes

The Dirichlet convolution

Dirichlet series

Methods of complex and harmonic analysis

An explicit formula for counting primes

The Riemann zeta function

The Perron inversion formula

The Prime Number Theorem

Dirichlet characters

Fourier analysis on finite abelian groups

Dirichlet $L$functions

The Prime Number Theorem for arithmetic progressions

Multiplicative functions and the anatomy of integers

Primes and multiplicative functions

Evolution of sums of multiplicative functions

The distribution of multiplicative functions

Large deviations

Sieve methods

Twin primes

The axioms of sieve theory

The Fundamental Lemma of Sieve Theory

Applications of sieve methods

Selberg’s sieve

Sieving for zerofree regions

Bilinear methods

Vinogradov’s method

Ternary arithmetic progressions

Bilinear forms and the large sieve

The BombieriVinogradov theorem

The least prime in an arithmetic progression

Local aspects of the distribution of primes

Small gaps between primes

Large gaps between primes

Irregularities in the distribution of primes

Appendices

The RiemannStieltjes integral

The Fourier and the Mellin transforms

The method of moments

...this is an excellent book introducing the reader to a wealth of modern techniques for studying prime numbers. There is a lot of new material here that has never appeared before in book form. The author took great care in explaining both the intuition behind this very technical subject and in providing the 'best' proofs, especially proofs that are short and understandable. The book will be an excellent introduction to anybody interested in primes at a research level (or rather, interested in quickly reaching this level).
Maksym Radziwi, University of Texas at Austin 
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 wellpaced 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)