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The Distribution of Prime Numbers

Dimitris Koukoulopoulos Université de Montréal, Montréal, QC, Canada
Available Formats:
Softcover ISBN: 978-1-4704-6285-7
Product Code: GSM/203.S
List Price: $85.00 MAA Member Price:$76.50
AMS Member Price: $68.00 Electronic ISBN: 978-1-4704-5420-3 EPUB ISBN: 978-1-4704-6830-9 Product Code: GSM/203.E List Price:$85.00
MAA Member Price: $76.50 AMS Member Price:$68.00
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List Price: $127.50 MAA Member Price:$114.75
AMS Member Price: $102.00 Click above image for expanded view The Distribution of Prime Numbers Dimitris Koukoulopoulos Université de Montréal, Montréal, QC, Canada Available Formats:  Softcover ISBN: 978-1-4704-6285-7 Product Code: GSM/203.S  List Price:$85.00 MAA Member Price: $76.50 AMS Member Price:$68.00
 Electronic ISBN: 978-1-4704-5420-3 EPUB ISBN: 978-1-4704-6830-9 Product Code: GSM/203.E
 List Price: $85.00 MAA Member Price:$76.50 AMS Member Price: $68.00 Please Note: Purchasing the electronic version includes access to both a PDF and EPUB version Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$127.50 MAA Member Price: $114.75 AMS Member Price:$102.00
• Book Details

Volume: 2032019; 356 pp
MSC: 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.

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 zero-free regions
• Bilinear methods
• Ternary arithmetic progressions
• Bilinear forms and the large sieve
• 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 Riemann-Stieltjes integral
• The Fourier and the Mellin transforms
• The method of moments

• 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 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)
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2032019; 356 pp
MSC: 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.

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 zero-free regions
• Bilinear methods
• Ternary arithmetic progressions
• Bilinear forms and the large sieve
• 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 Riemann-Stieltjes 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 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)
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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