The distribution of prime numbers

Author(s)

• Koukoulopoulos, Dimitris

Bibliographic Information

The distribution of prime numbers

Dimitris Koukoulopoulos

（Graduate studies in mathematics, v. 203）

American Mathematical Society, c2019

• : hardcover

Note

Includes bibliographical references (p. 344-353) and index

Description and Table of Contents

Description

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.

Table of Contents

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: Vinogradov's method Ternary arithmetic progressions Bilinear forms and the large sieve The Bombieri-Vinogradov 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 Riemann-Stieltjes integral The Fourier and the Mellin transforms The method of moments Bibliography Index

by "Nielsen BookData"

Details

• NCID: BB29377227
• ISBN:
  • 9781470447540
• LCCN: 2019028661
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Providence, R.I.
• Pages/Volumes: xii, 356 p.
• Size: 27 cm
• Classification:
  • LCC : QA246
  • DC23 : 512.7/3
• Subject Headings:
  • LCSH : Numbers, Prime
• Parent Bibliography ID:
  • BA20542214