The distribution of prime numbers
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 me…
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- Full title :The distribution of prime numbers / Dimitris Koukoulopoulos.
Summary
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.
Read More ↓Edition Statements
- Publication Date:2019
Publication Statements
- Publisher(s):American mathematical Society
- Place of Publication:Providence, Rhode Island
- Language:eng
- Publication Date:2019
The Physical Object
- Description:xii, 356 pages ; : 26 cm.
Edition Identifiers
- ISBN:9781470447540, 1470447541
- oclc:on1113143297
- LC Classification:QA246
- Dewey Decimal:512.7/3
Table of Contents
Asymptotic estimates -- Combinatorial ways to count primes -- The Dirichlet convolution -- Dirichlet series -- 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 -- Primes and multiplicative functions -- Evolution of sums of multiplicative functions -- The distribution of multiplicative functions -- Large deviations -- 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 -- Vinogradov's method -- Ternary arithmetic progressions -- Bilinear forms and the large sieve -- The Bombieri-Vinogradov theorem -- The least prime in an arithmetic progression -- Small gaps between primes -- Large gaps between primes -- Irregularities in the distribution of primes -- The Riemann-Stieltjes integral -- The Fourier and the Mellin transforms -- The method of moments.
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