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The Distribution of Prime Numbers

Dimitris Koukoulopoulos (Autor)

Buch | Softcover

356 Seiten

2020
American Mathematical Society (Verlag)
978-1-4704-6285-7 (ISBN)

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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.

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.

Erscheinungsdatum	16.12.2020
Reihe/Serie	Graduate Studies in Mathematics
Verlagsort	Providence
Sprache	englisch
Gewicht	650 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
ISBN-10	1-4704-6285-0 / 1470462850
ISBN-13	978-1-4704-6285-7 / 9781470462857
Zustand	Neuware