The Distribution of Prime Numbers

Dimitris Koukoulopoulos, Universite de Montreal, QC, Canada.

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