Analytic Number Theory Revealed

Andrew Granville, Universite de Montreal, Canada.

Background in analytic number theory How many primes are there? Unconditional estimates for sums over primes Partial summation, and consequences of the Prime Number Theorem What should be true about primes? The modified Gauss-Cramer heuristic Multiplicative functions and Dirichlet series Anatomies of mathematical objects Counting irreducibles The average number of indecomposables The typical number of indecomposables Normal distributions The multiplication table With two or more parts Poisson and beyond Sieves and primes The Chinese Remainder Theorem as a sieve A first look at sieve methods Background in analysis Fourier series, Fourier analysis, and Poisson summation Complex analysis Analytic continuation of the Riemann zeta-function Perron's formula The use of Perron's formula The proof of the Prime Number Theorem Riemann's plan for proving the Prime Number Theorem Technical remarks Zeros of $\zeta(s)$ with $\textrm{Re}(s)=1$ Proof of the Prime Number Theorem The Riemann Hypothesis without zeros of $\zeta(s)$ Primes in arithmetic progressions Primes in arithmetic progressions Dirichlet characters Dirichlet $L$-functions The Prime Number Theorem for arithmetic progressions The Generalized Riemann Hypothesis A dozen and one different directions Exceptional zeros and primes in arithmetic progressions Selberg's small sieve Equidistribution in arithmetic progressions? Distribution of the error in the Prime Number Theorem Chebyshev's bias Primes in short intervals Smooths, factoring, and large gaps between primes Short gaps between primes The circle method Primes missing digits Towards the prime $k$-tuplets conjecture Prime values of higher-degree polynomials Primes in sparse sequences Probability primer Couting prime factors with multiplicity Analytic continuation for certain Dirichlet series Different proofs of the Prime Number Theorem Sketch of technical proofs Circle method primer Image credits Bibliography Index