This is an introduction to analytic number theory developed through the study of the distribution of prime numbers, highlighting how analytic number theorists think. The central focus is on the Prime Number Theorem, presented through a proof selected to balance conceptual understanding with technical depth, alongside a sketch of Riemann's classical approach to highlight the subject's elegance. Providing a wide range of further directions (e.g., sieve methods, the anatomy of integers, primes in arithmetic progressions, prime gaps, smooth numbers, and extensive discussion of probabilistic heuristics which play an important role in guiding research goals), the emphasis throughout the book is on clarity of argument and the development of technique using a conversational style of writing. The book ends with 13 short introductions to hot topics. The book assumes familiarity with elementary number theory and basic complex analysis, though it provides helpful review material. Boxed equations highlight the most memorable formulas; exercises, some embedded directly in the proofs, are designed to deepen understanding without becoming overwhelming. Its flexible structure makes the book suitable for various course designs, whether emphasizing core theory or incorporating optional sections on combinatorics, arithmetic progressions, or open research problems. By blending classical results with current perspectives, this book prepares advanced undergraduates and beginning graduate students to not just learn analytic number theory, but to acquire contemporary ways of thinking about the subject.
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
