Analytic number theory studies the whole numbers — above all the primes — using the tools of analysis: limits, infinite series, integrals and, most powerfully of all, functions of a complex variable. It is the branch of number theory where calculus meets counting, and where a question as concrete as "how many primes are there below a million?" is answered by studying the zeros of a function in the complex plane.

This is the graduate-level continuation of the analytic stage of the number-theory course. Where those pages give the pictures and the intuition, these turn them into proofs, and press on all the way to the modern research frontier — sieve methods, bounded gaps between primes, the circle method, automorphic L-functions and the computational state of the art.

Follow it as a course

These lessons are curated, in order, as the master's course Analytic Number Theory: The Distribution of the Primes — eleven modules from the analytic toolkit to the automorphic frontier. Start there for the guided path, or dip into any topic below.

Open the course →