Analytic number theory

Analytic number theory is the branch of number theory that uses methods from mathematical analysis. Its first major success was Dirichlet's application of analysis to prove Dirichlet's theorem on arithmetic progressions, stating the existence of infinitely many primes in arithmetic progressions of the form a + nb, where a and b are relatively prime. The proofs of the prime number theorem based on the Riemann zeta function are another milestone.

The outline of the subject remains similar to the heyday of the subject in the 1930s. Multiplicative number theory deals with the distribution of the prime numbers, applying Dirichlet series as generating functions. It is assumed that the methods will eventually apply to the general L-function, though that theory is still largely conjectural. Additive number theory has as typical problems Goldbach's conjecture and Waring's problem.

Methods have changed somewhat. The circle method of

From Wikipedia, the free encyclopedia