Chair of Analytic Number Theory
Welcome to the chair of Analytic Number Theory !
Analytic Number Theory is the branch of number theory that studies properties of integers (and especially prime number) using principally methods from analysis. Its origins go back at least to Euler’s proof of the infiniteness of the set of prime numbers using the zeta function — to be called Riemann’s zeta function in the 19th. century — and was pursued notably by, Dirichlet (through his work on primes in arithmetic progression in which he introduced systematically L-functions) and, in a different direction, by Gauss when he counted the number of integral points within a circle of large radius.
Another milestone is Riemann’s memoir in which Riemann considered the Zeta function as a function of the complex variable: he established the basic analytic properties of Zeta, related its zeros to the distribution of prime numbers and finally formulated his Riemann hypothesis.
Eventually, Riemann’s work led to the proof by Hadamard/de la Vallée-Poussin of the Prime Number Theorem. Nowadays, Analytic Number Theory builds on a variety of very different techniques (L-functions, the Hardy-Littelwood circle method, Sieve methods…) and also make heavy use of deep methods from outside fields: arithmetic algebraic geometry, the theory of automorphic forms or ergodic theory…
The purpose of the TAN project is to bring these very different methods together so as to permit further advances in number theory.