Abstract

A famous result in number theory is Dirichlet’s theorem that there exist infinitely many prime numbers in any given arithmetic progression a, a N, a 2 N, … where a, N are coprime. In fact, a stronger statement holds: the primes are equidistributed in the different residue classes modulo N. In order to prove his theorem, Dirichlet introduced Dirichlet L-functions, which are analogues of the Riemann zeta function which depend on a choice of character of the group of units modulo N. More general L-functions appear throughout number theory and are closely connected with equidistribution questions, such as the Sato—Tate conjecture (concerning the number of solutions to y2 = x3 a x b in the finite field with p elements, as the prime p varies). L-functions also play a central role in both the motivation for and the formulation of the Langlands conjectures in the theory of the automorphic forms. I will give a gentle introduction to some of these ideas and discuss some recent theorems in the area.

A wine reception in the central core will follow the lecture