Abstract:

Riemann's zeta function encodes deep properties of prime numbers. Conjecturally Langlands L-functions do the same for primes in number fields. They are defined in terms of Euler products. Like the Riemann zeta function, they should satisfy functional equations. The functional equations are far from obvious and have been established only in special cases.

Steve Miller and I have introduced a new method for proving functional equations. I shall first show how the method works in the case of the zeta function. I shall then introduce the notion of an automorphic distribution and explain how it can be used to prove new results about L-functions.