Research Seminar in Mathematics - Universality of zeta and L-functions

Speaker

Johan Andersson, Örebro Universitet.

Abstract

Voronin proved in the seventies that any zero-free analytic function f(s) on a disc |s-3/4|<r<1/4 which is continuous up to its boundary may be uniformly approximated as closely as desired by imaginary shifts of the Riemann zeta-function. We say that the Riemann zeta-function is universal. In the last 40 years this property has been proved for a wide range of zeta and L-functions, such as automorphic L-functions and the Selberg zeta-function. I will talk about some of my recent work in the field. In particular I have recently proved (arXiv:1809.03444) that the Euler-Zagier multiple zeta-function is universal in several complex variables, thus giving the first example of a Dirichlet series that is universal in more than one complex variable. I will also talk about recent work in progress where I give the first universality theorem for the Hurwitz zeta-function with an algebraic irrational parameter.