New Directions in Value-Distribution Theory of Zeta and L-Functions

These are the proceedings of the conference New Directions in the Theory of Universal Zeta- and L-Functions, held at the Department of Mathematics, Würzburg University, in October 2008. Most of the papers deal with zeta- and L-functions which are very powerful tools in Number Theory that encode interesting information about the underlying arithmetical objects in their value-distribution. The famous yet unproved Riemann hypothesis is a prototypical example and one of the seven millennium problems. One of the main topics of contributions in these proceedings is the spectacular universality theorem of Voronin from 1975 which claims that any non-vanishing analytic function can be approximated uniformly by certain purely imaginary shifts of the Riemann zeta-function in the right half of the critical strip. In the meantime several significant extensions and generalizations of this remarkable theorem were found. Proofs of universality theorems for Dirichlet series involve methods and results from Number Theory, Complex Analysis, and, often, from Probability Theory, resp. Measure Theory.