Uniqueness Theorems for Zeta-Functions

Dirichlet series and their analytic properties play a central role in analytic number theory. In this talk we are concerned with the distribution of values of Dirichlet series with periodic coefficients, resp. their meromorphic continuations (including Dirichlet $L$-functions, for example). We prove an analogue of Rolf Nevanlinna's classical five point theorem (from 1926) for this family of functions and answer the question how many values can two such Dirichlet series share? Moreover, we discuss the origin of Nevanlinna's value distribution theory and highlight in particular the role played by George Pólya.