Discrete Universality in the Selberg Class

The Selberg class S consists of functions L ( s ) that are defined by Dirichlet series and satisfy four axioms (Ramanujan conjecture, analytic continuation, functional equation, and Euler product). It has been known that functions in S that satisfy the mean value condition on primes are universal in the sense of Voronin, i.e., every function in a sufficiently wide class of analytic functions can be approximated by the shifts L ( s + iτ ), τ ∈ R. In this paper we show that every function in the same class of analytic functions can be approximated by the discrete shifts L ( s + ikh ), k = 0, 1,..., where h > 0 is an arbitrary fixed number.