Continuous and discrete universality of zeta-functions: Two sides of the same coin?

Abstract

In 1975 Voronin proved the universality theorem for the Riemann zeta-function $\zeta (s)$ which roughly says that any admissible function $f(s)$ is approximated by $\zeta (s)$. A few years later Reich proved a discrete analogue of this result. The proofs of these theorems are almost identical but it is not known whether one of them implies the other. We will see that if we translate the question in the language of linear dynamics then there is a link which we exploit to obtain in a straightforward way a big variety of discrete universality results appearing in the literature.