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

TL;DR

This paper explores the connection between continuous and discrete universality properties of zeta-functions, revealing that linear dynamics provides a unifying framework to derive various universality results.

Contribution

It demonstrates that the relationship between continuous and discrete universality can be understood through linear dynamics, simplifying the derivation of numerous universality theorems.

Findings

Established a link between continuous and discrete universality via linear dynamics.

Derived a variety of discrete universality results from this connection.

Provided a unified approach to understanding zeta-function universality.

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.