On the interface of universal zeta-functions and frequently hypercyclic vectors of translation operators

Abstract: If $\Omega \subseteq \mathbb{C}$ is a simply connected domain and $H(\Omega)$ is the Fréchet space of holomorphic functions on $\Omega$, then it is well-known from linear dynamics that there is a dense $G_{\delta}$ set of $H(\Omega)$ such that for any element $f$ from this set, the set $\{f( \cdot + i\tau ) : \tau \in \mathbb{R} \}$ is dense in $H(\Omega)$. However, no information is provided on how these elements $f$ look like. Despite that, in the special case when $\Omega$ is the vertical strip of complex numbers with real part between $1/2$ and $1$, Sergei Voronin proved using analytic number theoretical methods that the Riemann zeta-function $\zeta$ is universal in the sense that the set $\{\log \zeta( \cdot + i\tau ) : \tau \in \mathbb{R} \}$ is dense in $H(\Omega)$. This phenomenon has now been observed for a large class of “zeta-functions”. In these talks I will discuss about the aforementioned results and draw connections between the theory of translation operators and the theory of universal zeta-functions in the form of applications and research questions.

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar