## Athanasios Kouroupis (e Norwegian University of Science and Technology)

will speak on

*Examples of Universal Dirichlet series*

**Time: **3:00PM

**Date: **Mon 22nd April 2024

**Location: **To be announced

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Further information**Abstract: **Universality refers to the phenomenon where an object, via a countable process, yields approximations to all members of some collection of interest.

In our case this object will be a Dirichlet series of the form $\sum_n a_n e^{-s\lambda_n}$ and the approximation will happen via vertical translations.

In 1975 Voronin proved that the Riemann zeta function, $\zeta(s)=\sum _{n\geq 1}n^{-s},\, \Re s>1$

is universal in the critical strip $\{\frac{1}{2}<\Re s<1 \}$. After the seminal work of Voronin on the Riemann zeta function a lot of authors studied Dirichlet series in terms of their universal properties.

The aim of this talk is to give \textbf{ examples of convergent universal objects} such as the alternating prime zeta function $P(s)=\sum(-1)^np_n^{-s}$ on the critical strip.

Essentially, we are going to prove that for every compact subset with connected complement $K$ of $\Omega$, the vertical translations of the form $P(\cdot+it)$ can approximate uniformly in $K$ every function $f$, which is continuous on $K$ and holomorphic in the interior.

This is joint work with Frédéric Bayart.

*(This talk is part of the Analysis series.)*

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