Discrete universality in the Selberg class

A. Laurinčikas ; R. Macaitienė

Informatics and Automation, Analytic number theory, Tome 299 (2017), pp. 155-169.

Résumé

The Selberg class $\mathcal S$ 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 $\mathcal S$ 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 $L(s+i\tau )$, $\tau \in \mathbb R$. 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,\dots $, where $h>0$ is an arbitrary fixed number.

Détail

Keywords: Selberg class, limit theorem, weak convergence, universality.

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