On the Hurwitz Zeta-Function with Algebraic Irrational Parameter. II

A. Laurinčikas

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 134-144

Résumé

It is known that the Hurwitz zeta-function $\zeta (s,\alpha )$ with transcendental or rational parameter $\alpha $ has a discrete universality property; i.e., the shifts $\zeta (s+ikh,\alpha )$, $k\in \mathbb N_0$, $h> 0$, approximate a wide class of analytic functions. The case of algebraic irrational $\alpha $ is a complicated open problem. In the paper, some progress in this problem is achieved. It is proved that there exists a nonempty closed set $F_{\alpha ,h}$ of analytic functions such that the functions in $F_{\alpha ,h}$ are approximated by the above shifts. Also, the case of certain compositions $\Phi (\zeta (s,\alpha ))$ is discussed.