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

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Abstract

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.

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Funding

The research is supported by the European Social Fund (project no. 09.3.3-LMT-K-712-01-0037) under grant agreement with the Research Council of Lithuania (LMT LT).

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Authors and Affiliations

  1. Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225, Vilnius, Lithuania

    A. Laurinčikas

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  1. A. Laurinčikas
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Correspondence to A. Laurinčikas.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 314, pp. 134–144 https://doi.org/10.4213/tm4165.

On the occasion of the 130th birthday of Ivan Matveevich Vinogradov

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Laurinčikas, A. On the Hurwitz Zeta-Function with Algebraic Irrational Parameter. II. Proc. Steklov Inst. Math. 314, 127–137 (2021). https://doi.org/10.1134/S0081543821040076

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