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On Second Moment of Selberg Zeta-Function for \(\sigma =1\)

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Abstract

Let Z(s) be the Selberg zeta-function for the modular group. We consider the existence of the second moments of Z(s) and of its reciprocal on \(\sigma =1\). The existence of such moments is related to the properties of certain Beurling natural numbers. Here the behavior of the counting function and the distribution of minimal gaps between these Beurling natural numbers are important. We also obtain unconditional upper bounds for these moments.

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Funding

The research of the second named author is funded by European Social Fund (Project No. 09.3.3-LMT-K-712-01-0037) under grant agreement with the Research Council of Lithuania (LMTLT).

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Authors and Affiliations

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

    Paulius Drungilas, Ramūnas Garunkštis & Aivaras Novikas

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  1. Paulius Drungilas
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  2. Ramūnas Garunkštis
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  3. Aivaras Novikas
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All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

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Correspondence to Ramūnas Garunkštis.

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Drungilas, P., Garunkštis, R. & Novikas, A. On Second Moment of Selberg Zeta-Function for \(\sigma =1\). Results Math 76, 184 (2021). https://doi.org/10.1007/s00025-021-01492-5

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