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Dirichlet Series with Periodic Coefficients and Their Value-Distribution near the Critical Line

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Abstract

The class of Dirichlet series associated with a periodic arithmetical function \(f\) includes the Riemann zeta-function as well as Dirichlet \(L\)-functions to residue class characters. We study the value-distribution of these Dirichlet series \(L(s;f)\) and their analytic continuation in the neighbourhood of the critical line (which is the axis of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number \(a\neq 0\), we find for an even or odd periodic \(f\) the number of \(a\)-points of the \(\Delta\)-factor of the functional equation, prove the existence of the mean of the values of \(L(s;f)\) taken at these points, show that the ordinates of these \(a\)-points are uniformly distributed modulo one and apply this to show a discrete universality theorem.

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Acknowledgments

The authors are grateful for the comments of the anonymous referee.

Funding

The first author is supported by the Austrian Science Fund, project Y-901, and the third author is supported by JSPS KAKENHI grant no. 18K13400.

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Authors and Affiliations

  1. Institute of Analysis and Number Theory, TU Graz, Steyrergasse 30, 8010, Graz, Austria

    Athanasios Sourmelidis

  2. Institute of Mathematics, Würzburg University, Emil-Fischer-Str. 40, 97074, Würzburg, Germany

    Jörn Steuding

  3. Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Ade Irma Suriajaya

Authors
  1. Athanasios Sourmelidis
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  2. Jörn Steuding
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  3. Ade Irma Suriajaya
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Correspondence to Athanasios Sourmelidis.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 314, pp. 248–274 https://doi.org/10.4213/tm4188.

To the memory of Ivan Matveevich Vinogradov on the occasion of his 130th birthday

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Sourmelidis, A., Steuding, J. & Suriajaya, A.I. Dirichlet Series with Periodic Coefficients and Their Value-Distribution near the Critical Line. Proc. Steklov Inst. Math. 314, 238–263 (2021). https://doi.org/10.1134/S0081543821040118

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