TY - JOUR

T1 - Dirichlet Series with Periodic Coefficients and Their Value-Distribution near the Critical Line

AU - Sourmelidis, Athanasios

AU - Steuding, Jörn

AU - Suriajaya, Ade Irma

N1 - Funding Information:
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.
Publisher Copyright:
© 2021, Pleiades Publishing, Ltd.

PY - 2021/9

Y1 - 2021/9

N2 - 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 ≠ 0, we find for an even or odd periodic f the number of a-points of the Δ-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.

AB - 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 ≠ 0, we find for an even or odd periodic f the number of a-points of the Δ-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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U2 - 10.1134/S0081543821040118

DO - 10.1134/S0081543821040118

M3 - Article

AN - SCOPUS:85116749833

SN - 0081-5438

VL - 314

SP - 238

EP - 263

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

IS - 1

ER -