# Dirichlet Series with Periodic Coefficients and their Value-Distribution Near the Critical Line

Source: [Dirichlet Series with Periodic Coefficients and their Value-Distribution Near the Critical Line](<https://www.sota2.com/research/paper/68fa59671e209131ce9b6864>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T15:52:13.471Z

Authors: Athanasios Sourmelidis, J\\"orn Steuding, Ade Irma Suriajaya

Published: 2020-07-28T06:24:28.000Z

## 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 $L(s;f)$, resp. their analytic continuation in the neighborhood of the critical line (which is the abscissa of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number $a\\neq 0$, we prove 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-value 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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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2007.14008>)

[PDF](<https://arxiv.org/pdf/2007.14008.pdf>)