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

TL;DR

This paper investigates the value-distribution of Dirichlet series with periodic coefficients near the critical line, revealing properties like uniform distribution of certain points and a universality theorem.

Contribution

It establishes new results on the distribution of $a$-points, mean-values, and universality for Dirichlet series with periodic coefficients near the critical line.

Findings

Number of $a$-points of the $ riangle$-factor is proven.

Existence of mean-values of $L(s;f)$ at these points is shown.

Ordinates of $a$-points are uniformly distributed modulo one.

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.