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

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.