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), 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≠ 0, we prove for an even or odd periodic f the number of a-points of the -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.