On universality and simple a-points of a family of Dirichlet L-functions and their derivatives

Abstract

In 1975, Voronin established the celebrated universality theorem for the Riemann zeta-function, and later, Bagchi and Gonek independently obtained its analogue for a certain family of Dirichlet L-functions in the q-aspect. In this paper, we generalize their results by establishing a simultaneous universality theorem for the family of Dirichlet L-functions and their derivatives in the q-aspect. Furthermore, as an application of this simultaneous universality, we investigate the value-distribution of these functions. Specifically, we show that for arbitrarily given complex numbers a_j and open disks in the right half of the critical strip, there exists a positive proportion of Dirichlet characters x modulo a large prime q such that the Dirichlet L-function L (s,x) and its derivatives simultaneously have simple a_j -points in the prescribed disks.