Universality theorems of the Selberg zeta functions for arithmetic groups

Abstract

After Voronin proved the universality theorem of the Riemann zeta function in the 1970s, universality theorems have been proposed for various zeta and L-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universality theorem of the Selberg zeta function for the modular group is one of them and is probably the first universality theorem of the zeta function of order greater than one. Recently, Mishou (2021) extended it by proving the joint universality theorem for the principal congruence subgroups. In the present paper, we further extend these works by proving the (joint) universality theorem for subgroups of the modular group and co-compact arithmetic groups derived from indefinite quaternion algebras, which is available in the region wider than the regions in the previous two works.