Universality theorems of Selberg zeta functions for arithmetic groups

Abstract

We prove a universality theorem for the Selberg zeta function of subgroups of $\mathrm{SL}_2(\mathbb{Z})$ or co-compact arithmetic groups derived from quaternion algebras, in the strip $\{5/6 \lt \mathrm{Re}{s} \lt 1\}$, improving the range compared with a previous work by Drungilas–Garunkštis–Kačenas. We also obtain the same range for a joint universality theorem for congruence subgroups, which improves a result by Mishou.