Square integrals of the logarithmic derivatives of Selberg's zeta functions in the critical strip

TL;DR

This paper investigates the growth of the square integral of the logarithmic derivatives of Selberg's zeta functions within the critical strip for various modular and arithmetic groups, extending previous bounds to broader contexts.

Contribution

It provides new estimates on the square integral growth of the logarithmic derivatives of Selberg's zeta functions for multiple classes of groups, including co-compact arithmetic groups.

Findings

Established bounds for the square integral growth in the critical strip.

Extended previous results to a wider class of groups.

Analyzed the behavior for both modular and quaternionic derived groups.

Abstract

In our previous work (https://doi.org/10.1002/mana.202000268, Math. Nachr., 2021), we proposed an upper bound of the logarithmic derivative of Selberg's zeta function for the modular groups in the critical strip. The present paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups.