Perturbation of zeros of the Selberg zeta-function for $\Gamma_0(4)$

TL;DR

This paper investigates how the zeros of the Selberg zeta-function for (4) behave asymptotically when a family of characters approaches the trivial character, supported by both numerical and theoretical analysis.

Contribution

It provides rigorous theorems describing the zero perturbation phenomena of the Selberg zeta-function for (4) as characters vary, complementing numerical observations.

Findings

Zeros exhibit specific asymptotic behavior as characters tend to trivial

Theoretical results confirm numerical patterns of zero perturbation

Comparison between proven theorems and numerical data enhances understanding

Abstract

We study the asymptotic behavior of zeros of the Selberg zeta-function for the congruence subgroup $\Gamma_0(4)$ as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems that we prove and compare with the original numerical results.