Zeros of the Selberg zeta function for symmetric infinite area hyperbolic surfaces

TL;DR

This paper provides a mathematical framework for understanding the zeros of Selberg zeta functions on symmetric infinite area hyperbolic surfaces, especially three-funneled surfaces, explaining empirical zero distributions through complex almost periodic functions.

Contribution

It introduces a simple mathematical foundation for the zeros of Selberg zeta functions on symmetric infinite area surfaces, with a focus on three-funneled cases, and explains empirical zero patterns via convergence to standard curves.

Findings

Selberg zeta function is a complex almost periodic function.

Zeta function can be approximated by complex trigonometric polynomials.

Zeros converge to standard curves under affine scaling.

Abstract

In the present paper we give a simple mathematical foundation for describing the zeros of the Selberg zeta functions $Z_X$ for certain very symmetric infinite area surfaces $X$. For definiteness, we consider the case of three funneled surfaces. We show that the zeta function is a complex almost periodic function which can be approximated by complex trigonometric polynomials on large domains (in Theorem 4.2). As our main application, we provide an explanation of the striking empirical results of Borthwick (arXiv:1305.4850) (in Theorem 1.5) in terms of convergence of the affinely scaled zero sets to standard curves $\mathcal C$.