Hierarchy of the Selberg zeta functions

TL;DR

This paper introduces a two-variable Selberg zeta function that unifies several higher Selberg zeta functions, providing new insights into its analytic properties and connections to the Laplacian on Riemann surfaces.

Contribution

It presents a novel two-variable Selberg zeta function with analytic continuation, functional equation, and determinant expression related to the Laplacian, extending the theory of Selberg zeta functions.

Findings

Established the analytic continuation of the new zeta function

Derived the functional equation for the two-variable zeta function

Expressed the zeta function as a determinant involving the Laplacian

Abstract

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.