Milnor-Selberg zeta functions and zeta regularizations

TL;DR

This paper introduces higher depth determinants of the Laplacian on compact Riemann surfaces, expressing them via multiple gamma functions and Milnor-Selberg zeta functions, and establishes their analytic properties including continuation, functional equations, and Euler products.

Contribution

It generalizes the determinant expression of the Selberg zeta function using higher depth determinants and Milnor-Selberg zeta functions, with proofs of their key analytic properties.

Findings

Higher depth determinants expressed as products of multiple gamma functions.

Milnor-Selberg zeta functions admit analytic continuation and functional equations.

Establishment of Euler product representations for these zeta functions.

Abstract

By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor-Selberg zeta function. It is shown that the Milnor-Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.