Turing's method for the Selberg zeta-function

TL;DR

This paper adapts Turing's method, originally for the Riemann zeta-function, to Selberg zeta-functions, demonstrating its rigorous application in the context of the modular surface.

Contribution

It extends Turing's method to Selberg zeta-functions, providing a rigorous framework for verifying zeros in this setting.

Findings

Successfully adapted Turing's method to Selberg zeta-functions

Proved the method's rigor in the modular surface case

Provides a foundation for zero verification in automorphic forms

Abstract

In one of his final research papers, Alan Turing introduced a method to certify the completeness of a purported list of zeros of the Riemann zeta-function. In this paper we consider Turing's method in the analogous setting of Selberg zeta-functions, and we demonstrate that it can be carried out rigorously in the prototypical case of the modular surface.