Euler products of Selberg zeta functions in the critical strip

TL;DR

This paper extends the convergence region of Euler products of Selberg zeta functions for congruence subgroups, linking it to eigenvalues of the Laplacian and prime geodesic theorem error estimates.

Contribution

It establishes new regions of convergence for Euler products of Selberg zeta functions based on eigenvalues, and relates partial Euler product behavior to prime geodesic theorem errors.

Findings

Region of convergence extends beyond Re s=1 for nontrivial representations.

Convergence region reaches Re s ≥ 3/4 under Selberg's eigenvalue conjecture.

Connection between partial Euler products and prime geodesic theorem error term.

Abstract

For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary Re s = 1, if they are attached with a nontrivial irreducible unitary representation. The region is determined by the size of the lowest eigenvalue of the Laplacian, and it extends to Re s $\geqslant$ 3/4 under Selberg's eigenvalue conjecture. More generally, for any unitary representation we establish the relation between the behavior of partial Euler products in the critical strip and the estimate of the error term in the prime geodesic theorem. For the trivial representation, the proof essentially exploits the idea of the celebrated work of Ramanujan.