Selberg Zeta Functions Have Second Moment At $\sigma = 1$

TL;DR

This paper proves the existence of the second moment of the Selberg zeta function at  = 1, extending to Beurling zeta-functions and Dirichlet series, using recent methods to avoid previous limitations.

Contribution

It establishes the second moment of the Selberg zeta function at =1 and extends the result to broader classes of zeta-functions and Dirichlet series with minimal restrictions.

Findings

Second moment of Selberg zeta function at =1 proven.

Extension to Beurling zeta-functions under weak Riemann hypothesis.

Application of Broucke and Hilberdink's approach avoids separation condition.

Abstract

In this paper, we demonstrate the existence of the second moment of the Selberg zeta function for a Fuchsian group of the first kind at $\sigma = 1$. The prime geodesic theorem plays a crucial role in this context. The proof extends to Beurling zeta-functions satisfying a weak form of the Riemann hypothesis and to general Dirichlet series with positive coefficients, the partial sums of which are well-behaved. Note that by employing the recent approach of Broucke and Hilberdink in proving the second moment theorem, we can circumvent the separation condition introduced by Landau for general Dirichlet series.