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

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.