Second moments of general Dirichlet series and Selberg zeta functions at $\sigma =1$

Abstract

In this paper, we prove the existence of the second moment at $\sigma =1$ for a general Dirichlet series with positive coefficients whose partial sum of the coefficients up to $x$ satisfies $\varrho x+O(x/(\log x{)}^{1+\epsilon })$ for some $\epsilon >0$. The result extends a theorem of Broucke and Hilberdink. As applications, we establish the existence of the second moment for Beurling zeta functions satisfying a prime number theorem with a power-saving error term and for the Selberg zeta function for every cofinite Fuchsian group.

arXiv comment14 pages, 0 figures