Selberg Zeta Functions Have Second Moment At $σ= 1$
Ramūnas Garunkštis, Jokūbas Putrius
TL;DR
This paper proves the unconditional existence of the second moment of the Selberg zeta function $Z(s)$ at $\sigma=1$ for Fuchsian groups of the first kind, by leveraging the prime geodesic theorem and a Beurling zeta-function perspective. It shows that $\lim_{T\to\infty}\frac{1}{T}\int_0^T|Z(1+it)|^2 dt=\sum_{n}\frac{b_n^2}{y_n^2}$ and $\lim_{T\to\infty}\frac{1}{T}\int_1^T|Z(1+it)|^{-2} dt=\sum_{n}\frac{c_n^2}{y_n^2}$, where $Z(s)=\sum b_n/y_n^s$ and $1/Z(s)=\sum c_n/y_n^s$, with both series convergent. The authors implement a Beurling zeta-function framework via $Z_1(s)=Z(s)/Z(s+1)$ and extend the second-moment results to Beurling generalized number systems and to general Dirichlet series with positive coefficients, even without a separation condition on the underlying sequence. A key methodological advance is adapting Broucke and Hilberdink’s approach to handle second moments without separation, and decomposing $Z(s)$ into $Z_1(s)Z_2(s)$ to reduce to tractable moment problems for $Z_1$. The results broaden the scope of second-moment phenomena from classical zeta to Beurling and general Dirichlet contexts, with potential implications for spectral theory and analytic number theory.
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 $σ= 1$. 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.