Bounds for special values of Selberg zeta functions of Riemann surfaces (Q2757778)

The authors give lower and upper bounds for the constant term \(c_M\) of the logarithmic derivative of the Selberg zeta function \(Z_M(s)\) at \(s=1\) for Riemann surfaces \(M= \Gamma\setminus\mathbb{H}\), where \(\Gamma \subseteq \text{PSL}_2(\mathbb{R})\) is any Fuchsian group of the first kind. The lower bound for \(c_M\) is essentially of the form \(O(\log \operatorname {vol}(M))\) with \(\operatorname {vol}(M)\) denoting the hyperbolic volume of \(M\). The upper bounds are of a more complicated nature; they involve the length spectrum of \(M\) and the small eigenvalues of the hyperbolic Laplacian. If \(M\) is a finite cover of a fixed base space \(M_0\) and \(\Gamma\) is an arithmetic subgroup, the upper bound for \(c_M\) can be simplified to \(O_{M_0} (\operatorname {vol}(M)^{1+\varepsilon})\) for any \(\varepsilon>0\). In particular, for the congruence subgroups \(\Gamma_0(N)\) with square-free \(N\), the upper bound can be improved to \(O(N^\varepsilon)\).