Parabolic components of zeta functions (Q1101499)

Let \(\Gamma\) be a cofinite subgroup of \(PSL_ 2({\mathbb{R}})\). The Selberg zeta-function for \(\Gamma\) decomposes as a product \[ Z(s)=Z_{id}(s) Z_{ell}(s) Z_{par}(s) Z_{hyp}(s), \] where the various factors are associated with the identity and the conjugacy classes of elliptic, parabolic and hyperbolic elements of \(\Gamma\), respectively [see \textit{M.- F. Vignéras}, Astérisque 61, 235-249 (1979; Zbl 0401.10036) and \textit{J. Fischer}, An approach to the Selberg trace formula via the Selberg zeta-function. Lect. Notes Math. 1253, 184 pp. (1987; Zbl 0618.10029)]. Then the functional equation simply is \(Z(s)=Z(1-s).\) The function \(Z_{id}\) is known in terms of the Barnes double gamma function. For \(\Gamma =PSL_ 2({\mathbb{Z}})\) the functions \(Z_{ell}\) and \(Z_{par}\) have simple expressions in terms of the gamma function and the Riemann zeta-function. The author also computes the determinant of the Laplacian in terms of Z(s) and obtains a simple formula for \(Z'_{hyp}(1)\) (for \(\Gamma =PSL_ 2({\mathbb{Z}}))\). Moreover he proves (for \(\Gamma =PSL_ 2({\mathbb{Z}}))\) that the spectral zeta-function \(\zeta (s,\Delta)=\sum^{\infty}_{n=1}\lambda_ n^{-s}\) has a meromorphic continuation to the entire complex plane, and he determines precisely the asymptotic behaviour of the function \(\sum^{\infty}_{n=1}e^{- \lambda_ nt}\) as \(t\to +0\). The author indicates that similar results hold for some higher rank non-compact locally symmetric spaces, and he gives a number-theoretic analogue of the determinant relation.