Euler's constants for the Selberg and the Dedekind zeta functions
Let as usual \(\zeta(s)\) be the Riemann zeta-function and \(\gamma\) be the Euler constant. The Laurent expansions around \(s=1\) are \[ \zeta(s)=\frac{1}{s-1}+\gamma+\gamma_1(s-1)+\gamma_2(s-1)^2+\dots \] and \[ \frac{\zeta'(s)}{\zeta(s)}=-\frac{1}{s-1}+\gamma +(2\gamma_1-\gamma^2)(s-1)+(3\gamma_2-3\gamma_1\gamma+\gamma^3)(s-1)^2+\dots. \tag{\(1\)} \] B. Riemann and C. J. de la Vallée-Poussin discovered identities \[ \sum_{\rho}\frac1\rho=1+\frac{\gamma}2-\frac12\log\pi-\log2, \tag{\(2\)} \] where \(\rho\) runs over all nontrivial zeros of \(\zeta(s)\), and \[ \gamma=\lim_{x\to\infty}\left(\log x-\sum_{\substack{ p<x\\ p:\text{prime}}} \frac{\log p}{p-1}\right) \tag{\(3\)} \] The Selberg zeta-function \(Z_{\Gamma}(s)\) associated to a discrete co-compact torsion free subgroup \(\Gamma\) of \(\text{SL}(2, \mathbb{R})\) is defined by \[ Z_{\Gamma}(s)= \prod_{P\in\text{Prim}(\Gamma)}\;\prod_{n=0}^\infty (1-N(P)^{-s-n}), \] where \(\text{Prim}(\Gamma)\) is the set of primitive hyperbolic conjugacy classes of \(\Gamma\), and \(N(P)\) denotes the norm of \(P\). The Euler-Selberg constant \(\gamma_{\Gamma}^{(0)}\) and higher Euler-Selberg constants \(\gamma_{\Gamma}^{(n)}\) are given by the following Laurent expansion around \(s=1\), \[ \frac{Z'_{\Gamma}}{Z_{\Gamma}}=\frac{1}{s-1}+\gamma_{\Gamma}^{(0)} +\gamma_{\Gamma}^{(1)}(s-1)+\gamma_{\Gamma}^{(2)}(s-1)^2+\dots. \] In the paper the authors establish similar to \((2)\) and \((3)\) formulas: \[ \sum_{n>0}\frac{1}{(\frac14+r_n^2)^2} =2\gamma_{\Gamma}^{(0)}-\gamma_{\Gamma}^{(1)}+(g-1)\frac{\pi^2}{3}-3, \] where \(\{r_n\}\) are imaginary parts of distinct nontrivial zeros of \(Z_{\Gamma}(s)\) (\(\{\frac14+r_n^2\}\) are the eigenvalues of the Laplacian on the Riemann surface associated to \(\Gamma\)); and \[ \gamma_{\Gamma}^{(0)}=\lim_{x\to\infty}\left(\sum_{N(\gamma)<x} \frac{\log N(P_\gamma)}{N(P_\gamma)-1}-\log x\right), \] where \(\gamma\) denotes a hyperbolic conjugacy class of \(\Gamma\) and \(P_\gamma\in\text{Prim}(\Gamma)\) is a generator of the cyclic group, which includes \(\gamma\). Similar formulas are obtained for higher Euler-Selberg constants \(\gamma_{\Gamma}^{(n)}\) and for Euler type gamma constant for Dedekind zeta-function of an algebraic number field. Further, considering coefficients of the expansion \((1)\), they obtain expressions (similar to \((2)\)) for \(\sum_{\rho}\frac1{\rho^j}\), \(j=2,3\).
scientific article; zbMATH DE number 2186490