Selberg zeta function

Abstract: The functional determinant of an eigenvalue sequence, as defined by zeta regularization, can be simply evaluated by quadratures. We apply this procedure to the Selberg trace formula for a compact Riemann surface to find a factorization of the Selberg zeta function into two functional determinants, respectively related to the Laplacian on the compact surface itself, and on the sphere. We also apply our formalism to various explicit eigenvalue sequences, reproducing in a simpler way classical results about the gamma function and the BarnesG-function. Concerning the latter, our method explains its connection to the Selberg zeta function and evaluates the related Glaisher-Kinkelin constantA.

Abstract: The solution of a stationary k-dimensional Schrodinger equation H Psi =E Psi in the semiclassical limit h(cross) to 0 is reduced to a discrete (k-1)-dimensional quantum map psi '=T psi where the integral kernel (the matrix) T is built through classical trajectories corresponding to the classical Poincare map of the given problem. High-excited energy eigenvalues obey the quantization condition zeta s(E)=0 where the function zeta s(E)=det(1-T) coincides with the Selberg zeta function defined as the product over primitive periodic orbits. Different properties of the constructed Poincare map are discussed, in particular the Riemann-Siegel relation for the dynamical zeta function.

Abstract: Determinants of Laplacians on tensors and spinors of arbitrary weights on compact hyperbolic Riemann surfaces are computed in terms of values of Selberg zeta functions at half integer points.

Abstract: LetZ(s, R) be the Selberg zeta function of a compact Riemann surfaceR. We study the behavior ofZ(s, R) asR tends to infinity in the moduli space of stable curves. The main result is an estimate forZ(s, R) valid fors in a neighborhood, depending only on the genus, ofs=1. Our analysis gives an alternate proof of the Belavin-Knizhnik double pole result, [5].