ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$ . Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$ , where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$ . Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)