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W. Luo has investigated the distribution of zeros of the derivative of theSelberg zeta function associated to compact hyperbolic Riemann surfaces. Inessence, the main results in Luo's article involve the following three points:Finiteness for the number of zeros in the half plane to the left of thecritical line; an asymptotic expansion for the counting function measuring thevertical distribution of zeros; and an asymptotic expansion for the countingfunction measuring the horizontal distance of zeros from the critical line. Inthe present article, we study the more complicated setting of distribution ofzeros of the derivative of the Selberg zeta function associated to anon-compact, finite volume hyperbolic Riemann surface. There are numerousdifficulties which exist in the non-compact case that are not present in thecompact setting, beginning with the fact that in the non-compact case theSelberg zeta function does not satisfy the analogue of the Riemann hypothesis.To be more specific, we actually study the zeros of the derivative of ZH, whereZ is the Selberg zeta function and H is the Dirichlet series component of thescattering matrix, both associated to an arbitrary finite-volume hyperbolicRiemann surface. Our main results address finiteness of zeros in the half planeto the left of the critical line, an asymptotic count for the verticaldistribution of zeros, and an asymptotic count for the horizontal distance ofzeros.

On the distribution of zeros of the derivative of Selberg's zeta function associated to finite volume Riemann surfaces