  W. Luo has investigated the distribution of zeros of the derivative of the
Selberg zeta function associated to compact hyperbolic Riemann surfaces. In
essence, 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 the
critical line; an asymptotic expansion for the counting function measuring the
vertical distribution of zeros; and an asymptotic expansion for the counting
function measuring the horizontal distance of zeros from the critical line. In
the present article, we study the more complicated setting of distribution of
zeros of the derivative of the Selberg zeta function associated to a
non-compact, finite volume hyperbolic Riemann surface. There are numerous
difficulties which exist in the non-compact case that are not present in the
compact setting, beginning with the fact that in the non-compact case the
Selberg 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, where
Z is the Selberg zeta function and H is the Dirichlet series component of the
scattering matrix, both associated to an arbitrary finite-volume hyperbolic
Riemann surface. Our main results address finiteness of zeros in the half plane
to the left of the critical line, an asymptotic count for the vertical
distribution of zeros, and an asymptotic count for the horizontal distance of
zeros.
