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Abstract

We study the zeros of the derivative of the unmodified Selberg zeta-function associated to a finite volume Riemann surface. The first main result is that the derivative has approximately the same number of non-trivial zeros as the function itself in the region of the complex plane bounded by the horizontal lines $t = 0$ and $t = T$ for any $T \gt 0$. We also obtain an asymptotic formula for the number of non-trivial zeros of the derivative of the Selberg zeta-function to the left of the critical line $\sigma = 1/2$ where $s = \sigma + it \in \mathbb {C}$ in the case of the modular group and its congruence subgroups.