Note on zeros of the derivative of the Selberg zeta-function
In [Am. J. Math. 127, No. 5, 1141--1151 (2005; Zbl 1109.11040)] \textit{W. Luo} gave an asymptotic formula for the distribution of non-trivial zeros of the derivative of the Selberg zeta function \(Z_X\) attached to a hyperbolic surface \(X\). Then he related the distribution of zeros to the multiplicities of Laplace-eigenvalues. His calculations were based on the estimate \[ Z_X(\sigma +it) \ll \exp\left(\frac 32\,\text{area}(X)t\right) \] for \(\sigma\geq -1\). In the paper [On the Bäcklund equivalent for the Lindelöf hypothesis. Probability and number theory -- Kanazawa 2005. Proceedings of the international conference on probability and number theory, Kanazawa, Japan, June 20--24, 2005. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 49, 91--104 (2007; Zbl 1228.11134)] the present author used the Riemann hypothesis for \(Z_X\) to show \[ Z_X\left(\frac12+it\right) \ll \exp(\varepsilon t) \] for every \(\varepsilon >0\). In the present paper he uses this estimate to improve the error terms in Luo's results. Editorial note: For a corrigendum to this article see Arch. Math. 93, No. 2, 143--145 (2009; Zbl 1251.11061).
scientific article; zbMATH DE number 5500572