Note on zeros of the derivative of the Selberg zeta-function
type: Item
instance of: scholarly article
title: Note on zeros of the derivative of the Selberg zeta-function
zbMATH Open document ID: 1162.11046
author: Ramūnas Garunkštis
published in: Archiv der Mathematik
publication date: 2009-01-29T00:00:00Z
review text: 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).
reviewed by: Anton Deitmar
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 5500572
zbMATH Keywords: Selberg zeta-function
zbMATH Keywords: zero distribution
zbMATH Keywords: Lindelöf hypothesis
MaRDI profile type: Publication
full work available at URL: s00013-008-2768-8
OpenAlex ID: W2168391453
DOI: 10.1007/S00013-008-2768-8
Recommended article: On zeros of the derivative of the Selberg zeta function
Recommended article: Q3633569
Recommended article: Zeros of the extended Selberg class zeta-functions and of their derivatives
Recommended article: The zero-free region of the derivative of Selberg zeta functions
Recommended article: On zeros of the derivative of the three-dimensional Selberg zeta-function
Recommended article: Zero-free regions for derivatives of the Selberg zeta-function
Recommended article: Zeros of derivatives of 𝐿-functions in the Selberg class on ℜ(𝑠)<1/2
Recommended article: Zeros of the higher-order derivatives of the functions belonging to the extended Selberg class
Recommended article: Q3710593
Recommended article: A note on S(t ) and the zeros of the Riemann zeta-function
description: scientific article; zbMATH DE number 5500572