On the zeros of the Riemann zeta function of large multiplicity
type: Item
instance of: scholarly article
title: On the zeros of the Riemann zeta function of large multiplicity
zbMATH Open document ID: 1315.11071
DOI: 10.1134/S0001434611050026
author: R. N. Boyarinov
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: The author obtains a new upper bound for the number of zeros of the Riemann zeta function of a given multiplicity lying in a given rectangle of the critical strip thereby refining results of \textit{M. A. Korolev} [Izv. Math. 70, No. 3, 427--446 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 3, 3--22 (2006; Zbl 1187.11027)]. Theorem 3. Suppose that \(\varepsilon\) is an arbitrarily small fixed number, \(0<\varepsilon<0.001\), \(T>T_0(\varepsilon)>0\) and \(H=T^{27/82+\varepsilon}\). Then, for any integer \(j\ge j_0\), the following estimate holds: \[ N_j(T+H)-N_j(T)\le \frac{2\kappa e^{3.1}}{(1-e^{-3})^2} (N(T+H)-N(T)) \exp(-\kappa j),\] where \(\kappa=2\varepsilon\sqrt{\varepsilon}/(3e\sqrt{B})\), \(B=e^{37}\pi^{-2}\), and \(j_0=(3+ 0.1\kappa+2(1.5\varepsilon e)^{2/3})/\kappa\). Theorem 4. For any \(j\ge 3.1/\alpha\) and \(T>T_0>0\), the following estimate holds: \[ N_j(T)\le beta N(T)\exp(-\alpha j),\] where \(\alpha=\frac{\pi\sqrt{5}}{3} 10^{-5}e^{-19.5}\), \(\beta=\frac{2\alpha e^{3.2}}{(1-e^{-3})^2}\).
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 5980809
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: zeros of the Riemann zeta function
zbMATH Keywords: Fujii's inequality
zbMATH Keywords: Lagrange's mean-value theorem
MaRDI profile type: Publication
cites work: On multiple zeros of the Riemann zeta function
cites work: The argument of the Riemann zeta function
cites work: On the distribution of the zeros of the Riemann zeta function in short intervals
full work available at URL: s0001434611050026
OpenAlex ID: W2005425224
description: scientific article; zbMATH DE number 5980809