A Carleman-Nevanlinna theorem and summation of the Riemann zeta-function logarithm
type: Item
instance of: scholarly article
title: A Carleman-Nevanlinna theorem and summation of the Riemann zeta-function logarithm
zbMATH Open document ID: 1157.11313
DOI: 10.1007/BF03321076
published in: Computational Methods and Function Theory
publication date: 2005-07-06T00:00:00Z
review text: In this paper the authors proves a Carleman-Nevanlinna theorem for a rectangle. This result is is applied to the summation of \(\log| \zeta(s)| \) on the critical and other vertical lines, where \(\zeta(s)\) means the Riemann zeta-function. In particular, let \[ I(\varepsilon)=\int_0^\infty e^{-\varepsilon t}\log\left| \zeta\left(\frac{1}{2}+it\right) \right|\, dt,\, \varepsilon>0, \] and let \(\{\rho_j\}\) be the non-trivial zeros of \(\zeta(s)\). Then \[ \frac{\pi}{2}\sum_{j}\left| \Re \rho_j-\frac{1}{2}\right| =I(+0)+\frac{\pi}{2}, \] where \(I(+0):=\lim_{\varepsilon\to 0}I(\varepsilon)\). Hence, the Riemann hypothesis for \(\zeta(s)\) holds if and only if \(I(+0)=-\pi/2\).
reviewed by: Mirela Kohr
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 2186786
zbMATH Keywords: Carleman-Nevanlinna theorem
zbMATH Keywords: Euler product
zbMATH Keywords: meromorphic function
zbMATH Keywords: summation
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Riemann hypothesis
author: Andrij Kondratyuk
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03321076
OpenAlex ID: W1973987283
description: scientific article; zbMATH DE number 2186786