Note on the Riemann \(\zeta\)-function. III
type: Item
instance of: scholarly article
title: Note on the Riemann \(\zeta\)-function. III
zbMATH Open document ID: 1008.11032
author: Luis Báez-Duarte
author: Michel Balazard
author: Bernard Landreau
author: Éric Saïas
published in: Advances in Mathematics
publication date: 2000-05-11T00:00:00Z
review text: In this continuation on the work on the zeros of the Riemann zeta-function [Part I, Adv. Math. 139, 310-321 (1998; Zbl 0920.11062) and ibid. 143, 284-287 (1999; Zbl 0937.11042)] the authors investigate the function \(D(\lambda)\), the distance in \(\mathcal H\) between \(\chi\) and \({\mathcal B}_\lambda\), the subspace of \(\mathcal B\) of functions \(f\) such that \(\theta_k \geq \lambda\). Here \(\mathcal H\) is the Hilbert space \(L^2(0, \infty)\), \(\mathcal B\) the subspace of \(\mathcal H\) of functions \(f(t) = \sum_{k=1}^n c_k\rho(\theta_k/t)\), where \(\rho(x)\) denotes the fractional part of \(x\), the \(c_k\)'s are complex numbers, and \(\chi\) is the characteristic function of \((0,1]\). A classic theorem of Beurling and Nyman asserts that the Riemann Hypothesis (RH; all complex zeros of \(\zeta(s)\) have real parts equal to \(1\over 2\)) is equivalent to the statement that \(\lim_{\lambda\to 0} D(\lambda) = 0\). Here the authors prove that \(\liminf_{\lambda\to 0} D(\lambda)\sqrt{\log(1/\lambda)} > 0\), implying that \(D(\lambda) \gg 1/\sqrt{\log(2/\lambda)}\) for \(0 < \lambda \leq 1\). Moreover, they conjecture (this is stronger than RH) that \[ \liminf_{\lambda\to 0}D(\lambda)\sqrt{\log(1/\lambda)} = \sqrt{2+\gamma - \log(4\pi)}, \] where \(\gamma = -\Gamma'(1)\) is Euler's constant. The constant on the right-hand side appears as \[ \sum_\beta {2\Re \beta\over|\beta|^2} = 2 + \gamma - \log(4\pi), \] where summation is over complex zeros \(\beta\) of \(\zeta(s)\) with real parts equal to \(1\over 2\) such that each zero is counted only once, regardless of its possible multiplicity.
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 1422992
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Beurling-Nyman theorem
zbMATH Keywords: Riemann hypothesis
MaRDI profile type: Publication
full work available at URL: aima.1999.1861
OpenAlex ID: W4210343647
cites work: On Beurling's real variable reformulation of the Riemann Hypothesis
cites work: A class of invariant unitary operators
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cites work: Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann \(\zeta\)-function
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DOI: 10.1006/AIMA.1999.1861
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description: scientific article; zbMATH DE number 1422992