Generalisation of the Beurling-Nyman criterion for the Riemann Hypothesis
type: Item
instance of: scholarly article
title: Generalisation of the Beurling-Nyman criterion for the Riemann Hypothesis
zbMATH Open document ID: 1063.11029
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2005-02-24T00:00:00Z
review text: The Beurling-Nyman criterion for the Riemann \(\zeta\)-function is generalized to a class of Dirichlet series. Suppose \(F(s)=\sum_{n=1}^{\infty}{a_{n}\over n^s }\), with \(a_{n}\ll n^{\varepsilon}\) for every \(\varepsilon >0\), is a Dirichlet series which is absolutely convergent for \(\Re s > 1\), and which admits a meromorphic continuation to \(\Re s \geq {1\over 2}\) with a unique pole of finite order \(m_{F}\) at \(s=1\). Let \(\Psi_{F}: \mathbb{R}^{+} \to \mathbb{C}\), \[ \Psi_{F}(x) = \text{Res} \biggl({x^{s}\over s}F(s),1\biggr)- \sum_{n\leq x}a_{n}, \] and \(\Psi_{F}^{(1)}(x)=\Psi_{F}(1/x)\). Let \(\widetilde{B_{F}}\) be the space of functions such that \[ f(t)=\sum_{k=1}^{n}c_{k}\Psi_{F} \biggl({\alpha_{k}\over t}\biggr), \] \( c_{k}\in \mathbb{C}\), where \(\alpha_{k}\in (0,1]\) satisfy \(\forall \ell \in [0,m_{F}-1],\, \sum_{k=1}^{n}c_{k}\alpha_{k}(\ln \alpha_{k})^{\ell} = 0\). The main result announced in this paper is as follows. Let \(F\) be such that \(\Psi_{F}^{(1)} \in L^{2}(0,\infty)\). Then the following statements are equivalent: (i) \(F\) does not vanish in \(\Re s > 1/2\); (ii) \(\widetilde{B_{F}}\) is dense in \(L^{2}(0,1)\); (iii) The characteristic function of the interval \((0,1]\) is in the closure of \(\widetilde{B_{F}}\) in \(L^{2}(0,1)\). The main result allows the formulation of similar conditions for the validity of the Generalized Riemann Hypothesis for functions in the Selberg class. The proofs are due to appear in forthcoming articles by the author.
reviewed by: C. Y. Yildirim
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 30B50
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 2138103
zbMATH Keywords: Beurling-Nyman criterion
zbMATH Keywords: generalized Riemann hypothesis
author: Anne de Roton
MaRDI profile type: Publication
full work available at URL: j.crma.2004.11.023
OpenAlex ID: W2071760167
cites work: The Nyman-Beurling equivalent form for the Riemann hypothesis
cites work: A CLOSURE PROBLEM RELATED TO THE RIEMANN ZETA-FUNCTION
cites work: Q5796220
cites work: On the mean square of the error term for an extended Selberg class
cites work: Généralisation du critère de Beurling-Nyman pour l’hypothèse de Riemann
cites work: Q3267555
cites work: Q5803461
DOI: 10.1016/J.CRMA.2004.11.023
description: scientific article; zbMATH DE number 2138103