A lower bound for the de Bruijn-Newman constant \(\Lambda\)
type: Item
instance of: scholarly article
title: A lower bound for the de Bruijn-Newman constant \(\Lambda\)
zbMATH Open document ID: 0663.65017
DOI: 10.1007/BF01400887
author: Richard S. Varga
author: George Csordas
author: Timothy S. Norfolk
published in: Numerische Mathematik
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133248
review text: Let \[ H_{\lambda}(x):=\int^{\infty}_{0}e^{\lambda t^ 2}\Phi (t)\cos (xt)\,dt\quad (\lambda \in \mathbb{R}), \] where \[ \Phi (t):=\sum^{\infty}_{n=1}(2n^ 4\pi^ 2e^{9t}-3n^ 2\pi e^{5t})\exp (-n^ 2\pi e^{4t})\quad (0\le t<\infty). \] For any real \(\lambda\), the function \(H_{\lambda}(x)\) is an entire function of order one, which is real fo real x. \textit{N. G. de Bruijn} [Duke Math. J. 17, 197--226 (1950; Zbl 0038.23302)] has shown that (i) \(H_{\lambda}(x)\) has only real zeros for \(\lambda\geq 1/2\), and (ii) if \(H_{\lambda}(x)\) has only real zeros for some real \(\lambda\), then \(H_{\mu}(x)\) also has only real zeros for any \(\mu >\lambda\). \textit{M. Newman} [Proc. Am. Math. Soc. 61, 245--251 (1976; Zbl 0342.42007)] proved that there exists a smallest value of \(\lambda\) for which \(H_{\lambda}(x)\) has only real zeros, in the following sense: there exists a real number \(\Lambda\) such that \(H_{\lambda}(x)\) has only real zeros for \(\lambda\geq \Lambda\), whereas \(H_{\lambda}(x)\) has some nonreal zero for any \(\lambda <\Lambda\). De Bruijn's result i) implies that \(\Lambda\leq 1/2\). The truth of the Riemann hypothesis would imply that \(\Lambda\leq 0\). Newmann (loc. cit.) conjectures that \(\Lambda\geq 0\). In the paper under review \(\Lambda\) is baptized the de Bruijn-Newman constant. (Not only) Because of the relation of \(\Lambda\) with the Riemann hypothesis, it would be interesting to find a lower bound for \(\Lambda\), which is as large as possible. The main purpose of the present paper is to present a constructive method to find such a lower bound. The method is rather involved, and too complicated to be described here. It requires high-precision numerical calculations (with an accuracy of up to 110 decimal digits) on a VAX- 11/780 combined with precise error estimates, to yield the result \(\Lambda >-50\). The authors expect that this bound can be substantially increased at the expense of higher precision calculations and more compute…
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 30D10
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11Y60
zbMATH DE Number: 4084837
zbMATH Keywords: zeros of polynomial
zbMATH Keywords: de Bruijn-Newman constant
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: lower bound
zbMATH Keywords: high-precision numerical calculations
zbMATH Keywords: error estimates
describes a project that uses: Algorithm 524
describes a project that uses: Algorithm 493
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4084837