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An improved lower bound for the de Bruijn-Newman constant


Authors: Yannick Saouter, Xavier Gourdon and Patrick Demichel
Journal: Math. Comp. 80 (2011), 2281-2287
MSC (2010): Primary 11-04, 11M26, 11Y35, 11Y60
DOI: https://doi.org/10.1090/S0025-5718-2011-02472-5
Published electronically: March 9, 2011
MathSciNet review: 2813360
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Abstract: In this article, we report on computations that led to the discovery of a new Lehmer pair of zeros for the Riemann $ \zeta$ function. Given this new close pair of zeros, we improve the known lower bound for de Bruijn-Newman constant $ \Lambda$. The Riemann hypothesis is equivalent to the assertion $ \Lambda \leq 0$. In this article, we establish that in fact we have $ \Lambda > -1.14541 \times 10^{-11}$. This new bound confirms the belief that if the Riemann hypothesis is true, it is barely true.


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Additional Information

Yannick Saouter
Affiliation: Institut Telecom Brest, Bretagne, France
Email: Yannick.Saouter@telecom-bretagne.eu

Xavier Gourdon
Affiliation: Dassault Systemes, Velizy-Villacoublay, France
Email: xgourdon@gmail.com

Patrick Demichel
Affiliation: Hewlett-Packard France, Les Ulis, France
Email: patrick.demichel@hp.com

DOI: https://doi.org/10.1090/S0025-5718-2011-02472-5
Received by editor(s): May 7, 2009
Received by editor(s) in revised form: August 6, 2010
Published electronically: March 9, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.