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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Généralisation du critère de Beurling-Nyman pour l'hypothèse de Riemann


Author: Anne de Roton
Journal: Trans. Amer. Math. Soc. 359 (2007), 6111-6126
MSC (2000): Primary 37A45
Published electronically: June 4, 2007
MathSciNet review: 2336318
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Abstract: Nous généralisons dans cet article le critère de Beurling-Nyman, qui concerne la fonction $ \zeta$ de Riemann, à une large classe de séries de Dirichlet. Nous établissons donc une correspondance entre la densité d'un certain sous-espace de fonctions dans $ L^2(0,1)$ et la localisation des zéros d'une série de Dirichlet. Nous utilisons pour obtenir ce résultat la structure de l'espace de Hardy du demi-plan.


ABSTRACT. We generalise Beurling-Nyman's criterion, already known for the Riemann $ \zeta$ function, to a larger class of Dirichlet series. We reveal a link between the density of some subspace of functions in $ L^2(0,1)$ and the localization of the zeros of a Dirichlet series. To do so, we use the structure of the Hardy space of the half-plan.


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Additional Information

Anne de Roton
Affiliation: IECN - Université Henri Poincaré: Nancy 1, BP 239, 54506 Vandoeuvre-lès-Nancy, France
Email: deroton@iecn.u-nancy.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04261-4
PII: S 0002-9947(07)04261-4
Keywords: S\'eries de Dirichlet, crit\`ere de Beurling-Nyman, hypoth\`ese de Riemann g\'en\'eralis\'ee, classe de Selberg, espace de Hardy
Received by editor(s): September 2, 2004
Received by editor(s) in revised form: January 6, 2006
Published electronically: June 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.