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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-07-04261-4
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.


References [Enhancements On Off] (What's this?)

  • [BD] L. Báez-Duarte, A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis, Rendiconti Accad. Lincei 23 (2003), 5-11. MR 2057270 (2005b:11135)
  • [BS] M. Balazard, E. Saias, The Nyman-Beurling equivalent form for the Riemann hypothesis, Expo. Math. 18 (2000), 131-138. MR 1759315 (2001i:11109)
  • [BF] H. Bercovici, C. Foias, A real variable restatement of Riemann's hypothesis, Israel J. of Maths. 48 (1984), 57-68. MR 768266 (86d:11065)
  • [Be] A. Beurling, A closure problem related to the Riemann Zeta-function, Proc. Nat. Acad. Sci. 41 (1955), 312-314. MR 0070655 (17:15a)
  • [CG1] J. B. Conrey, A. Ghosh, Remarks on the generalized Lindelöf Hypothesis, Preprint.
  • [HIP] G. H. Hardy, A. E. Ingham, G. Pólya, Theorems concerning mean values of analytic functions, Proc. of the Royal Soc. 113 (1927), 542-569.
  • [KP0] J. Kaczorowski, A. Perelli, The Selberg class: a survey, Acta Math. 192 (1999), 953-992. MR 1689554 (2001g:11141)
  • [Lev] N. Levinson, Gap and Density Theorems, A.M.S. Coll. Publ. XXVI, 1940. MR 0003208 (2:180d)
  • [PW] R. E. A. C. Paley, N. Wiener, The Fourier Transform in the Complex Domain, A.M.S. Coll. Publ. XIX, 1934.
  • [dR] A. de Roton, On the mean square of an error term for an extended Selberg's class, Acta Arith. 126 (2007), no. 1, 27-55. MR 2284311
  • [Sch] L. Schwartz, Étude des sommes d'exponentielles, Hermann, Paris, 1959. MR 0106383 (21:5116)
  • [Sel] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Almafi Conference on Analytic Number Theory, Università di Salerno (1992). MR 1220477 (94f:11085)
  • [Sz] O. Szász, Über die Approximation stetiger Funktionnen durch lineare Aggregate von Potenzen, Math. Ann. 77 (1916), 482-496. MR 1511875
  • [Te] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Société mathématique de France, 1995. MR 1366197 (97e:11005a)
  • [T1] E. C. Titchmarsh, The theory of functions, Second edition, Oxford University Press, 1939. MR 882550 (88c:11049)
  • [T2] E. C. Titchmarsh, The theory of the Riemann Zeta-function, Oxford University Press, 1951. MR 0046485 (13:741c)

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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: https://doi.org/10.1090/S0002-9947-07-04261-4
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.

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