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

de Roton, Anne

doi:10.1090/S0002-9947-07-04261-4

Généralisation du critère de Beurling-Nyman pour l’hypothèse de Riemann
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Trans. Amer. Math. Soc. 359 (2007), 6111-6126 Request permission

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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