Absolutely continuous restrictions of a Dirac measure and non-trivial zeros of the Riemann zeta function

Julio Alcántara-Bode

doi:10.1016/j.crma.2011.03.002

Number Theory/Functional Analysis

Absolutely continuous restrictions of a Dirac measure and non-trivial zeros of the Riemann zeta function
[Restrictions absolument continues dʼune mesure de Dirac et zéros non triviaux de la fonction zêta de Riemann]

Présenté par : Jean-Pierre Kahane

Julio Alcántara-Bode ¹

¹ Pontificia Universidad Católica del Perú and Instituto de Matemática y Ciencias Afines, Lima, Peru

Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 357-359.

Résumé
Abstract

Nous montrons que la mesure de Dirac $δ (f) = f (1)$ définie sur lʼespace de Banach $C ([0, 1])$ de fonctions continues à valeurs complexes définies sur lʼintervalle $[0, 1]$ , possède une restriction absolument continue sur un sous-espace de dimension infinie R de $C ([0, 1])$ , cʼest-à-dire

f (1) = \int_{0}^{1} l (x) f (x) d x, \forall f \in R .

Chaque zéro non trivial de la fonction zêta de Riemann détermine une densité de Radon–Nikodym différente

l \in L^{1} ([0, 1])

. Lʼhypothèse de Riemann est vérifiée si et seulement si aucune de ces densités appartient à

L^{2} ([0, 1])

, ou si et seulement si R est dense dans lʼespace

L^{2} ([0, 1])

.

It is shown that the Dirac measure $δ (f) = f (1)$ defined on the Banach space $C ([0, 1])$ of complex valued continuous functions defined on the interval $[0, 1]$ , has an absolutely continuous restriction to an infinite dimensional subspace R of $C ([0, 1])$ , that is

f (1) = \int_{0}^{1} l (x) f (x) d x, \forall f \in R .

Each non-trivial zero of the Riemann zeta function determines a different Radon–Nikodym density

l \in L^{1} ([0, 1])

. The Riemann Hypothesis holds if and only if none of these densities belongs to

L^{2} ([0, 1])

or if and only if R is dense in

L^{2} ([0, 1])

.

Reçu le : 2009-10-02
Accepté le : 2011-03-08
Publié le : 2011-03-31

DOI : 10.1016/j.crma.2011.03.002

Affiliations des auteurs :

Julio Alcántara-Bode ¹

¹ Pontificia Universidad Católica del Perú and Instituto de Matemática y Ciencias Afines, Lima, Peru

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     author = {Julio Alc\'antara-Bode},
     title = {Absolutely continuous restrictions of a {Dirac} measure and non-trivial zeros of the {Riemann} zeta function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {357--359},
     publisher = {Elsevier},
     volume = {349},
     number = {7-8},
     year = {2011},
     doi = {10.1016/j.crma.2011.03.002},
     language = {en},
}

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Julio Alcántara-Bode. Absolutely continuous restrictions of a Dirac measure and non-trivial zeros of the Riemann zeta function. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 357-359. doi : 10.1016/j.crma.2011.03.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.03.002/

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