no. 7-8
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

Alcántara-Bode, Julio1
1 Pontificia Universidad Católica del Perú and Instituto de Matemática y Ciencias Afines, Lima, Peru
Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 357-359.

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)=∫01l(x)f(x)dx,∀f∈R.
Chaque zéro non trivial de la fonction zêta de Riemann détermine une densité de Radon–Nikodym différente l∈L1([0,1]). Lʼhypothèse de Riemann est vérifiée si et seulement si aucune de ces densités appartient à L2([0,1]), ou si et seulement si R est dense dans lʼespace L2([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)=∫01l(x)f(x)dx,∀f∈R.
Each non-trivial zero of the Riemann zeta function determines a different Radon–Nikodym density l∈L1([0,1]). The Riemann Hypothesis holds if and only if none of these densities belongs to L2([0,1]) or if and only if R is dense in L2([0,1]).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.03.002
Alcántara-Bode, Julio 1

1 Pontificia Universidad Católica del Perú and Instituto de Matemática y Ciencias Afines, Lima, Peru 
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Alcántara-Bode, Julio. Absolutely continuous restrictions of a Dirac measure and non-trivial zeros of the Riemann zeta function. Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 357-359. doi : 10.1016/j.crma.2011.03.002. http://www.numdam.org/articles/10.1016/j.crma.2011.03.002/

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