On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function

Sébastien Darses; Erwan Hillion

doi:10.5802/cml.71

Sébastien Darses ; Erwan Hillion

Confluentes Mathematici, Tome 13 (2021) no. 1, pp. 43-59.

Résumé

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on $(0, \infty)$ , which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors $θ_{k} \in (0, 1)$ , $k \geq 1$ . We develop probabilistic extensions of the Nyman-Beurling criterion by considering these $θ_{k}$ as random: this yields new structures and criteria, one of them having a significant overlap with the general strong Báez-Duarte criterion.

The main goal of the present paper is the study of the interplay between these probabilistic Nyman-Beurling criteria and the Riemann hypothesis. We are able to obtain equivalences in two main classes of examples: dilated structures as exponential $ℰ (𝓀)$ distributions, and random variables $Z_{k, n}$ , $1 \leq k \leq n$ , concentrated around $1 / k$ as $n$ is growing. By means of our probabilistic point of view, we bring an answer to a question raised by Báez-Duarte in 2005: the price to pay to consider non compactly supported kernels is a controlled condition on the coefficients of the involved approximations.

Reçu le : 2019-12-18
Révisé le : 2021-05-14
Accepté le : 2021-05-25
Publié le : 2021-10-20

DOI : https://doi.org/10.5802/cml.71
Classification : 41A30, 46E20, 60E05, 11M26
Mots clés : Number theory; Probability; Zeta function; Nyman-Beurling criterion; Báez-Duarte criterion

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Sébastien Darses; Erwan Hillion. On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function. Confluentes Mathematici, Tome 13 (2021) no. 1, pp. 43-59. doi : 10.5802/cml.71. https://cml.centre-mersenne.org/articles/10.5802/cml.71/

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