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On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function
Sébastien Darses1; Erwan Hillion1
1 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France
Confluentes Mathematici, Volume 13 (2021) no. 1, pp. 43-59.

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on (0,∞), which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors θ k ∈(0,1), k≥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≤k≤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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/cml.71
Classification: 41A30, 46E20, 60E05, 11M26
Keywords: Number theory; Probability; Zeta function; Nyman-Beurling criterion; Báez-Duarte criterion

Sébastien Darses 1; Erwan Hillion 1

1 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Sébastien Darses; Erwan Hillion. On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function. Confluentes Mathematici, Volume 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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