On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function
Confluentes Mathematici, Tome 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), k1. 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 (𝓀) distributions, and random variables Z k,n , 1kn, 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.

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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
@article{CML_2021__13_1_43_0,
     author = {S\'ebastien Darses and Erwan Hillion},
     title = {On probabilistic generalizations of the {Nyman-Beurling} criterion for the zeta function},
     journal = {Confluentes Mathematici},
     pages = {43--59},
     publisher = {Institut Camille Jordan},
     volume = {13},
     number = {1},
     year = {2021},
     doi = {10.5802/cml.71},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.71/}
}
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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