The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on , which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors , . We develop probabilistic extensions of the Nyman-Beurling criterion by considering these 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 distributions, and random variables , , concentrated around as 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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@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/}
}
TY - JOUR AU - Sébastien Darses AU - Erwan Hillion TI - On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function JO - Confluentes Mathematici PY - 2021 SP - 43 EP - 59 VL - 13 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.71/ DO - 10.5802/cml.71 LA - en ID - CML_2021__13_1_43_0 ER -
%0 Journal Article %A Sébastien Darses %A Erwan Hillion %T On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function %J Confluentes Mathematici %D 2021 %P 43-59 %V 13 %N 1 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.71/ %R 10.5802/cml.71 %G en %F CML_2021__13_1_43_0
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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