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.
@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 TI - On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function JO - Confluentes Mathematici PY - 2021 DA - 2021/// SP - 43 EP - 59 VL - 13 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.71/ UR - https://doi.org/10.5802/cml.71 DO - 10.5802/cml.71 LA - en ID - CML_2021__13_1_43_0 ER -
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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