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

Abstract

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 $\theta_k\in(0,1)$, $k\ge1$. We develop probabilistic extensions of the Nyman-Beurling criterion by considering these $\theta_k$ as random: this yields new structures and criteria, one of them having a significant overlap with the general strong B\'aez-Duarte criterion. We start here the study of these criteria, with a special focus on exponential and gamma distributions. 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 $\cal E(k)$ distributions, and random variables $Z_{k,n}$, $1\le k\le n$, 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 B\'aez-Duarte in 2005: the price to pay to consider non compactly supported kernels is a controlled condition on the coefficients of the involved approximations.