  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.
