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 ) , k ≥ 1 . 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 ℰ ( 𝓀 ) distributions, and random variables Z k , n , 1 ≤ k ≤ 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 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.