On a probabilistic Nyman-Beurling criterion
Résumé
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 dilation of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. We develop a probabilistic extension of the Nyman-Beurling criterion by considering these $\theta_k$ as random: this yields new structures and criteria, which have some relationships with the general strong B\'aez-Duarte criterion (gBD). 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 this probabilistic Nyman-Beurling criterion and the Riemann hypothesis. By means of a probabilistic point of view, we partially solve an open problem raised by B\'aez-Duarte for gBD in 2005. Finally, we focus on the particular example of independent dilated exponential random variables, which provides an analytic autocorrelation function. This regularization effect is illustrated via the elimination of an arithmetical complexity within the Vasyunin formula. The involved reciprocity formula for cotangent sums is of independent interest.