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,∞), 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.