On generalized Nyman-Beurling criteria

One of the innocent reformulations of terrifying conjecture RH is the Nyman-Beurling-Baez-Duarte criterion : the indicator function of (0,1) can be linearly approximated in a L^2 space by dilations of the fractional part function. We will discuss the interest of randomizing these dilations, which generates entirely new structures and criteria, and answers an open question raised by Baez-Duarte in 2005. I will eventually tell you a little story. Joint work with Erwan Hillion.