  The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is
an approximation problem in the space of square integrable functions on
$(0,\infty)$, involving dilations of the fractional part function by factors
$\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new
structures and criteria. One of them is a sufficient condition for RH that
splits into (i) showing that the indicator function can be approximated by
convolution with the fractional part, (ii) a control on the coefficients of the
approximation. This self-contained paper generalizes conditions (i) and (ii)
that involve a $\sigma_0\in(1/2,1)$, and imply $\zeta(\sigma+it)\neq 0$ in the
strip $1/2<\sigma\le\sigma_0<1$. We then identify functions for which (i) holds
unconditionally, by means of polynomial approximations. This yields in passing
a short probabilistic proof of a known consequence of Wiener's Tauberian
theorem. In this context, the difficulty for proving RH is then reallocated in
(ii), which heavily relies on the corresponding Gram matrices, for which two
remarkable structures are obtained. We show that a particular tuning of the
approximating sequence leads to a striking simplification of the second Gram
matrix, then reading as a block Hankel form.
