A general strong Nyman-Beurling Criterion for the Riemann Hypothesis

For each $f:[0,\mathrm{\infty })\to \text{Com}$ formally consider its co-Poisson or M"{u}ntz transform $g(x)={\sum }_{n\ge 1}f(nx)-\frac{1}{x}{\int }_0^{\mathrm{\infty }}f(t)dt$. For certain $f$’s with both $f,g\in L_2(0,\mathrm{\infty })$ it is true that the Riemann hypothesis holds if and only if $f$ is in the $L_2$ closure of the vector space generated by the dilations $g(kx)$, $k\in \text{Nat}$. Such is the case for example when $f={\chi }_{(0,1]}$ where the above statement reduces to the strong Nyman criterion already established by the author. In this note we show that the necessity implication holds for any continuously differentiable function $f$ vanishing at infinity and satisfying ${\int }_0^{\mathrm{\infty }}t\mathrm{\mid }{f}^{\mathrm{\prime }}(t)\mathrm{\mid }dt<\mathrm{\infty }$. If in addition $f$ is of compact support then the sufficiency implication also holds true. It would be convenient to remove this compactness condition.