A General Strong Nyman--beurling Criterion for the Riemann Hypothesis


For each $f:[0,\infty)\to\mathbb C$ formally consider its Müntz transform $g(x)=\sum_{n\geq 1}f(nx)-\frac1x\int_0^\infty f(t)dt$. For certain $f$'s with both $f,g\in L_2(0,\infty)$ 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 $x\mapsto g(kx)$, $k\in\mathbb N$. 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 $\int_0^\infty t|f'(t)|\,dt<\infty$. If in addition $f$ is of compact support, then the sufficiency implication also holds true. It would be convenient to remove this compactness condition.