A general strong Nyman-Beurling Criterion for the Riemann Hypothesis

Abstract

For each $f:[0,\infty)\to\Com$ formally consider its co-Poisson or Müntz transform $g(x)={\sum }_{n\ge 1}f(nx)-\frac{1}{x}{\int }_0^{\infty }f(t)dt$. 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 $g(kx)$, $k\in\Nat$. 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 ${\int }_0^{\infty }t|f^{\prime }(t)|dt<\infty$. If in addition $f$ is of compact support then the sufficiency implication also holds true. It would be convenient to remove this compactness condition.

10 pages