A general strong Nyman-Beurling criterion for the Riemann hypothesis

Abstract

For each [FORMULA] formally consider its Miintz transform [FORMULA]. For certain ?'s with both [FORMULA] it is true that the Riemann hypothesis holds if and only if ? is in the L2 closure of the vector space generated by the dilations [FORMULA]. Such is the case for example when ? = X(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 ? vanishing at infinity and satisfying [FORMULA]. If in addition ? is of compact support, then the sufficiency implication also holds true. It would be convenient to remove this compactness condition .