PUBLICATIONS DE L'INSTITUT MATHEMATIQUE (BEOGRAD) (N.S.) Vol. 78(92), pp. 117–125 (2005) 

A GENERAL STRONG NYMAN–BEURLING CRITERION FOR THE RIEMANN HYPOTHESISLuis BaezDuarteDepartamento de Matematicas, Instituto Venezolano de Investigaciones Cientificas, Caracas, VenezuelaAbstract: 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 $\int_0^\infty tf'(t) dt<\infty$. If in addition $f$ is of compact support, then the sufficiency implication also holds true. It would be convenient to remove this compactness condition. Keywords: Riemann zetafunction; Riemann hypothesis; strong Nyman–Beurling theorem; Müntz's formula Classification (MSC2000): 11M26 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 2 Mar 2006. This page was last modified: 27 Oct 2006.
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