# A general strong Nyman-Beurling criterion for the Riemann hypothesis

@article{BezDuarte2005AGS, title={A general strong Nyman-Beurling criterion for the Riemann hypothesis}, author={Luis B{\'a}ez-Duarte}, journal={Publications De L'institut Mathematique}, year={2005}, volume={78}, pages={117-125} }

For each [FORMULA] formally consider its Miintz transform [FORMULA]. For certain ƒ's with both [FORMULA] it is true that the Riemann hypothesis holds if and only if ƒ is in the L2 closure of the vector space generated by the dilations [FORMULA]. Such is the case for example when ƒ = 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…

## 10 Citations

On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function

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The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on $(0,\infty)$, which is equivalent to the Riemann hypothesis. This involves dilations of the…

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We establish generalizations of the Nyman-Beurling and Baez-Duarte criteria concerning lack of zeros of Dirichlet $L$-functions in the semi-plane $\Re(s) >1/p$ for $p\in (1,2]$. We pose and solve a…

On a probabilistic Nyman-Beurling criterion

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The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on $(0,\infty)$, which is equivalent to the Riemann hypothesis. This involves dilation of the…

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For coprime numbers p and q, we consider the Vasyunin–cotangent sum (0.1) V (q, p) = p−1 ∑ k=1 {kq p } cot (πk p ) . First, we prove explicit formula for the symmetric sum V (p, q)+V (q, p) which is…

New summation and transformation formulas of the Poisson, Müntz, Möbius and Voronoi type

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Starting from the classical summation formulas and basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta functions of…

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Applying the known Nyman--Beurling criterion, it is disproved the Riemann hypothesis on zeros of $\zeta -$function.

Riemann’s hypothesis

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Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and M\"untz's identity.

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The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and of Hardy and Littlewood are embedded in a general theorem for a class of entire functions, which in turn is seen to be a consequence of a rather transparent convolution criterion.

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