On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis

A Lower Bound in an Approximation Problem Involving the Zeros of the Riemann Zeta Function

We slightly improve the lower bound of Báez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the so-called “Hilbert-Pólya idea”. Université de NiceSophia Antipolis Laboratoire J.-A. Dieudonné Parc Valrose F-06108 Nice Cedex 02 France burnol@m...

Institute for Mathematical Physics the Nyman{beurling Equivalent Form for the Riemann Hypothesis the Nyman-beurling Equivalent Form for the Riemann Hypothesis

An Adelic Causality Problem Related to Abelian L–functions

We associate to the global field K a Lax–Phillips scattering which has the property of causality if and only if the Riemann Hypothesis holds for all the abelian L–functions of K. As a Hilbert space closure problem this provides an adelic variation on a theme initiated by Nyman and Beurling. The adelic aspects are related to previous work by Tate, Iwasawa and Connes. Université de NiceSophia Ant...

A Strengthening of the Nyman-beurling Criterion for the Riemann Hypothesis

Let ρ(x) = x − [x], χ = χ (0,1). In L2(0, ∞) consider the subspace B generated by {ρa|a ≥ 1} where ρa(x) := ρ 1 ax. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement χ ∈ B. For some time it has been conjectured, and proved in this paper, that the Riemann hypothesis is equivalent to the stronger statement that χ ∈ B nat where B nat is the much smaller subspace...

A Note on Nyman’s Equivalent Formulation of the Riemann Hypothesis

A certain subspace of L2((0, 1), dt) has been considered by Nyman, Beurling, and others, with the result that the constant function 1 belongs to it if and only if the Riemann Hypothesis holds. I show in this note that the product ∏ ζ(ρ)=0,Re(ρ)> 1 2 ∣∣1−ρ ρ ∣∣ is the norm of the projection of 1 to this subspace. This provides a quantitative refinement to Nyman’s theorem. 1 Nyman’s criterion for...