On the Nyman-Beurling criterion for the Riemann hypothesis. (English) Zbl 1226.11088

Summary: The Nyman-Beurling criterion states that the Riemann hypothesis is equivalent to the density in \(L^2(0,+\infty;t^{-2}\,dt)\) of a certain space. We introduce an orthonormal family in \(L^2(0,+\infty;t^{-2}\,dt)\), study the space generated by this family and reformulate the Nyman-Beurling criterion using this orthonormal basis. We then study three approximations that could lead to a proof of this criterion.


11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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