On a Hilbert Space Reformulation of Riemann Hypothesis

Abstract

We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, Báez-Duarte, et. al. with a weighted Bergman space H = A 2 1 (D), i.e., Riemann hypothesis holds if and only if the Hilbert subspace H 0 spanned by a certain family of functions coincides with H. A condition that a function does not belong to H ⊥ 0 is given. Moreover, it is proved that the von-Neumann algebra generated by a certain monoid T N = {T k : k ∈ N} of operators is exactly B(H). As a result, Riemann hypothesis is true if and only if H 0 is T * k -invariant for all k ∈ N.