On a Hilbert Space Reformulation of Riemann Hypothesis

2019/11/11 by Boqing Xue, Xue, Boqing
Mathematics · #Advanced Operator Algebra Research #Analytic Number Theory Research #FOS: Mathematics #Holomorphic and Operator Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1911.04029

openalex publication_date 2019/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, Báez-Duarte, et. al. with a weighted Bergman space H=A₁²(\mathbbD), i.e., Riemann hypothesis holds if and only if the Hilbert subspace H₀ spanned by a certain family of functions coincides with H. A condition that a function does not belong to H₀^\bot 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 H₀ is T_k^∗-invariant for all k∈ ℕ.

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