On a Hilbert Space Reformulation of Riemann Hypothesis

TL;DR

This paper investigates a Hilbert space reformulation of the Riemann Hypothesis using weighted Bergman spaces, establishing conditions under which the hypothesis is equivalent to certain invariance properties of subspaces and operator algebras.

Contribution

It introduces a new Hilbert space framework for the Riemann Hypothesis, linking it to invariance under specific operator monoids and von Neumann algebra properties.

Findings

Riemann Hypothesis holds iff a certain subspace equals the entire Hilbert space.

Characterization of functions outside the orthogonal complement of the subspace.

Von Neumann algebra generated by the operator monoid is the full algebra of bounded operators.

Abstract

We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, B\'{a}ez-Duarte, et. al. with a weighted Bergman space $\mathcal{H}=A_1^2(\mathbb{D})$, i.e., Riemann hypothesis holds if and only if the Hilbert subspace $\mathcal{H}_0$ spanned by a certain family of functions coincides with $\mathcal{H}$. A condition that a function does not belong to $\mathcal{H}_0^\bot$ is given. Moreover, it is proved that the von-Neumann algebra generated by a certain monoid $T_\mathbb{N}=\{T_k:\, k\in \mathbb{N}\}$ of operators is exactly $B(\mathcal{H})$. As a result, Riemann hypothesis is true if and only if $\mathcal{H}_0$ is $T_k^\ast$-invariant for all $k\in \mathbb{N}$.