A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis

TL;DR

This paper proves that the Riemann Hypothesis is equivalent to the inclusion of the characteristic function in a smaller subspace generated by rational parameters, strengthening the classical Nyman-Beurling criterion.

Contribution

The paper establishes that the Riemann Hypothesis is equivalent to the characteristic function being in the closure of a subspace generated by rational parameters, a significant strengthening of the Nyman-Beurling criterion.

Findings

Proves the equivalence of the Riemann Hypothesis with the inclusion in a smaller subspace.

Shows the subspace generated by rational parameters suffices for the criterion.

Strengthens the connection between the Riemann Hypothesis and functional analysis.

Abstract

Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a | a \geq 1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $\chi\in\bar{\B}$. For some time it has been conjectured, and proved in this paper, that the Riemann hypothesis is equivalent to the stronger statement that $\chi\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{\rho_a | a\in\Nat\}$.