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

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 $\{\rho_a | a\in\Nat\}$.