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

Abstract

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

7 pages, 3 typos corrected, one reference added