A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, 2

Abstract

Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a|a\geq1\}$ 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 the first version of this paper, posted in arXiv:math.NT/0202141 v2, that the Riemann hypothesis is equivalent to the stronger statement that $\chi\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{\rho_a|a\in\Nat\}$. This second version differs from the first in showing that under the Riemann hypothesis for some constant $c>0$ the distance between $\chi$ and $-\sum_{a=1}^n\mu(a)e^{-c\frac{\log a}{\log\log n}}\rho_a$ is of order $(\log\log n)^{-1/3}$.