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 $\{{\rho }_a|a\ge 1\}$ 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 the first version of this paper, posted in arXiv:math.NT/0202141 v2, that the Riemann hypothesis is equivalent to the stronger statement that $χ\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{ρ_a|a\in\Nat\}$. 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 $(\log \log n{)}^{-1/3}$.

9 pages

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