A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis (Q2569799)

Let \(\rho(x)=x-[x]\) be the fractional part of \(x\), and \(\chi\) the characteristic function of the interval \((0,1]\). Also let \(H\) be the Hilbert space \(L_2((0,\infty), dx)\). Let \(B\) be the subspace of Beurling functions which are defined to be the linear hull of the family \(\{\rho_a: 1\leq a \in R\}\) with \(\rho_a(x)=\rho(1/ax)\). Let \(B^{nat}\) be the subspace generated by \(\{\rho_a: a\in N\}\). The Nyman-Beurling criterion states that the Riemann hypothesis is equivalent to the assertion that \(\chi\in \overline{B}\). In this paper, the author proves that the Riemann hypothesis is equivalent to the statement that \(\chi\in \overline{B^{nat}}\).