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A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis. (English) Zbl 1097.11041
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 $$\chi\in \overline{B^{nat}}$$.
Reviewer: Jianya Liu (Jinan)

##### MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses