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

Luis Baez-Duarte
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Abstract

Let $ρ(x)=x-[x]$, $χ=χ_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{ρ_a | a \geq 1\}$ where $ρ_a(x):=ρ(\frac{1}{ax})$. 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 $\{ρ_a | a\in\Nat\}$.

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Luis Baez-Duarte. 2002-02-18. A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis. https://arxiv.org/abs/math/0202141

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