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A lower bound in an approximation problem involving the zeros of the Riemann zeta function. (English) Zbl 1029.11045
This paper is a contribution to the Buerling-Nyman formulation of the Riemann hypothesis as an approximation problem. Let $$K =L^2( (0,\infty), dt)$$, let $$\chi$$ be the characteristic function of the interval $$(0,1]$$, and let $$\rho$$ be the fractional part function. For $$0 < \lambda <1$$, let $$\mathcal{B}_\lambda$$ be the subspace of $$K$$ consisting of finite linear combinations of the functions $$t\mapsto \rho(\theta/t)$$ for $$\lambda \leq \theta \leq 1$$. The Nyman-Beurling theorem is that the Riemann Hypothesis is true if and only if $$\chi$$ is in the closure of the union of the $$\mathcal{B}_\lambda$$, $$0 < \lambda <1$$.
Let $$D(\lambda)$$ denote the Hilbert-space distance $$\inf_{f\in \mathcal{B}_\lambda} \|\chi-f\|$$. L. Báez-Duarte, M. Balazard, B. Landreau, and E. Saias [Adv. Math. 149, 130-144 (2000; Zbl 1008.11032)] proved that $$\liminf_{\lambda \to 0} D(\lambda) (\log(1/\lambda))^{1/2} \geq (\sum_{\rho} 1/|\rho|^2)^{1/2},$$ where the sum on the right-hand side is over non-trivial zeros $$\rho$$ of the Riemann zeta-function with each zero being counted only once, regardless of multiplicity. In this paper, the author shows that the right-hand side of the above may be replaced by $$(\sum_{\rho} m_\rho^2/|\rho|^2)^{1/2},$$ where $$m_\rho$$ is the multiplicity of the zero $$\rho$$. The proof depends on a construction of certain Hilbert space vectors that are used to control $$D(\lambda)$$.

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