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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
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