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Note on the Riemann \(\zeta\)-function. III. (Notes sur la fonction \(\zeta\) de Riemann. III.) (French) Zbl 1008.11032
In this continuation on the work on the zeros of the Riemann zeta-function [Part I, Adv. Math. 139, 310-321 (1998; Zbl 0920.11062) and ibid. 143, 284-287 (1999; Zbl 0937.11042)] the authors investigate the function \(D(\lambda)\), the distance in \(\mathcal H\) between \(\chi\) and \({\mathcal B}_\lambda\), the subspace of \(\mathcal B\) of functions \(f\) such that \(\theta_k \geq \lambda\). Here \(\mathcal H\) is the Hilbert space \(L^2(0, \infty)\), \(\mathcal B\) the subspace of \(\mathcal H\) of functions \(f(t) = \sum_{k=1}^n c_k\rho(\theta_k/t)\), where \(\rho(x)\) denotes the fractional part of \(x\), the \(c_k\)’s are complex numbers, and \(\chi\) is the characteristic function of \((0,1]\). A classic theorem of Beurling and Nyman asserts that the Riemann Hypothesis (RH; all complex zeros of \(\zeta(s)\) have real parts equal to \(1\over 2\)) is equivalent to the statement that \(\lim_{\lambda\to 0} D(\lambda) = 0\). Here the authors prove that \(\liminf_{\lambda\to 0} D(\lambda)\sqrt{\log(1/\lambda)} > 0\), implying that \(D(\lambda) \gg 1/\sqrt{\log(2/\lambda)}\) for \(0 < \lambda \leq 1\). Moreover, they conjecture (this is stronger than RH) that \[ \liminf_{\lambda\to 0}D(\lambda)\sqrt{\log(1/\lambda)} = \sqrt{2+\gamma - \log(4\pi)}, \] where \(\gamma = -\Gamma'(1)\) is Euler’s constant. The constant on the right-hand side appears as \[ \sum_\beta {2\operatorname{Re} \beta\over|\beta|^2} = 2 + \gamma - \log(4\pi), \] where summation is over complex zeros \(\beta\) of \(\zeta(s)\) with real parts equal to \(1\over 2\) such that each zero is counted only once, regardless of its possible multiplicity.

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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