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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.

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
Full Text: DOI
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