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The Beurling-Nyman criterion for the Riemann hypothesis: numerical aspects. (Le critère de Beurling et Nyman pour l’hypothèse de Riemann: aspects numérique.) (French) Zbl 1117.11305

Summary: Let \(\mathcal B\) be the subspace of \(\mathcal H=L^2(0,+\infty)\) consisting of the functions \(f\) such that \[ f(t)=\sum_{k=1}^{n} c_{k} \rho \left( \frac{\theta_{k}}{t} \right), \; n\in\mathbb N,\; c_{k}\in\mathbb C,\; 0<\theta_{k}\leq 1,\;\text{ for } 1 \leq k \leq n, \] where \(\rho(t)\) denotes the fractional part of \(t\). We also denote by \(\chi\) the characteristic function of \((0,1]\). A well known result of Nyman and Beurling implies that the Riemann hypothesis holds if and only if \(d(\chi,\mathcal B) = 0\). We present several numerical results about the approximation of \(\chi\) by elements of \(\mathcal B\).

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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References:

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