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On a criterion of Báez-Duarte for the Riemann hypothesis. (Sur un critère de Báez-Duarte pour l’hypothèse de Riemann.) (French. English summary) Zbl 1201.11088
The Báez-Duarte criterion says that the Riemann Hypothesis (that all complex zeros of \(\zeta(s)\) in the critical strip have real part \(1/2\)) is equivalent to the statement that \[ \inf_\varphi\int_0^\infty|\chi(x)-\varphi(x)|^2\,{dx\over x^2} = 0. \] Here \(\chi(x)\) is the indicator function of the interval \((0,1]\), while for \(\{x\} = x - [x]\) \[ \varphi(x) = \sum_{n=1}^Nc_n\{x/n\}\qquad(N\in\mathbb N,\; c_n\in \mathbb R). \] An equivalent form of this is as follows. Define \(e_n(t) = \{t/n\}\). Let \(d_N\) denote the distance in \(L^2(0,\infty;t^{-2}{\roman d}t)\) between \(\chi\) and the vector space generated by \(e_1,\ldots, e_N\). Then the Riemann Hypothesis is true if and only if \(\lim_{N\to\infty}d_N =0\). The authors make a significant contribution to this subject by showing that, if the Riemann Hypothesis is true, then if \(N\geq3\) one has, for any given \(\delta>0\), \[ d_N^2 \ll_\delta (\log\log N)^{5/2+\delta}(\log N)^{-1/2} \] This should be compared with the lower bound of J.-F. Burnol [Adv. Math. 170, 56–70 (Zbl 1029.11045)], namely \[ d_N^2 \geq {C+o(1)\over \log N},\quad C = \sum_{\rho} m_\rho^2/|\rho|^2, \] where \(m_\rho\) is the multiplicity of the complex zero \(\rho\) of \(\zeta(s)\) and \(N\to\infty\).

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