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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 \(N\to\infty\).

MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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References:
[1] DOI: 10.1006/aima.1993.1038 · Zbl 0795.11035
[2] DOI: 10.1006/aima.1998.1801 · Zbl 0978.47025
[3] DOI: 10.1006/aima.1999.1861 · Zbl 1008.11032
[4] Báez-Duarte L., Rend. Mat. Acc. Lincei (9) 14 pp 5–
[5] DOI: 10.1006/aima.2001.2066 · Zbl 1029.11045
[6] Burnol J.-F., Acta Cientifica Venezolana 54 pp 210–
[7] Davenport H., Multiplicative Number Theory (2000) · Zbl 1002.11001
[8] DOI: 10.1007/BFb0096465 · Zbl 0477.10007
[9] DOI: 10.1112/blms/bdm032 · Zbl 1127.11058
[10] Littlewood J. E., C. R. Acad. Sci. Paris 154 pp 263–
[11] DOI: 10.1112/blms/bdn119 · Zbl 1241.11121
[12] DOI: 10.1090/cbms/084
[13] Soundararajan K., J. Reine Angew. Math. 631 pp 141–
[14] Tchebichef P., J. Maths Pures Appl. (Ser. I) 17 pp 366–
[15] Titchmarsh E. C., The Theory of the Riemann Zeta-Function (1986) · Zbl 0601.10026
[16] Tenenbaum G., Introduction à la Théorie Analytique et Probabiliste des Nombres (2008)
[17] Whittaker E. T., A Course of Modern Analysis (1927)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.