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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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