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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.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
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