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The Nyman-Beurling equivalent form for the Riemann hypothesis. (English) Zbl 0954.11029
In this article based on a talk given during the 1998 Vienna Conference on the Riemann Hypothesis, the authors collect many results and open questions about the Riemann zeta functions [for example a result of L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias, Adv. Math. 149, 130-144 (2000)]; in particular, a proof of the following result of H. Bercovici and C. Foias [Isr. J. Math. 48, 57-68 (1984; Zbl 0569.46011)], related to the Nyman-Beurling equivalent form of Riemann’s hypothesis, is outlined:
Denote by $$\rho_\alpha(t)$$ the function $$\{{\alpha \over t}\}- \alpha\cdot \{ {1 \over t}\}$$, where $$\{\beta\} = \beta - [\beta]$$ is the fractional part of $$\beta$$, and define (for $$f \in L^2(0,1)$$) the Mellin transform by $(Mf)(s) = {1 \over \sqrt{2\pi}} \cdot \int^1_0 f(t) \cdot t^{s-1} dt.$ Then $\text{span}_{L^2(0,1)} \left\{ \rho_\alpha, 0 < \alpha < 1 \right\} = \left\{ f \in L^2(0,1), {(Mf)(s) \over \zeta(s)} \text{ is holomorphic in } \text{Re}(s) > {1 \over 2} \right\}.$

##### MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses
##### Citations:
Zbl 0991.25536; Zbl 0569.46011