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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 \[ \text{span}_{L^2(0,1)} \left\{ \rho_\alpha, 0 < \alpha < 1 \right\} = \left\{ f \in L^2(0,1), {(Mf)(s) \over \zeta(s)} \text{ is holomorphic in } \text{Re}(s) > {1 \over 2} \right\}. \]

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