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New versions of the Nyman-Beurling criterion for the Riemann hypothesis. (English) Zbl 1069.11037

The paper contains a detailed discussion of the Nyman-Beurling criterion for the truth of the quasi-Riemann hypothesis. \[ \zeta(s)\neq 0 \quad\text{for}\quad\sigma=\text{Re}\,s>\frac 1p \] for some \(p\in (1, \infty)\) [Proc. Natl. Acad. Sci. USA 41, 312–314 (1955; Zbl 0065.30303)]. There exist several extensions of the original statement [H. Bercovici and C. Foias, Isr. J. Math 48, 57–68 (1984; Zbl 0569.46011), M. Balazard and E. Saias, Adv. Math 139, No. 2, 310–321 (1998; Zbl 0920.11062)]. The author adds some further version of the Nyman-Beurling criterion.

MSC:

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