A characterization of the zero-free region of the Riemann zeta function and its applications. (English) Zbl 1503.30123
Summary: We characterize the zero-free regions of a class of functions (including the Riemann zeta function) in half-planes in terms of closures of ranges of the corresponding multiplication operators on Hardy spaces. We give an explicit characterization of these closures. As applications, we obtain a weaker version of the Nyman-Beurling-Báez-Duarte criterion, and provide some investigations on a problem relating to the Riemann hypothesis proposed by L. Báez-Duarte et al. [Adv. Math. 149, No. 1, 130–144 (2000; Zbl 1008.11032)].
MSC:
| 30H10 | Hardy spaces |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
Keywords:
Riemann zeta function; Hardy spaces; Nyman-Beurling-Báez-Duarte criterion; Dirichlet polynomials; \(L\)-functionsCitations:
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