A characterization of the zero-free region of the Riemann zeta function and its applications

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 Báez-Duarte et al. [Adv. Math. 149 (2000) 130-144].