Almost sure-sign convergence of Hardy-type Dirichlet series

Abstract

Hartman proved in 1939 that the width of the largest possible strip in the complex plane on which a Dirichlet series ∑ nann− s is uniformly a.s.- sign convergent (i.e., ∑ nεnann− s converges uniformly for almost all sequences of signs εn = ±1) but does not convergent absolutely, equals 1/2. We study this result from a more modern point of view within the framework of so-called Hardytype Dirichlet series with values in a Banach space.