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 $\sum_n a_n n^{-s}$ is uniformly a.s.-sign convergent (i.e., $\sum_n \varepsilon_n a_n n^{-s}$ converges uniformly for almost all sequences of signs $\varepsilon_n =\pm 1$) 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 Hardy-type Dirichlet series with values in a Banach space.