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., ${\sum }_n{\epsilon }_n{a}_n{n}^{-s}$ converges uniformly for almost all sequences of signs ${\epsilon }_n=\pm 1$) 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.

References in corpus (1)

• Fourier multipliers for Hardy spaces of Dirichlet series