Convergence and almost sure properties in Hardy spaces of Dirichlet series

Given a frequency $$\lambda $$ , we study general Dirichlet series $$\sum a_n e^{-\lambda _n s}$$ . First, give new condition on which ensures that somewhere convergent defining bounded holomorphic function in the right half-plane converges uniformly this half-plane, improving classical results of Bohr and Landau. Then, following recent works Defant Schoolmann, investigate Hardy spaces these series. We get almost sure convergence have an harmonic analysis flavour. Nevertheless, also exhibit examples showing it seems hard to as functions.