On generalized Hardy classes of Dirichlet series

We generalize the Hardy class H^2 of Dirichlet series studied by Hedenmalm, Lindqvist, Olofsson, Olsen, Saksman, Seip and others to consider more general Dirichlet series. We prove some results on this class, such as estimates for its logarithmic L^1-norm in short intervals. We relate this to, and use these results to make a recent nonvanishing result of Dirichlet series of ours more explicit. In particular we give an application on the Hurwitz zeta-function.