On generalized Hardy classes of Dirichlet series

TL;DR

This paper extends the Hardy class H^2 of Dirichlet series to more general classes, providing estimates and applications, including explicit nonvanishing results and implications for the Hurwitz zeta-function.

Contribution

It introduces a generalized Hardy class for Dirichlet series and derives new estimates and applications, expanding the understanding of these functions.

Findings

Estimates for the logarithmic L^1-norm in short intervals

Explicit nonvanishing results for Dirichlet series

Applications to the Hurwitz zeta-function

Abstract

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.