Convergence and almost sure properties in Hardy spaces of Dirichlet series

TL;DR

This paper investigates convergence properties of Dirichlet series in Hardy spaces, providing new conditions for uniform convergence and exploring almost sure convergence with harmonic analysis techniques.

Contribution

It introduces a new condition on the frequency sequence ensuring uniform convergence of bounded Dirichlet series and analyzes their Hardy space properties.

Findings

New condition on frequency sequence for uniform convergence

Results on almost sure convergence in Hardy spaces

Examples showing limitations in holomorphic function space results

Abstract

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