# Convergence and almost sure properties in Hardy spaces of Dirichlet series

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.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-03103584
Contributor : Frédéric Bayart <>
Submitted on : Friday, January 8, 2021 - 11:20:34 AM
Last modification on : Thursday, February 25, 2021 - 10:32:06 AM

### Files

gendir5.pdf
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### Identifiers

• HAL Id : hal-03103584, version 1
• ARXIV : 2101.02990

### Citation

Frédéric Bayart. Convergence and almost sure properties in Hardy spaces of Dirichlet series. 2021. ⟨hal-03103584⟩

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