Convergence and almost sure properties in Hardy spaces of Dirichlet\n series

2021/01/08 by Frédéric Bayart, Bayart, Frédéric
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2101.02990

openalex publication_date 2021/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a frequency \λ, we study general Dirichlet series \∑ a_n≠^{-\λ_n s}. First, we give a new condition on \λ which ensures\nthat a somewhere convergent Dirichlet series defining a bounded holomorphic\nfunction in the right half-plane converges uniformly in this half-plane,\nimproving classical results of Bohr and Landau. Then, following recent works of\nDefant and Schoolmann, we investigate Hardy spaces of these Dirichlet series.\nWe get general results on almost sure convergence which have an harmonic\nanalysis flavour. Nevertheless, we also exhibit examples showing that it seems\nhard to get general results on these spaces as spaces of holomorphic functions.\n

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