Almost periodicity and boundary values of Dirichlet series

2024/05/06 by Ole Fredrik Brevig, Brevig, Ole Fredrik, Athanasios Kouroupis +3 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Nonlinear Differential Equations Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2405.03522

openalex publication_date 2024/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We employ almost periodicity to establish analogues of the Hardy--Stein identity and the Littlewood--Paley formula for Hardy spaces of Dirichlet series. A construction of Saksman and Seip shows that the limits in this Littlewood--Paley formula cannot be interchanged. We apply this construction to show that the limits in the definition of the mean counting function for Dirichlet series cannot be interchanged. These are essentially statements about the two different kinds of boundary values that we associate with Dirichlet series that converge to a bounded analytic function in a half-plane. The treatment of the mean counting function also involves an investigation of the zero sets and Blaschke products of such Dirichlet series.

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