Almost periodicity and boundary values of Dirichlet series

TL;DR

This paper explores boundary behaviors and almost periodicity in Dirichlet series, establishing new identities and examining boundary value limits, zero sets, and Blaschke products in Hardy spaces.

Contribution

It introduces analogues of classical identities for Hardy spaces of Dirichlet series and analyzes boundary value limits and zero set properties.

Findings

Limits in Littlewood--Paley formula cannot be interchanged

Limits in mean counting function definition cannot be interchanged

Boundary value types relate to convergence and boundedness

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.