Carlson's theorem and vertical limit functions

TL;DR

This work extends Carlson's moment theorem for Dirichlet series from $p=2$ to all $1\le p<\infty$, and couples it with ergodic theory for the Kronecker flow to analyze vertical limit functions. By linking Dirichlet series in $\mathscr{H}^p$ to $H^p(\mathbb{T}^\infty)$ and employing Riesz means, the authors establish an almost sure analytic continuation of vertical limits to $\mathbb{C}_0$, provide a method to compute the $H^p$-norm via these limits, and deduce a Fatou-type theorem for the vertical limits. The main contributions are (i) a Carlson-type extension ensuring existence of $M_p(\sigma,f)$ and norm convergence as $\sigma\to0^+$, (ii) a structural identification of $f$ via its vertical limits $f_\chi$ with controlled $L^p$-norms, and (iii) almost-sure convergence and Fatou-type results for $f_\chi$ across a full-measure set of characters. This yields a unified ergodic-analytic framework for the boundary behavior of Dirichlet-series in $H^p$ spaces and enables practical norm computations and continuation results.

Abstract

We extend a classical theorem of Carlson on moments of Dirichlet series from $p=2$ to $1 \leq p < \infty$. When combined with the ergodic theorem for the Kronecker flow, a coherent approach to almost sure properties of vertical limit functions in $H^p$ spaces of Dirichlet series is obtained. This allows us to establish an almost sure analytic continuation of vertical limit functions to the right half-plane that can be used to compute the $H^p$ norm and to prove a version of Fatou's theorem.

Theorems & Definitions (23)

• Theorem 1
• Definition
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5: Titchmarsh Titchmarsh1958*§9.55
• Corollary 6
• Lemma 7
• proof
• Lemma 8