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arXiv · 2510.05793

Carlson's theorem and vertical limit functions

Ole Fredrik Brevig·Athanasios Kouroupis
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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.

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Ole Fredrik Brevig, Athanasios Kouroupis. 2025-10-07. Carlson's theorem and vertical limit functions. https://arxiv.org/abs/2510.05793

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