# Carlson's Theorem for Different Measures

## TL;DR

This paper interprets Carlson's theorem through ergodic theory and constructs measures ensuring its validity on the imaginary axis for certain Dirichlet series functions.

## Contribution

It connects Carlson's theorem to ergodic theory and introduces measures that make the theorem hold on the imaginary axis for specific function classes.

## Key findings

- Carlson's theorem can be viewed as an ergodic theorem on the polytorus.
- Constructed measures extend the validity of Carlson's theorem to the imaginary axis.
- The approach links Dirichlet series and power series via Bohr's observation.

## Abstract

We use an observation of Bohr connecting Dirichlet series in the right half plane $\mathbb{C}_+$ to power series on the polydisk to interpret Carlson's theorem about integrals in the mean as a special case of the ergodic theorem by considering any vertical line in the half plane as an ergodic flow on the polytorus. Of particular interest is the imaginary axis because Carlson's theorem for Lebesgue measure does not hold there. In this note, we construct measures for which Carlson's theorem does hold on the imaginary axis for functions in the Dirichlet series analog of the disk algebra $\mathcal{A}(\mathbb{C}_+)$.