# An extension of Bohr's theorem

## TL;DR

This paper extends Bohr's theorem to Dirichlet series with specific analytic continuations, showing uniform convergence in a broader half-plane under certain mapping conditions, and establishes the optimality of this extension.

## Contribution

It introduces a new extension of Bohr's theorem for Dirichlet series with two omitted points, proving uniform convergence in a larger half-plane and demonstrating the extension's optimality.

## Key findings

- Extension holds for two omitted points, not one.
- Uniform convergence in half-plane beyond original boundary.
- Extension is proven to be optimal.

## Abstract

The following extension of Bohr's theorem is established: If a somewhere convergent Dirichlet series $f$ has an analytic continuation to the half-plane $\mathbb{C}_\theta = \{s = \sigma+it\,:\, \sigma>\theta\}$ that maps $\mathbb{C}_\theta$ to $\mathbb{C} \setminus \{\alpha,\beta\}$ for complex numbers $\alpha \neq \beta$, then $f$ converges uniformly in $\mathbb{C}_{\theta+\varepsilon}$ for any $\varepsilon>0$. The extension is optimal in the sense that the assertion no longer holds should $\mathbb{C}\setminus \{\alpha,\beta\}$ be replaced with $\mathbb{C}\setminus \{\alpha\}$.