An extension of Bohr's theorem

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 $\mathbb{C}\setminus \{\alpha\}$.