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