\section{Introduction} Let $\mathfrak{D}$ denote the class of Dirichlet series
\[f(s) = \sum_{n=1}^\infty a_n n^{-s}\]
that converge in at least one point $s=\sigma+it$ in the complex plane. Associated to each Dirichlet series $f$ in $\mathfrak{D}$ is a number $\sigmac(f)$, called the \emph{abscissa of convergence}, with the property that $f$ converges if $\sigma>\sigmac(f)$ and $f$ does not converge if $\sigma<\sigmac(f)$. This note concerns an extension of Bohr's classical theorem on uniform convergence of Dirichlet series~\cite{Bohr1913}. We therefore define the \emph{abscissa of uniform convergence} $\sigmau(f)$ as the infimum of the real numbers $\theta$ such that $f$ converges uniformly in the half-plane $\mathbb{C}_\theta$. Here and in what follows, we set
\[\mathbb{C}_\theta = \{s = \sigma+it\,:\, \sigma>\theta\}.\]

Our starting point reads as follows. 
\begin{bohr}
	Let $f$ be in $\mathfrak{D}$. If there is a real number $\theta$ and a bounded set $\Omega$ such that $f$ has an analytic continuation to $\mathbb{C}_\theta$ that maps $\mathbb{C}_\theta$ to $\Omega$, then $\sigmau(f) \leq \theta$. 
\end{bohr}

Queff\'{e}lec and Seip~\cite{QS2015} (see also~\cite[Theorem~8.4.1]{QQ2020}) showed that the assumption that $\Omega$ is a bounded set may be replaced with the weaker assumption that $\Omega$ is a half-plane. This extension of Bohr's theorem was applied obtain the canonical formulation of the Gordon--Hedenmalm characterization of composition operators~\cite{GH1999}, which has proven to be essential for further developments (see e.g.~\cite[Section~6]{BP2021}).

Replacing $f$ by $a+bf$ for constants $a$ and $b\neq0$, we may assume without loss of generality that $\Omega = \mathbb{C}_0$ in their extension. In addition to Bohr's theorem, their proof is based on Harnack's inequality for positive harmonic functions. Consequently, their argument uses the assumption that $\Omega$ is a half-plane in a rather strong sense. 

We have the following result. 
\begin{theorem}\label{thm:bohrext} 
	Let $f$ be in $\mathfrak{D}$. If there is a real number $\theta$ and a simply connected domain $\Omega \neq \mathbb{C}$ such that $f$ has an analytic continuation to $\mathbb{C}_\theta$ that maps $\mathbb{C}_\theta$ to $\Omega$, then $\sigmau(f)\leq \theta$. 
\end{theorem}

Our proof of Theorem~\ref{thm:bohrext} is arguably simpler than the Queff\'{e}lec--Seip proof in the special case that $\Omega$ is a half-pane. In addition to Bohr's theorem and the Riemann mapping theorem, we will only use some elementary properties of Dirichlet series and the Hadamard three-lines theorem.

\subsection*{Acknowledgements.} The author thanks Herv\'{e} Queff\'{e}lec for providing helpful comments.


\section{Proof of Theorem~\ref{thm:bohrext}} In addition to the abscissa of convergence and the abscissa of uniform convergence discussed above, every Dirichlet series $f$ in $\mathfrak{D}$ has an \emph{abscissa of absolute convergence} $\sigma_{\mathrm{a}}(f)$ with the property that $f$ converges absolutely for $\sigma>\sigma_{\mathrm{a}}(f)$ and $f$ does not converge absolutely for $\sigma<\sigma_{\mathrm{a}}(f)$. It is well-known that $\sigma_{\mathrm{a}}(f) \leq \sigmac(f)+1$, so every Dirichlet series in $\mathfrak{D}$ is absolutely convergent in some possibly remote half-plane.

To facilitate certain computations below, we introduce the Wiener norm
\[\|f\|_{W_\sigma} = \sum_{n=1}^\infty |a_n| n^{-\sigma},\]
where $f$ is the Dirichlet series with coefficient sequence $(a_n)_{n\geq1}$. Note that if $\|f\|_{W_\sigma} < \infty$, then $\sigma\geq\sigma_{\mathrm{a}}(f)$. In addition to the usual norm properties, we will have use of the submultiplicative property 
\begin{equation}\label{eq:wienerprod} 
	\|f g\|_{W_\sigma} \leq \|f\|_{W_\sigma} \|g\|_{W_\sigma}. 
\end{equation}
We also require a lemma for the proof of Theorem~\ref{thm:bohrext}. It is probably well-known to experts, but we have not found the formulation we need explicitly stated in the literature. Since the proof is quite short, we include the details. To set the stage, suppose that $f$ is a function defined in some half-plane $\mathbb{C}_\theta$ and consider
\[M(f,\sigma) = \sup_{t \in \mathbb{R}} |f(\sigma+it)|\]
for $\sigma>\theta$. Note that if $f$ is a M\"obius transformation from the half-plane $\mathbb{C}_\theta$ to the unit disc $\mathbb{D}$, then $M(f,\sigma) = 1$ for all $\theta<\sigma<\infty$. The following result illustrates how uniformly convergent Dirichlet series cannot exhibit the same behavior.
\begin{lemma}\label{lem:3lines} 
	Suppose that $f$ is in $\mathfrak{D}$ and that $f$ is not identically equal to a constant. The function
	\[\sigma \mapsto M(f,\sigma)\]
	is strictly decreasing on the interval $(\sigmau(f),\infty)$. 
\end{lemma}
\begin{proof}
	The Hadamard three-lines theorem states that 
	\begin{equation}\label{eq:h3l} 
		M(f,\sigma) \leq \big(M(f,\sigma_1)\big)^{\alpha} \big(M(f,\sigma_2)\big)^{1-\alpha}, 
	\end{equation}
	for $\sigmau(f)<\sigma_1 < \sigma_2$ and $\sigma = \alpha \sigma_1 + (1-\alpha)\sigma_2$ for some $0\leq \alpha \leq 1$. We think of $\sigma_1$ and $\sigma$ as fixed, and aim to demonstrate that $M(f,\sigma)<M(f,\sigma_1)$ by choosing a sufficiently large $\sigma_2$. The fact that $f$ converges uniformly on the line $s=\sigma+it$ allows us to move the limit and the integral inside the Dirichlet series to infer that
	\[a_1 = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T f(\sigma+it) \,dt.\]
	It follows from this formula, the triangle inequality, and the assumption that $f$ is not identically equal to a constant that $|a_1|<M(f,\sigma)$. Since $f$ is in $\mathfrak{D}$, it is absolutely convergent for sufficiently large $\sigma$ and, consequently,
	\[|a_1| = \lim_{\sigma \to \infty} M(f,\sigma).\]
	This shows that if $\sigma_2$ is sufficiently large, then $M(f,\sigma_2)<M(f,\sigma)$. Inserting this estimate into \eqref{eq:h3l} yields the stated result. 
\end{proof}
\begin{proof}
	[Final part in the proof of Theorem~\ref{thm:bohrext}] If $f$ is identically equal to a constant, then there is nothing to do. If $f$ is not identically equal to a constant, then there is a smallest integer $n>1$ such that $a_n \neq 0$. By absolute convergence, we have
	\[f(s) = a_1 + a_n n^{-s} + O\big((n+1)^{-\sigma}\big)\]
	as $\sigma \to \infty$. By the assumption that $f$ maps $\mathbb{C}_\theta$ to $\Omega$, it follows that $\Omega$ contains a small punctured disc centered at $a_1$. Since $\Omega$ is simply connected, this means that $a_1$ lies in the interior of $\Omega$.
	
	Let $\Theta$ be a Riemann map from $\mathbb{D}$ to $\Omega$ with $\Theta(0)=a_1$. The function $g = \Theta^{-1} \circ f$ is analytic and bounded by $1$ in $\mathbb{C}_\theta$. We may formally expand $g$ as a Dirichlet series, but to show that $g$ is in $\mathfrak{D}$ we need to prove that the series converges somewhere. Since $a_1$ lies in the interior of $\Omega$, there is some $\varrho>0$ such that the Taylor series
	\[\Theta^{-1}(w) = \sum_{k=0}^\infty \frac{(\Theta^{-1})^{(k)}(a_1)}{k!} (w-a_1)^k\]
	converges absolutely when $|w-a_1| \leq \varrho$. Appealing to absolute convergence as above, we find that there is some sufficiently large $\sigma$ such that $\|f-a_1 \|_{W_\sigma} \leq \varrho$. Thus
	\[\|g\|_{W_\sigma} \leq \sum_{k=0}^\infty \frac{\left|(\Theta^{-1})^{(k)}(a_1)\right|}{k!} \big\|(f-a_1)^k\big\|_{W_\sigma} \leq \sum_{k=0}^\infty \frac{\left|(\Theta^{-1})^{(k)}(a_1)\right|}{k!} \varrho^k < \infty,\]
	where we in the second estimate used the submultiplicative property \eqref{eq:wienerprod}. This shows that $\sigma_{\mathrm{a}}(g) \leq \sigma$. Consequently, $g$ is in $\mathfrak{D}$ and, by Bohr's theorem, $\sigmau(g) \leq \theta$. 
	
	It remains to demonstrate that the uniform convergence of $g$ implies the uniform convergence of $f$. Since $\Theta$ is an analytic function in the unit disc, the Taylor series
	\[\Theta(z) = \sum_{k=0}^\infty \frac{\Theta^{(k)}(0)}{k!} z^k\]
	is absolutely convergent when $|z| \leq r$ for any $r<1$. Since $f$ is not identically equal to a constant, it is plain that $g = \Theta^{-1} \circ f$ is also not identically equal to a constant. Using that $g$ maps $\mathbb{C}_\theta$ to $\mathbb{D}$, we infer from Lemma~\ref{lem:3lines} that there for any $\varepsilon>0$ is an $r<1$ such that $|g(s)| \leq r$ for all $s$ in $\mathbb{C}_{\theta+\varepsilon}$. For such $s$ we find that
	\[|f(s)| = |(\Theta \circ g)(s)| \leq \sum_{k=0}^\infty \frac{\left|\Theta^{(k)}(0)\right|}{k!} |g(s)|^k \leq \sum_{k=0}^\infty \frac{\left|\Theta^{(k)}(0)\right|}{k!} r^k < \infty.\]
	Since $f$ is in $\mathfrak{D}$ and $f$ is a bounded analytic function in $\mathbb{C}_{\theta+\varepsilon}$ for any $\varepsilon>0$, it follows from Bohr's theorem that $\sigmau(f)\leq \theta$. 
\end{proof}


\section{Concluding remarks} Gordon and Hedenmalm~\cite{GH1999} (see also \cite[Theorem~8.4.3]{QQ2020}) proved that if $f$ satisfies the assumptions of Theorem~\ref{thm:bohrext} with $\Omega=\mathbb{C}_\vartheta$, then for any $\varepsilon>0$ there is a $\delta>0$ such that $f$ maps $\mathbb{C}_{\theta+\varepsilon}$ to $\mathbb{C}_{\vartheta+\delta}$. The proof of this result is based on an argument similar to the one we used above to conclude that $a_1$ lies in the interior of $\Omega$ and the Hadamard three-lines theorem applied to the function $\exp{f}$.

If the above-mentioned result of Gordon and Hedenmalm is combined with with the extension of Bohr's theorem due to Queff\'{e}lec and Seip, which in particular implies that $f(\mathbb{C}_{\theta+\varepsilon})$ is a bounded set for any $\varepsilon>0$, then it follows that $f(\mathbb{C}_{\theta+\varepsilon})$ is contained in a compact subset of $\mathbb{C}_{\vartheta}$ for any $\varepsilon>0$. Our proof of Theorem~\ref{thm:bohrext} also yields the extension of this assertion to simply connected domains.
\begin{theorem}
	Let $f$ satisfy the assumptions of Theorem~\ref{thm:bohrext}. Then $f(\mathbb{C}_{\theta+\varepsilon})$ is contained in a compact subset of $\Omega$ for any $\varepsilon>0$. 
\end{theorem}
\begin{proof}
	Following the proof of Theorem~\ref{thm:bohrext}, we find that for any $\varepsilon>0$ there is some $r<1$ such that $f(\mathbb{C}_{\theta+\varepsilon})$ is contained in $\Theta(r\mathbb{D})$, where $r\mathbb{D} = \{z \in \mathbb{C}\,:\, |z|<r\}$. Since $\Theta$ is a Riemann map from $\mathbb{D}$ to $\Omega$, the stated result follows. 
\end{proof}

A \emph{general} Dirichlet series is a series of the form
\[f(s) = \sum_{n=1}^\infty a_n e^{-\lambda_n s},\]
where the \emph{frequency} $\lambda=(\lambda_n)_{n\geq1}$ is a strictly increasing and unbounded sequence of nonnegative real numbers. In the present note we have restricted our attention to the classical case $\lambda_n = \log{n}$.

The century-old problem of determining which frequencies satisfy Bohr's theorem has recently enjoyed a resurgence of interest (see e.g. \cite{Bayart2022,Schoolmann2020}). Our proof carries over to the general case provided there is absolute convergence in some possibly remote half-plane. This is not always the case: The frequency defined by $\lambda_n = \sqrt{\log{n}}$ satisfies Bohr's theorem, but there are convergent general Dirichlet series of this frequency which do not converge absolutely anywhere. It would be interesting to know whether our extension of Bohr's theorem holds for such frequencies.

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