The following extension of Bohr’s theorem is established: If a somewhere convergent Dirichlet series
f
f
has an analytic continuation to the half-plane
C
θ
=
{
s
=
σ
+
i
t
:
σ
>
θ
}
\mathbb {C}_\theta = \{s = \sigma +it\,:\, \sigma >\theta \}
that maps
C
θ
\mathbb {C}_\theta
to
C
∖
{
α
,
β
}
\mathbb {C} \setminus \{\alpha ,\beta \}
for complex numbers
α
≠
β
\alpha \neq \beta
, then
f
f
converges uniformly in
C
θ
+
ε
\mathbb {C}_{\theta +\varepsilon }
for any
ε
>
0
\varepsilon >0
. The extension is optimal in the sense that the assertion no longer holds should
C
∖
{
α
,
β
}
\mathbb {C}\setminus \{\alpha ,\beta \}
be replaced with
C
∖
{
α
}
\mathbb {C}\setminus \{\alpha \}
.