Theorem: Uniform Convergence of General Dirichlet Series

Let $\map \arg z$ denote the argument of the complex number $z \in \C$.
Let $\ds \map f s = \sum_{n \mathop = 1}^\infty a_n e^{-\map {\lambda_n} s}$ be a general Dirichlet series.
Let $\map f s$ converge at $s_0 = \sigma_0 + i t_0$.
Then $\map f s$ converges uniformly for all $s$ such that:
:$\cmod {\map \arg {s - s_0} } \le a < \dfrac \pi 2$

Proof:
Let $s = \sigma + i t$
Let $s_0 \in \C$ be such that $\map f {s_0}$ converges. 
Let $\map S {m, n} = \ds \sum_{k \mathop = n}^m a_k e^{-\lambda_k s_0}$
We may create a new Dirichlet series that converges at $0$ by writing:
{{begin-eqn}}
{{eqn | l = \map g s
      | r = \map f {s + s_0}
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n \paren {s + s_0} }
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n s_0} e^{-\lambda_n s}
}}
{{end-eqn}}
Thus it suffices to show $\map g s$ converges uniformly for for $\cmod {\map \arg s} \le  a < \frac \pi 2$
By Cauchy's Convergence Criterion, it suffices to show that for all $\epsilon > 0$ there exists an $N$ independent of $s$ such that for all $m, n > N$:
:$\ds \cmod {\sum_{k \mathop = n}^m a_n e^{-\lambda_k s_0} e^{-\lambda_k s} } < \epsilon$
By Abel's Lemma: Formulation 2 we may write:
{{begin-eqn}}
{{eqn | l = \cmod {\sum_{k \mathop = n}^m a_k e^{-\lambda_k s_0} e^{-\lambda_k s} }
      | r = \cmod {\sum_{k \mathop = n}^m \paren {\map S {k, n} - \map S {k - 1, n} } e^{-\lambda_k s} }
}}
{{eqn | r = \cmod {\map S {m, n} e^{-\lambda_m s} + \sum_{k \mathop = n}^{m - 1} \map S {k, n} \paren {e^{-\lambda_k s} - e^{-\lambda_{k + 1} s} } }
}}
{{eqn | o = \le
      | r = \cmod {\map S {m, n} e^{-\lambda_m s} } + \sum_{k \mathop = n}^{m - 1} \cmod {\map S {k, n} \paren {e^{-\lambda_k s} - e^{-\lambda_{k + 1} s} } }
      | c = Triangle Inequality
}}
{{end-eqn}}
Because $\map S {k, j}$ is the difference of two terms of a convergent, and thus cauchy, sequence, we may pick $N$ large enough so that for $j > N$:
:$\ds \cmod {\map S {k, j} } < \frac {\epsilon \cos a} 3$
which gives us:
{{begin-eqn}}
{{eqn | l = \cmod {\map S {m, n} e^{-\lambda_m s} } + \sum_{k \mathop = n}^{m - 1} \cmod {\map S {k, n} \paren {e^{-\lambda_k s} - e^{-\lambda_{k + 1} s} } }
      | o = \le
      | r = \frac {\epsilon \cos a} 3 \paren {\cmod {e^{-\lambda_m s} } + \sum_{k \mathop = n}^{m - 1} \cmod {\paren {e^{-\lambda_k s} - e^{-\lambda_{k + 1} s} } } }
}}
{{end-eqn}}
We see that:
{{begin-eqn}}
{{eqn | l = \cmod {e^{-\lambda_k s} - e^{-\lambda_{k + 1} s} }
      | r = \cmod {\int_{\lambda_k}^{\lambda_{k + 1} } -s e^{-x s} \rd x}
}}
{{eqn | o = \le
      | r = \int_{\lambda_k}^{\lambda_{k + 1} } \cmod {-s e^{-x s} } \rd x
      | c = Modulus of Complex Integral
}}
{{eqn | r = \int_{\lambda_k}^{\lambda_{k + 1} } \cmod s e^{-x \sigma} \rd x
      | c = 
}}
{{eqn | r = \cmod s \int_{\lambda_k}^{\lambda_{k + 1} } e^{-x \sigma} \rd x
      | c = 
}}
{{eqn | r = \frac {\cmod s} \sigma \paren {e^{-\lambda_k \sigma} - e^{-\lambda_{k + 1} \sigma} }
      | c = 
}}
{{eqn | r = \map \sec {\map \arg s} \paren {e^{-\lambda_k \sigma} - e^{-\lambda_{k + 1} \sigma} }
      | c = 
}}
{{end-eqn}}
From Shape of Secant Function, we have that on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$:
:$\cmod {\map \arg s} \le a \implies \map \sec {\map \arg s} \le \sec a$
which gives us:
:$\map \sec {\map \arg s} \paren {e^{-\lambda_k \sigma} - e^{-\lambda_{k + 1} \sigma} } \le \sec a \paren {e^{-\lambda_k \sigma} - e^{-\lambda_{k + 1} \sigma} }$
Hence:
{{begin-eqn}}
{{eqn | l = \frac {\epsilon \cos a} 3 \paren {\cmod {e^{-\lambda_m s} } + \sum_{k \mathop = n}^{m - 1} \cmod {\paren {e^{-\lambda_k s} - e^{-\lambda_{k + 1} s} } } }
      | o = \le
      | r = \frac {\epsilon \cos a} 3 \paren {e^{-\lambda_m \sigma} + \sec a \sum_{k \mathop = n}^{m - 1} e^{-\lambda_k \sigma} - e^{-\lambda_{k + 1} \sigma} }
}}
{{eqn | r = \frac {\epsilon \cos a} 3 \paren {e^{-\lambda_m \sigma} + \sec a \paren {e^{-\lambda_n \sigma} - e^{-\lambda_m \sigma} } }
      | c = Telescoping Sum
}}
{{end-eqn}}
Because $\sigma>0$, we have that $ - \lambda_k \sigma <0$ and hence:
:$e^{-\lambda_k \sigma} < 1 \le \sec a$
which gives us:
{{begin-eqn}}
{{eqn | l = \cmod {\sum_{k \mathop = n}^m a_n e^{-\lambda_k s_0} e^{-\lambda_k s} }
      | o = \le
      | r = \frac {\epsilon \cos a} 3 \paren {e^{-\lambda_m\ sigma} + \sec a \paren {e^{-\lambda_n \sigma} - e^{-\lambda_{m}\sigma} } }
}}
{{eqn | o = \le
      | r = \frac {\epsilon \cos a} 3 \paren {\sec a + \sec a \paren {1 + 1} }
}}
{{eqn | r = \frac {\epsilon \cos a} 3 \paren {3 \sec a}
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
{{qed}}