Theorem: Dirichlet Series Convergence Lemma/General

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 $\map f s$ converge for all $s = \sigma + i t$ where $\sigma > \sigma_0$.

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 for $\sigma > 0$.
By Cauchy's Convergence Criterion, it suffices to show that for all $\epsilon>0$ there exists an $N$ 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} } }
}}
{{end-eqn}}
Because $\map S {m, n}$ is the difference of partial sums of a convergent, and thus cauchy, sequence, its modulus, $\cmod {\map S {m, n} }$, is bounded, say by $Q$.
Thus we have:
{{begin-eqn}}
{{eqn | l = \cmod {\map S {m, n}^{-\lambda_m s} + \sum_{k \mathop = n}^{m - 1} \map S {k, n} \paren {e^{-\lambda_k s} - e^{-\lambda_{k + 1} s} } }
      | 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} } }
}}
{{eqn | o = \le
      | r = Q e^{-\lambda_m \sigma} + Q \sum_{k \mathop = n}^{m - 1} \cmod {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 = 
}}
{{end-eqn}}
Thus we have:
{{begin-eqn}}
{{eqn | l = Q e^{-\lambda_m \sigma} + Q \sum_{k \mathop = n}^{m - 1} \cmod {e^{-\lambda_k s} - e^{-\lambda_{k + 1} s} }
      | o = \le
      | r = Q e^{-\lambda_m \sigma} + Q \sum_{k \mathop = n}^{m - 1} \frac {\cmod s} {\sigma} \paren {e^{-\lambda_k \sigma} - e^{-\lambda_{k + 1} \sigma} }
}}
{{eqn | r = Q e^{-\lambda_m \sigma} + Q \frac {\cmod s} \sigma \paren {e^{-\lambda_n \sigma} - e^{-\lambda_m \sigma} }
      | c = Telescoping Series
}}
{{end-eqn}}
Because $\lambda_n$ tends to infinity, both summands tend to $0$ as $n$ goes to $\infty$ if $\sigma > 0$.
Thus we can pick $N$ large enough such that both terms are less than $\dfrac \epsilon 2$ for $n, m > N$, giving us the desired result.
{{qed}}
Category:General Dirichlet Series
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