Dirichlet Series is Analytic

Theorem
Let $(a_n)$ be sequence of complex numbers.

Let


 * $\displaystyle f \left({z}\right) = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^z}$

be the associated Dirichlet Series, which is defined at the points where the series converges.

Then $f$ is analytic in every open set such that the sum converges in the set.