Dirichlet Series is Analytic


This article needs to be linked to other articles.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.


Theorem

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

Let:

$\ds \map f z = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^z}$

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.


Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.