Logarithm of Dirichlet L-Functions

Theorem
Let $\chi$ be a Dirichlet character modulo $q$.

The Dirichlet series


 * $\map f s = \displaystyle \sum_{n \mathop \ge 1} \sum_p \frac {\map \chi p^n} {n p^{n s} }$

converges absolutely to an analytic function, where $p$ ranges over the primes.

Moreover, $\map f s$ defines a branch of $\ln \map L {s, \chi}$

Proof
By Convergence of Dirichlet Series with Bounded Coefficients, $\map f s$ converges absolutely on $\map \Re s > 1$ to an analytic function.

For fixed $s \in \set {\map \Re s > 1}$:

Hence $\map f s$ is an analytic branch of $\ln$ on $\map \Re s > 1$.