Logarithm of Dirichlet L-Functions

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

The Dirichlet series


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

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

Moreover, $f(s)$ defines a branch of $\log L (s, \chi)$

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

For fixed $s \in \{ \Re(s) > 1 \}$,

Hence $f (s)$ is an analytic branch of $\log$ on $\Re(s) > 1$.