Logarithm of Dirichlet L-Functions

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

\(\displaystyle \sum_{n \mathop \ge 1} \sum_p \frac{\chi (p)^n} {n p^{n s} }\) \(=\) \(\displaystyle \sum_p \left({\frac {\chi(p)} {p^s} + \frac {\chi(p)^2} {2 p^{2 s} } + \cdots}\right)\) $\quad$ Manipulation of Absolutely Convergent Series $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_p \log\left({\frac 1 {1 - \chi (p) p^{-s} } }\right)\) $\quad$ Taylor Series of Logarithm; this is the branch of $\log$ with $\log(1 + x) = x - x^2/2 + \cdots$ for small $x$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \log \prod_p \left({\frac 1 {1 - \chi (p) p^{-s} } }\right)\) $\quad$ Logarithm is Continuous; this not necessarily the same branch of $\log$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \log L (s, \chi)\) $\quad$ Euler Product $\quad$

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

$\blacksquare$