Logarithm of Dirichlet L-Functions

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

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

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

$\blacksquare$