Logarithm of Dirichlet L-Functions

Theorem

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

The Dirichlet series:

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

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

\(\ds \sum_{n \mathop \ge 1} \sum_p \frac {\map \chi p^n} {n p^{n s} }\)

\(=\)

\(\ds \sum_p \paren {\frac {\map \chi p} {p^s} + \frac {\map \chi p^2} {2 p^{2 s} } + \cdots}\)

Manipulation of Absolutely Convergent Series

\(\ds \)

\(=\)

\(\ds \sum_p \map \ln {\frac 1 {1 - \map \chi p p^{-s} } }\)

Taylor Series of Logarithm; this is the branch of $\ln$ with $\map \ln {1 + x} = x - \dfrac {x^2} 2 + \cdots$ for small $x$

\(\ds \)

\(=\)

\(\ds \ln \prod_p \paren {\frac 1 {1 - \map \chi p p^{-s} } }\)

Logarithm is Continuous; this is not necessarily the same branch of $\ln$

\(\ds \)

\(=\)

\(\ds \ln \map L {s, \chi}\)

Definition of Euler Product

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

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$\blacksquare$