Analytic Continuation of Dirichlet L-Function

Theorem
Let $\chi : G := \paren {\Z / q \Z}^\times \to \C^\times$ be a Dirichlet character modulo $q$.

Let $\map L {s, \chi}$ be the Dirichlet $L$-function for $\chi$.

Let $\chi$ be the trivial character.

Then $\map L {s, \chi}$ has an analytic continuation to $\C$ except for a simple pole at $s = 1$.

Let $\chi$ be non-trivial.

Then $\map L {s, \chi}$ is analytic on $\map \Re s > 0$.

Proof
Let $\chi$ be the trivial character.

Then by Dirichlet L-Function from Trivial Character:


 * $\ds \map L {s, \chi} = \map \zeta s \cdot \prod_{p \mathop \divides q} \paren {1 - p^{-s} }$

where $\divides$ denotes divisibility.

Also, by Poles of Riemann Zeta Function, $\zeta$ is analytic on $\C$ except for a simple pole at $s = 1$.

Since $\map L {s, \chi}$ is just $\zeta$ times some constant, the same holds for this function.

If $\chi$ is non-trivial, then by the Orthogonality Relations for Characters:


 * $\ds \sum_{x \mathop \in G} \map \chi x = 0$

By definition, $\chi$ is $q$-periodic, and zero on integers not coprime to $q$.

So for any $M \in \N$:


 * $\ds \sum_{n \mathop = M + 1}^{M + Q} \map \chi n = 0$

Let $M, N \in \N$ be arbitrary.

Let $d$ be such that $M + q d \le N \le M + q \paren {d + 1}$.

Then:

So the coefficients $\map \chi n$ of $\map L {s, \chi}$ have bounded partial sums.

Therefore, by Convergence of Dirichlet Series with Bounded Partial Sums, $\map L {s, \chi}$ converges locally uniformly to an analytic function on $\map \Re s > 0$.