Analytic Continuation of Dirichlet L-Functions

Theorem

Let $\chi : G := \left(\Z/q\Z\right)^\times \to \C^\times$ be a Dirichlet character modulo $q$.

Let $L(s,\chi)$ be the Dirichlet $L$-function for $\chi$.

If $\chi$ is the trivial character then $L(s,\chi)$ has an analytic continuation to $\C$ except for a simple pole at $s = 1$.

If $\chi$ is non-trivial then $L(s,\chi)$ is analytic on $\Re(s) > 0$.

Proof

If $\chi$ is the trivial character, then by Dirichlet L-Function from Trivial Character,

$\displaystyle L(s,\chi) = \zeta(s) \cdot \prod_{p\mid q}\left(1 - p^{-s}\right)$

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

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

If $\chi$ is nontrivial, then by the Orthogonality Relations for Characters,

$\displaystyle \sum_{x \in G} \chi(x) = 0$

By definition, $\chi$ is $q$-periodic, and zero on integers not coprime to $q$, so for any $M \in \N$,

$\displaystyle \sum_{n = M + 1}^{M+Q}\chi(n) = 0$

Let $M,N \in \N$ be arbitrary, and let $d$ be such that $M + qd \leq N \leq M + q(d+1)$. Then

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sum_{n = M}^N \chi(n)\)	\(=\)	\(\displaystyle \sum_{k = 0}^{d-1} \sum_{n = 0}^{q-1}\chi(M + kq + n) + \sum_{n = M + qd}^N\chi(n)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{n = M + qd}^N\chi(n)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Because $\chi$ is $q$-periodic, and zero on integers not coprime to $q$.
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\leq\)	\(\displaystyle q\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Because $\left\vert N - M + qd \right\vert \leq q$

So the coefficients $\chi(n)$ of $L(s,\chi)$ have bounded partial sums.

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

$\blacksquare$