Sorry for the lengthy question. But is this statement true? Because in all papers on the Dirichlet character I read, they assume this to be true but don't really provide an explanation.
For example, I've seen:
$|{\chi(n) \over n^s}|={1\over |n^s|} \space$ for all $\space s\in\mathbb{C} \space$ with real part greater than $\space 1 \space$.
Sorry if this is obvious, but I can't seem to get my head around it.
More generally, a character is a group homomorphism $\chi:G\to\mathbb C^\times$ and so induces a group homomorphism $\phi:G\to\mathbb R_+^\times$ given by $\phi(g)=|\chi(g)|$, since $z \mapsto |z|$ is a group homomorphism $\mathbb C^\times\to\mathbb R_+^\times$.
Any group homomorphism takes torsion elements to torsion elements.
If $G$ is finite, then all elements in $G$ are torsion elements. Therefore, $\phi$ takes all elements of $G$ to $1$, the only torsion element of $\mathbb R_+^\times$, that is, $|\chi(g)|=1$ for all $g \in G$.