Definition:Dirichlet Character

Definition
Let $q \in \Z_{>1}$.

Let $\left({\Z / q \Z}\right)$ denote the ring of integers modulo $q$.

Let $G = \left({\Z / q \Z}\right)^\times$ be the group of units of $\left({\Z / q \Z}\right)$.

A Dirichlet character modulo q is a group homomorphism:
 * $\chi: G \to \C^\times$

By Multiplicative Group of Integers Modulo m, $a + q \Z \in G$ if and only if $\gcd \left({a, q}\right) = 1$.

It is standard practice to extend $\chi$ to a function on $\Z$ by setting:


 * $\displaystyle \chi \left({a}\right) = \begin{cases}

\chi \left({a + q \Z}\right) & : \gcd \left({a, q}\right) = 1 \\ 0 & : \text{otherwise} \end{cases}$

Primitive Character
Let $q^*$ be the least divisor of $q$ such that we can write $\chi = \chi_0 \chi^*$, where $\chi_0$ is the trivial character modulo $q$, and $\chi^*$ is some character modulo $q^*$.

If $q = q^*$ then $\chi$ is called primitive, otherwise $\chi$ is imprimitive.

Also see

 * Definition:Character (Number Theory)