Definition:Dirichlet Character

Definition
Let $G = \left( \Z / q\Z \right)^\times$ be the the multiplicative Group of Units of a the ring ring $\left( \Z / q\Z \right)$.

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 $\gcd(a,q) = 1$.

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


 * $\chi(a + q\Z) = 0, \quad \gcd(a,q) > 1$.

Furthermore, one extends $\chi$ to a $q$-periodic function on $\Z$ by setting


 * $\chi(a) = \chi(a + q\Z),\quad a \in \Z$.

Also See

 * Character (Number Theory)