Dirichlet character

Summary

A Dirichlet character modulo $q$ is a function $\chi \colon \Z \to \C$ that comes from a homomorphism $(\Z/q\Z)^\times \to \C^\times$. It is periodic with period $q$ and completely multiplicative. There are exactly $\varphi(q)$ characters modulo $q$, and their orthogonality relations isolate a single residue class. Dirichlet introduced them to prove Dirichlet's theorem on arithmetic progressions, through the Dirichlet L-functions $L(s, \chi)$.