Dirichlet character

Remarks

Example: for $q = 4$, let $\chi(n) = 0$ for even $n$, $\chi(n) = 1$ for $n \equiv 1$ and $\chi(n) = -1$ for $n \equiv 3 \pmod 4$. Clauses 1 and 2 are clear. For clause 3 it suffices to check odd classes: $1 \cdot 1 \equiv 1$, $1 \cdot 3 \equiv 3$ and $3 \cdot 3 \equiv 1 \pmod 4$ match $1 \cdot 1 = 1$, $1 \cdot (-1) = -1$ and $(-1)(-1) = 1$. Non-example: for $q = 3$, the function $f(n) = r$, where $r \in \set{0, 1, 2}$ is the remainder of $n$ modulo $3$, satisfies clauses 1 and 2 but not clause 3, since $f(2 \cdot 2) = f(4) = 1 \ne 4 = f(2) f(2)$. Equivalently, a Dirichlet character modulo $q$ is a group homomorphism $(\Z/q\Z)^\times \to \C^\times$ from the unit group of the integers modulo q, extended by $0$ on the classes not coprime to $q$. Hence each value $\chi(n)$ with $\gcd(n, q) = 1$ is a $\varphi(q)$-th root of unity, and there are exactly $\varphi(q)$ characters modulo $q$, where $\varphi$ is Euler's totient function. Each Dirichlet character is a completely multiplicative function. The Legendre symbol $n \mapsto (n/p)$ is a Dirichlet character modulo an odd prime $p$.