Dirichlet Character
Definition
Let \(f\) be a group character of \((\mathbb{Z}/n\mathbb{Z})^\ast\). We define a Dirichlet character modulo \(n\) by taking \(f\) and extending it to a function on \(\mathbb{Z}\) by taking
\[ \chi(a) = \begin{cases}
f(a + n\mathbb{Z}) & \gcd(a, n) = 1 \\
0 & \text{otherwise} \\
\end{cases}.\]