Orthogonality of Dirichlet Characters
The following are all direct corollaries of the results given in orthogonality of group characters.
Theorem
Let \(n \in \mathbb{Z}\), then and \(\chi\) a dirichlet character modulo \(n\), then
\[ \sum_{0 \leq a < n} \chi(a) = \begin{cases}
\varphi(n) & \chi = \chi_0 \\
0 & \text{otherwise} \\
\end{cases}\]
Theorem
Let \(n, a \in \mathbb{Z}\), then
\[ \sum_{\chi \pmod n} \chi(a) = \begin{cases}
\varphi(n) & a \equiv 1 \pmod n \\
0 & \text{otherwise} \\
\end{cases}\]
where the sum is taken over all dirichlet characters modulo \(n\).
Corollary
Let \(n, a, b \in \mathbb{Z}\) with \(\gcd(b, n) = 1\), then
\[ \sum_{\chi \pmod n} \overline{\chi(b)}\chi(a) = \begin{cases}
\varphi(n) & a \equiv b \pmod n \\
0 & \text{otherwise} \\
\end{cases}\]
where the sum is taken over all Dirichlet characters modulo \(n\).