Definition:Ramanujan Sum

Definition
For $\alpha \in \R$ let $e(\alpha) = \exp(2 \pi i \alpha)$.

For $q \in \N$, $n \in \N \cup\{0\}$, we define the Ramanujan sum:


 * $\displaystyle c_q(n) = \sum_{\substack{1 \leq a \leq q\\\gcd(a,q) = 1}} e\left( \frac{an}q \right)$