\begin{definition}[Definition:Ramanujan Sum]
Let $e: \R \to \C$ be the mapping defined as:
:$\forall \alpha \in \R: \map e \alpha := \map \exp {2 \pi i \alpha}$
For $q \in \N_{>0}$, $n \in \N$, the '''Ramanujan sum''' is defined as:
:$\ds \map {c_q} n = \sum_{\substack {1 \mathop \le a \mathop \le q \\ \gcd \set {a, q} \mathop = 1} } \map e {\frac {a n} q}$
\end{definition}