Ramanujan sum ($c_n(k)$)

The Ramanujan sum is $c_n(k)={\sum }_{1\le j\le n,(j,n)=1}{e}^{2\pi ijk/n}$. It is an integer: insert $1_{(j,n)=1}={\sum }_{d\mid (j,n)}\mu (d)$, sum the geometric series, and obtain $c_n(k)={\sum }_{r\mid (n,k)}r\mu (n/r)$. Here $\mu$ is the Möbius function. Such sums connect roots of unity with integer character of a representation values.