Ramanujan sum
= Ramanujan sum
{c}
{title2=$c_n(k)$}
{wiki}
The Ramanujan sum is $c_n(k)=\sum_{1\leq j\leq n,\,(j,n)=1}e^{2\pi i jk/n}$. It is an integer: insert $\mathbf1_{(j,n)=1}=\sum_{d\mid(j,n)}\mu(d)$, 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.