# Ramanujan sum

For positive integers $s$ and $n$, the complex number   $c_{s}(n)=\sum_{\begin{subarray}{c}0\leq k
 $c_{s}(n)=\sum_{k\in r(n)}e^{2\pi iks/n}$

where $r(n)$ is some reduced residue system  mod $n$, meaning any subset of $\mathbb{Z}$ containing exactly one element of each invertible    residue class   mod $n$.

 $\sum_{d|s}c_{s}(n)=\begin{cases}s&\textrm{if s|n}\\ 0&\text{otherwise.}\end{cases}$

Applying Möbius inversion, we get

 $c_{s}(n)=\sum_{\begin{subarray}{c}d|n\\ d|s\end{subarray}}\mu(n/d)d=\sum_{d|(n,s)}\mu(n/d)d$

which shows that $c_{s}(n)$ is a real number, and indeed an integer. In particular $c_{s}(1)=\mu(s)$. More generally,

 $c_{st}(mn)=c_{s}(m)c_{t}(n)\text{ if }(m,t)=(n,s)=1\;.$

If $m$ is invertible mod $n$, then the mapping $k\mapsto km$ is a permutation of the invertible residue classes mod $n$. Therefore

 $c_{s}(mn)=c_{s}(n)\text{ if }(m,s)=1\;.$

Remarks: Trigonometric sums often make convenient apparatus in number theory  , since any function on a quotient ring of $\mathbb{Z}$ defines a periodic function on $\mathbb{Z}$ itself, and conversely. For another example, see Landsberg-Schaar relation.

Some writers use different notation from ours, reversing the roles of $s$ and $n$ in the expression $c_{s}(n)$.

The name “Ramanujan sum” was introduced by Hardy.

Title Ramanujan sum RamanujanSum 2013-03-22 12:11:57 2013-03-22 12:11:57 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 11L03 msc 11T23 RootOfUnity Ramanujan trigonometric sum