# Ramanujan sum

For positive integers $s$ and $n$, the complex number^{}

$$ |

is referred to as a *Ramanujan sum ^{}*, or a

*Ramanujan trigonometric sum*. Since ${e}^{2\pi i}=1$, an equivalent

^{}definition is

$${c}_{s}(n)=\sum _{k\in r(n)}{e}^{2\pi iks/n}$$ |

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 $n$.

Using a symmetry^{} argument^{} about roots of unity^{}, one can show

$$\sum _{d|s}{c}_{s}(n)=\{\begin{array}{cc}s\hfill & \text{if}s|n\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}$$ |

Applying Möbius inversion, we get

$${c}_{s}(n)=\sum _{\begin{array}{c}d|n\\ d|s\end{array}}\mu (n/d)d=\sum _{d|(n,s)}\mu (n/d)d$$ |

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

$${c}_{st}(mn)={c}_{s}(m){c}_{t}(n)\text{if}(m,t)=(n,s)=1.$$ |

Using the Chinese remainder theorem^{}, it is not hard to show that
for any fixed $n$, the function $s\mapsto {c}_{s}(n)$ is multiplicative:

$${c}_{s}(n){c}_{t}(n)={c}_{st}(n)\text{if}(s,t)=1.$$ |

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 |
---|---|

Canonical name | RamanujanSum |

Date of creation | 2013-03-22 12:11:57 |

Last modified on | 2013-03-22 12:11:57 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 11 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 11L03 |

Classification | msc 11T23 |

Related topic | RootOfUnity |

Defines | Ramanujan trigonometric sum |