Ramanujan sum

For positive integers s and n, the complex numberMathworldPlanetmathPlanetmath


is referred to as a Ramanujan sumMathworldPlanetmath, or a Ramanujan trigonometric sum. Since e2πi=1, an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definition is


where r(n) is some reduced residue systemMathworldPlanetmath mod n, meaning any subset of containing exactly one element of each invertiblePlanetmathPlanetmathPlanetmathPlanetmath residue classMathworldPlanetmathPlanetmath mod n.

Using a symmetryPlanetmathPlanetmathPlanetmath argumentPlanetmathPlanetmath about roots of unityMathworldPlanetmath, one can show

d|scs(n)={sif s|n0otherwise.

Applying Möbius inversion, we get


which shows that cs(n) is a real number, and indeed an integer. In particular cs(1)=μ(s). More generally,

cst(mn)=cs(m)ct(n) if (m,t)=(n,s)=1.

Using the Chinese remainder theoremMathworldPlanetmathPlanetmathPlanetmath, it is not hard to show that for any fixed n, the function scs(n) is multiplicative:

cs(n)ct(n)=cst(n) if (s,t)=1.

If m is invertible mod n, then the mapping kkm is a permutation of the invertible residue classes mod n. Therefore

cs(mn)=cs(n) if (m,s)=1.

Remarks: Trigonometric sums often make convenient apparatus in number theoryMathworldPlanetmath, since any function on a quotient ring of defines a periodic function on 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 cs(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