The sum
(1)
where relatively prime to
relatively prime ),
then
(2)
For argument 1,
(3)
where Möbius function . For general
(4)
where totient function .
See also Möbius Function ,
Weyl's
Criterion
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References Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. Vardi, I. Computational
Recreations in Mathematica. Referenced on Wolfram|Alpha Ramanujan's Sum
Cite this as:
Weisstein, Eric W. "Ramanujan's Sum."
From MathWorld https://mathworld.wolfram.com/RamanujansSum.html
Subject classifications
runs through the residues relatively prime to
, which is important in the representation
of numbers by the sums of squares. If 
(i.e., 
and 
' are relatively prime),
then
is the Möbius function. For general 
,
is the totient function.
