The sum
(1)
where
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
(2)
For argument 1,
(3)
where
is the Möbius function . For general ,
(4)
where
is the 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. New York:
Chelsea, pp. 137-143, 1999. Vardi, I. Computational
Recreations in Mathematica. Redwood City, CA: Addison-Wesley, p. 254,
1991. Referenced on Wolfram|Alpha Ramanujan's Sum
Cite this as:
Weisstein, Eric W. "Ramanujan's Sum."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujansSum.html
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