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Ramanujan's Sum


The sum

 c_q(m)=sum_(h^*(q))e^(2piihm/q),
(1)

where h runs through the residues relatively prime to q, which is important in the representation of numbers by the sums of squares. If (q,q^')=1 (i.e., q and q' are relatively prime), then

 c_(qq^')(m)=c_q(m)c_(q^')(m).
(2)

For argument 1,

 c_b(1)=mu(b),
(3)

where mu(n) is the Möbius function. For general m,

 c_b(m)=mu(b/((b,m)))(phi(b))/(phi(b/((b,m)))),
(4)

where phi(n) 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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