Ramanujan's Sum

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.

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.