Ramanujan's Sum

The sum

$$c_q(m) = \sum_{h^*(q)} e^{2\pi i hm/q},$$ (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 $\mu$ is the Möbius Function, and for general $m$,

$$c_b(m) = \mu\left({b\over (b,m)}\right){\phi(b)\over \phi\left({b\over (b,m)}\right)}.$$ (4)

See also Möbius Function, Weyl's Criterion

References

Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, p. 254, 1991.