Möbius Inversion Formula

If $g(n)\equiv \sum_{d\vert n} f(d)$, then

$$f(n)=\sum_{d\vert n} \mu(d)g\left({n\over d}\right),$$

where the sums are over all possible Integers $d$ that Divide $n$ and $\mu(d)$ is the Möbius Function. The Logarithm of the Cyclotomic Polynomial

$$\Phi_n(x)=\prod_{d\vert n}(1-x^{n/d})^{\mu(d)}$$

is the Möbius inversion formula.

See also Cyclotomic Polynomial, Möbius Function

References

Hardy, G. H. and Wright, W. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 91-93, 1979.

Schroeder, M. R. Number Theory in Science and Communication, 3rd ed. New York: Springer-Verlag, 1997.

Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 7-8 and 223-225, 1991.