Concept# Möbius inversion formula

Summary

In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.
A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra.
Statement of the formula
The classic version states that if g and f are arithmetic functions satisfying
: g(n)=\sum_{d \mid n}f(d)\quad\text{for every integer }n\ge 1
then
:f(n)=\sum_{d \mid n}\mu(d)g\left(\frac{n}{d}\right)\quad\text{for every integer }n\ge 1
where μ is the Möbius function and the sums extend over all positive divisors d of n (indicated by d \mid n in the above formulae). In effect, the orig

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