Möbius transform

In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius.

Other Möbius inversion formulas are obtained when different locally finite partially ordered sets replace the classic case of the natural numbers ordered by divisibility; for an account of those, see incidence algebra.

The classic version states that if g and f are arithmetic functions satisfying

then

where μ is the Möbius function and the sums extend over all positive divisors d of n. In effect, the original f(n) can be determined given g(n) by using the inversion formula. The two sequences are said to be Möbius transforms of each other.

The formula is also correct if f and g are functions from the positive integers into some abelian group (viewed as a ℤ-module).

In the language of Dirichlet convolutions, the first formula may be written as

where ∗ denotes the Dirichlet convolution, and 1 is the constant function 1(n) = 1. The second formula is then written as

Many specific examples are given in the article on multiplicative functions.

The theorem follows because ∗ is (commutative and) associative, and 1 ∗ μ = ε, where ε is the identity function for the Dirichlet convolution, taking values ε(1) = 1, ε(n) = 0 for all n > 1. Thus

...
Wikipedia