structural:scholar_edition, linguistic:paraphrasing, information:classification
Statement of the formula

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 (indicated by d ∣ n {\displaystyle d\mid n} in the above formulae). 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

Replacing f, g {\displaystyle f,g} by ln f, ln g {\displaystyle \ln f,\ln g}, we obtain the product version of the Möbius inversion formula: