The classic version states that if and are arithmetic functions satisfying then where is the Möbius function and the sums extend over all positive divisors of . In effect, the original can be determined given by using the inversion formula. The two sequences are said to be Möbius transforms of each other. The formula is also correct if and are functions from the positive integers into some abelian group. In the language of Dirichlet convolutions, the first formula may be written as where denotes the Dirichlet convolution, and is the constant function. The second formula is then written as Many specific examples are given in the article on multiplicative functions. The theorem follows because is associative, and, where is the identity function for the Dirichlet convolution, taking values, for all. Thus There is a product version of the summation-based Möbius inversion formula stated above:
Series relations
Let so that is its transform. The transforms are related by means of series: the Lambert series and the Dirichlet series: where is the Riemann zeta function.
Repeated transformations
Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation. For example, if one starts with Euler's totient function, and repeatedly applies the transformation process, one obtains:
Both of these lists of functions extend infinitely in both directions. The Möbius inversion formula enables these lists to be traversed backwards. As an example the sequence starting with is: The generated sequences can perhaps be more easily understood by considering the corresponding Dirichlet series: each repeated application of the transform corresponds to multiplication by the Riemann zeta function.
Generalizations
A related inversion formula more useful in combinatorics is as follows: suppose and are complex-valued functions defined on the interval such that then Here the sums extend over all positive integers which are less than or equal to. This in turn is a special case of a more general form. If is an arithmetic function possessing a Dirichlet inverse, then if one defines then The previous formula arises in the special case of the constant function, whose Dirichlet inverse is. A particular application of the first of these extensions arises if we have functions and defined on the positive integers, with By defining and, we deduce that A simple example of the use of this formula is counting the number of reduced fractions, where and are coprime and. If we let be this number, then is the total number of fractions with, where and are not necessarily coprime. Here it is straightforward to determine, but is harder to compute. Another inversion formula is : As above, this generalises to the case where is an arithmetic function possessing a Dirichlet inverse : For example, there is a well known proof relating the Riemann zeta function to the prime zeta function that uses the series-based form of Möbius inversion in the previous equation when. Namely, by the Euler product representation of for These identities for alternate forms of Möbius inversion are found in. A more general theory of Möbius inversion formulas partially cited in the next section on incidence algebras is constructed by Rota in.
Multiplicative notation
As Möbius inversion applies to any abelian group, it makes no difference whether the group operation is written as addition or as multiplication. This gives rise to the following notational variant of the inversion formula:
Proofs of generalizations
The first generalization can be proved as follows. We use Iverson's convention that is the indicator function of the condition, being 1 if the condition is true and 0 if false. We use the result that that is,. We have the following: The proof in the more general case where replaces 1 is essentially identical, as is the second generalisation.
On posets
For a poset, a set endowed with a partial order relation, define the Möbius function of recursively by Then for, where is a field, we have if and only if
