where μ is the Möbius function and the sums extend over all positive divisorsd 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 -module).
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 in 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.
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 (complex-valued) functions f(n) and g(n) 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 0 < a/b < 1, where a and b are coprime and b≤n. If we let f(n) be this number, then g(n) is the total number of fractions 0 < a/b < 1 with b≤n, where a and b are not necessarily coprime. (This is because every fraction a/b with gcd(a,b) = d and b≤n can be reduced to the fraction (a/d)/(b/d) with b/d ≤ n/d, and vice versa.) Here it is straightforward to determine g(n) = n(n-1)/2, but f(n) is harder to compute.
Another inversion formula is (where we assume that the series involved are absolutely convergent):
As above, this generalises to the case where is an arithmetic function possessing a Dirichlet inverse :
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:
The first generalization can be proved as follows. We use Iverson's convention that [condition] 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, 1*μ=i.
We have the following:
The proof in the more general case where α(n) replaces 1 is essentially identical, as is the second generalisation.
The statement of the general Möbius inversion formula was first given independently by Weisner (1935) and Philip Hall (1936); both authors were motivated by group theory problems. Neither author seems to have been aware of the combinatorial implications of his work and neither developed the theory of Möbius functions. In a fundamental paper on Möbius functions, Rota showed the importance of this theory in combinatorial mathematics and gave a deep treatment of it. He noted the relation between such topics as inclusion-exclusion, classical number theoretic Möbius inversion, coloring problems and flows in networks. Since then, under the strong influence of Rota, the theory of Möbius inversion and related topics has become an active area of combinatorics.