quality:correction, structural:summarization
Generalizations

A related inversion formula more useful in combinatorics is as follows: suppose F (x) and G (x) are complex -valued functions defined on the interval [ 1, ∞) such that

then

Here the sums extend over all positive integers n which are less than or equal to x.

This in turn is a special case of a more general form. If α (n) is an arithmetic function possessing a Dirichlet inverse α (n), then if one defines

then

The previous formula arises in the special case of the constant function α (n) = 1, whose Dirichlet inverse is α (n) = μ (n).

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 F (x) = f (⌊ x ⌋) and G (x) = g (⌊ x ⌋), 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 α (n) is an arithmetic function possessing a Dirichlet inverse α (n):

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 s = 1 {\displaystyle s=1}. Namely, by the Euler product representation of ζ (s) {\displaystyle \zeta (s)} for ℜ (s) > 1 {\displaystyle \Re (s)>1}

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.