In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.
A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra.
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 (indicated by in the above formulae). 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 (viewed as a -module).
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 (commutative and) associative, and , where is the identity function for the Dirichlet convolution, taking values , for all . Thus
Replacing by , we obtain the product version of the Möbius inversion formula:
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.
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:
If the starting function is the Möbius function itself, the list of functions is:
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.
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 (complex-valued) 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. (This is because every fraction with and can be reduced to the fraction with , and vice versa.) Here it is straightforward to determine , but 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 :
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.
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, , where is the unit function.
We have the following:
The proof in the more general case where replaces 1 is essentially identical, as is the second generalisation.
For a poset , a set endowed with a partial order relation , define the Möbius function of recursively by
(Here one assumes the summations are finite.) Then for , where is a commutative ring, we have
if and only if
(See Stanley's Enumerative Combinatorics, Vol 1, Section 3.7.)
The classical arithmetic Mobius function is the special case of the poset P of positive integers ordered by divisibility: that is, for positive integers s, t, we define the partial order to mean that s is a divisor of t. On the power set of a set , ordered by (set inclusion), the Möbius inversion theorem reproduces the inclusionâÂÂexclusion principle, and on the set of natural numbers with their standard (total) ordering by the theorem coincides with a discrete version of the fundamental theorem of calculus (See Stanley's Enumerative Combinatorics, Vol 1, Section 3.8). Across the sciences, many measures of interaction can be formulated as Möbius inversions on different posets. Examples include Shapley values in game theory, maximum entropy interactions in statistical mechanics, epistasis in genetics, and the interaction information, total correlation, and partial information decomposition from information theory.