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.
where μ is the
Möbius function and the sums extend over all positive
divisorsd of n (indicated by 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).
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 by , we obtain the product version of the Möbius inversion formula:
Series relations
Let
so that
is its transform. The transforms are related by means of series: the
Lambert series
1 ∗ 1 = σ0 = d = τ, where d = τ is the number of divisors of n, (see
divisor function).
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 φ 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 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 inverseα−1(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−1(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 fractions0 < 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−1(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 . Namely, by the
Euler product representation of for
These identities for alternate forms of Möbius inversion are found in.[2]
A more general theory of Möbius inversion formulas partially cited in the next section on incidence algebras is constructed by Rota in.[3]
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 [condition] is the indicator function of the condition, being 1 if the condition is true and 0 if false. We use the result that
For a
posetP, a set endowed with a partial order relation , define the Möbius function of P recursively by
(Here one assumes the summations are finite.) Then for , where K 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.
Contributions of Weisner, Hall, and Rota
The statement of the general Möbius inversion formula [for partially ordered sets] 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.[4]
Ireland, K.; Rosen, M. (2010), A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics (Book 84) (2nd ed.), Springer-Verlag,
ISBN978-1-4419-3094-1