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.
Statement of the formula The classic version states that if g and f are arithmetic functions satisfying
g ( n ) = ∑ d ∣ n f ( d ) for every integer n ≥ 1 {\displaystyle g(n)=\sum _{d\mid n}f(d)\quad {\text{for every integer }}n\geq 1}
then
f ( n ) = ∑ d ∣ n μ ( d ) g ( n d ) for every integer n ≥ 1 {\displaystyle f(n)=\sum _{d\mid n}\mu (d)\,g\!\left({\frac {n}{d}}\right)\quad {\text{for every integer }}n\geq 1}
where μ is the Möbius function and the sums extend over all positive divisors d of n (indicated by d ∣ n {\displaystyle d\mid n} 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). In the language of Dirichlet convolutions, the first formula may be written as
g = 1 ∗ f {\displaystyle g={\mathit {1}}*f}
where ∗ denotes the Dirichlet convolution, and 1 is the constant function 1(n) = 1. The second formula is then written as
f = μ ∗ g . {\displaystyle f=\mu *g.}
Many specific examples are given in the article on multiplicative functions. 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
μ ∗ g = μ ∗ ( 1 ∗ f ) = ( μ ∗ 1 ) ∗ f = ε ∗ f = f {\displaystyle \mu *g=\mu *({\mathit {1}}*f)=(\mu *{\mathit {1}})*f=\varepsilon *f=f} . Replacing f , g {\displaystyle f,g} by ln f , ln g {\displaystyle \ln f,\ln g} , we obtain the product version of the Möbius inversion formula:
g ( n ) = ∏ d | n f ( d ) ⟺ f ( n ) = ∏ d | n g ( n d ) μ ( d ) , ∀ n ≥ 1. {\displaystyle g(n)=\prod _{d|n}f(d)\iff f(n)=\prod _{d|n}g\left({\frac {n}{d}}\right)^{\mu (d)},\forall n\geq 1.}
Series relations Let
a n = ∑ d ∣ n b d {\displaystyle a_{n}=\sum _{d\mid n}b_{d}}
so that
b n = ∑ d ∣ n μ ( n d ) a d {\displaystyle b_{n}=\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)a_{d}}
is its transform. The transforms are related by means of series: the Lambert series
∑ n = 1 ∞ a n x n = ∑ n = 1 ∞ b n x n 1 − x n {\displaystyle \sum _{n=1}^{\infty }a_{n}x^{n}=\sum _{n=1}^{\infty }b_{n}{\frac {x^{n}}{1-x^{n}}}}
and the Dirichlet series:
… excerpt ends here. Continue reading the full article.
