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Concept
Möbius inversion formula
Summary
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 g and f are arithmetic functions satisfying
then
where μ is the Möbius function and the sums extend over all positive divisors d 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).
In the language of Dirichlet convolutions, the first formula may be written as
where ∗ denotes the Dirichlet convolution, and 1 is the constant function 1(n) = 1. 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 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:
Let
so that
is its transform. The transforms are related by means of series: the Lambert series
and the Dirichlet series:
where ζ(s) 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.
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Möbius function
The Möbius function μ(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted μ(x). For any positive integer n, define μ(n) as the sum of the primitive nth roots of unity.
Dirichlet series
In mathematics, a Dirichlet series is any series of the form where s is complex, and is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.
Möbius inversion formula
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.
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