The classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. It was later generalized to other "Möbius inversion formulas"; see incidence algebra. The classic version states that if g(n) and f(n) are arithmetic functions satisfying Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... August Ferdinand Möbius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. ... In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ... In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. ...

then

where μ is the Möbius function and the sums extend over all positive divisors d of n. The classical Möbius function is an important multiplicative function in number theory and combinatorics. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

The formula is also correct if f and g are functions from the positive integers into some abelian group. In mathematics, an abelian group is a commutative group, i. ...

In the language of convolutions (see multiplicative function), the inversion formula can also be expressed as In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...

μ * 1 = ε.

An equivalent formulation of the 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 Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

then

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

The Möbius inversion treated above is the original Möbius inversion. When the partially ordered set of natural numbers ordered by divisibility one is replaced by other locally finite partially ordered sets, one has other Möbius inversion formulas; for an account of those, see incidence algebra. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ...

See also August Ferdinand Möbius. August Ferdinand Möbius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. ...