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Mbius Inversion FormulaThe classic Mbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Mbius. It was later generalized to other "Mbius inversion formulas"; see incidence algebra. The classic version states that if g(n) and f(n) are arithmetic functions satisfying -
then -
where μ is the Mbius function and the sums extend over all positive divisors d of n. The formula is also correct if f and g are functions from the positive integers into some abelian group. In the language of convolutions (see multiplicative function), the inversion formula can also be expressed as - 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 -
then -
Here the sums extend over all positive integers n which are less than or equal to x. The Mbius inversion treated above is the original Mbius 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 Mbius inversion formulas; for an account of those, see incidence algebra. See also August Ferdinand Mbius.
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