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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.
The Möbius function () is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. [i] [ii] [2] It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula.
This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The Möbius group is then a complex Lie group . The Möbius group is usually denoted Aut ( C ^ ) {\displaystyle \operatorname {Aut} ({\widehat {\mathbb {C} }})} as it is the automorphism group of the Riemann sphere.
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His interest in number theory led to the important Möbius function μ(n) and the Möbius inversion formula. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results. [6]
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The restriction of the divisors in the convolution to unitary, bi-unitary or infinitary divisors defines similar commutative operations which share many features with the Dirichlet convolution (existence of a Möbius inversion, persistence of multiplicativity, definitions of totients, Euler-type product formulas over associated primes, etc.).
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