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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.
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The most famous example of a Dirichlet series is = =,whose analytic continuation to (apart from a simple pole at =) is the Riemann zeta function.. Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write +:
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.
Linear fractional transformations are shown to be conformal maps by consideration of their generators: multiplicative inversion z → 1/z and affine transformations z → az + b. Conformality can be confirmed by showing the generators are all conformal. The translation z → z + b is a change of origin and makes no difference to angle.
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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]