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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 .
For example, if one starts with Euler's totient function φ, and repeatedly applies the transformation process, one obtains: φ the totient function; φ ∗ 1 = I, where I(n) = n is the identity function; I ∗ 1 = σ 1 = σ, the divisor function; If the starting function is the Möbius function itself, the list of functions is: μ, the Möbius ...
There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: /, which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), , which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and / which fixes 0 and swaps 1 with ∞ ...
An example of such linear fractional transformation is the Cayley transform, which was originally defined on the 3 × 3 real matrix ring. Linear fractional transformations are widely used in various areas of mathematics and its applications to engineering, such as classical geometry , number theory (they are used, for example, in Wiles's proof ...
Examples of this trope include Martin Gardner ' s "No-Sided Professor" (1946), Armin Joseph Deutsch ' s "A Subway Named Mobius" (1950) and the film Moebius (1996) based on it. An entire world shaped like a Möbius strip is the setting of Arthur C. Clarke 's "The Wall of Darkness" (1946), while conventional Möbius strips are used as clever ...
A two-dimensional representation of the Klein bottle immersed in three-dimensional space. In mathematics, the Klein bottle (/ ˈ k l aɪ n /) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.
The Möbius strip is one of the most famous objects in mathematics. Discovered in 1858 by two German mathematicians—August Ferdinand Möbius and Johann Benedict Listing—the Möbius strip is a ...
A combinatorial algorithm has been developed incrementally starting in 1870 by Ernst Meissel, [8] Lehmer, [9] Lagarias-Miller-Odlyzko, [10] and Deléglise-Rivat [11] that computes isolated values of M(x) in O(x 2/3 (log log x) 1/3) time; a further improvement by Harald Helfgott and Lola Thompson in 2021 improves this to O(x 3/5 (log x) 3/5+ε ...