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  2. Möbius function - Wikipedia

    en.wikipedia.org/wiki/Möbius_function

    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.

  3. Möbius inversion formula - Wikipedia

    en.wikipedia.org/wiki/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 ...

  4. Möbius transformation - Wikipedia

    en.wikipedia.org/wiki/Möbius_transformation

    The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions g 1, g 2, g 3, g 4 such that each g i is the inverse of f i.

  5. Mertens function - Wikipedia

    en.wikipedia.org/wiki/Mertens_function

    Mertens function to n = 10 000 Mertens function to n = 10 000 000. In number theory, the Mertens function is defined for all positive integers n as = = (), where () is the Möbius function. The function is named in honour of Franz Mertens.

  6. Cross-ratio - Wikipedia

    en.wikipedia.org/wiki/Cross-ratio

    As functions of , these are examples of Möbius transformations, which under composition of functions form the Mobius group PGL(2, C). The six transformations form a subgroup known as the anharmonic group, again isomorphic to S 3. They are the torsion elements (elliptic transforms) in PGL(2, C).

  7. Sieve theory - Wikipedia

    en.wikipedia.org/wiki/Sieve_theory

    The partial sum of the sifting function alternately over- and undercounts, so the remainder term will be huge. Brun's idea to improve this was to replace () in the sifting function with a weight sequence () consisting of restricted Möbius functions.

  8. Selberg sieve - Wikipedia

    en.wikipedia.org/wiki/Selberg_sieve

    In terms of sieve theory the Selberg sieve is of combinatorial type: that is, derives from a careful use of the inclusion–exclusion principle.Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem.

  9. Möbius ladder - Wikipedia

    en.wikipedia.org/wiki/Möbius_ladder

    In graph theory, the Möbius ladder M n, for even numbers n, is formed from an n-cycle by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of M 6 (the utility graph K 3,3), M n has exactly n/2 four-cycles [1] which link together by their shared edges to form a topological Möbius strip.