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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 .
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.
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 ...
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and = () whenever a and b are coprime.. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime.
MATLAB (an abbreviation of "MATrix LABoratory" [18]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
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).
The constant-coefficient linear recurrences which are periodic are precisely the power series coefficients of rational functions whose denominators are products of cyclotomic polynomials. In the theory of combinatorial generating functions , the denominator of a rational function determines a linear recurrence for its power series coefficients.
The zeta function of an incidence algebra is the constant function ζ(a, b) = 1 for every nonempty interval [a, b]. Multiplying by ζ is analogous to integration . One can show that ζ is invertible in the incidence algebra (with respect to the convolution defined above).