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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 .
It was originally known as "HECKE and Manin". After a short while it was renamed SAGE, which stands for ‘’Software of Algebra and Geometry Experimentation’’. Sage 0.1 was released in 2005 and almost a year later Sage 1.0 was released. It already consisted of Pari, GAP, Singular and Maxima with an interface that rivals that of Mathematica.
For example, there is a well known proof relating the Riemann zeta function to the prime zeta function that uses the series-based form of Möbius inversion in the previous equation when =. Namely, by the Euler product representation of ζ ( s ) {\displaystyle \zeta (s)} for ℜ ( s ) > 1 {\displaystyle \Re (s)>1}
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
Each model has a group of isometries that is a subgroup of the Mobius group: the isometry group for the disk model is SU(1, 1) where the linear fractional transformations are "special unitary", and for the upper half-plane the isometry group is PSL(2, R), a projective linear group of linear fractional transformations with real entries and ...
An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.
Dirac delta function: everywhere zero except for x = 0; total integral is 1. Not a function but a distribution, but sometimes informally referred to as a function, particularly by physicists and engineers. Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous.
In number theory, the gcd-sum function, [1] also called Pillai's arithmetical function, [1] is defined for every by = = (,)or equivalently [1] = (/)where is a divisor of and is Euler's totient function.
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