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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 ...
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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.
A Higher-Dimensional Sieve Method: with Procedures for Computing Sieve Functions. Cambridge Tracts in Mathematics. Vol. 177. With William F. Galway. Cambridge: Cambridge University Press. ISBN 978-0-521-89487-6. Zbl 1207.11099. Greaves, George (2001). Sieves in number theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 43.
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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 ...
The Mertens function M(x) is the sum function for the Möbius function, in the theory of arithmetic functions. The Mertens conjecture concerning its growth, conjecturing it bounded by x 1/2 , which would have implied the Riemann hypothesis , is now known to be false ( Odlyzko and te Riele , 1985).