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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 ...
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.
The central idea of the method is expressed by the following identity, sometimes called the Legendre identity: (,) =; = | |,where A is a set of integers, P is a product of distinct primes, is the Möbius function, and is the set of integers in A divisible by d, and S(A, P) is defined to be:
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell . Given an arithmetic function f {\displaystyle f} and a prime p {\displaystyle p} , define the formal power series f p ( x ) {\displaystyle f_{p}(x)} , called the Bell series ...
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).
Pages in category "Multiplicative functions" The following 13 pages are in this category, out of 13 total. This list may not reflect recent changes. ...