Möbius function

The Möbius functionMathworldPlanetmath of number theoryMathworldPlanetmathPlanetmath is the function μ:+{-1,0,1} defined by

μ(n)={1,if n=10,if p2|n for some prime p(-1)r,if n=p1p2pr, where the pi are distinct primes.

In other words, μ(n)=0 if n is not a square-free integer, while μ(n)=(-1)r if n is square-free with r prime factorsMathworldPlanetmath. The function μ is a multiplicative functionMathworldPlanetmath, and obeys the identity

d|nμ(d)={1if n=10if n>1

where d runs through the positive divisorsMathworldPlanetmathPlanetmath of n.

Title Möbius function
Canonical name MobiusFunction
Date of creation 2013-03-22 11:47:03
Last modified on 2013-03-22 11:47:03
Owner mps (409)
Last modified by mps (409)
Numerical id 11
Author mps (409)
Entry type Definition
Classification msc 11A25
Classification msc 55-00
Classification msc 55-01
Synonym Moebius function
Related topic SquareFreeNumber
Related topic SumOfFracmunn
Related topic MoebiusInversionFormula
Related topic ConvolutionMethod