Möbius function

The Möbius function of number theory is the function $\\mu:\\mathbb{Z}^{+}\\to\\{-1,0,1\\}$ defined by

\\mu(n)=\\begin{cases}1,&\\text{if $n=1$}\\\\0,&\\text{if $p^{2}|n$ for some prime $p$}\\\\(-1)^{r},&\\text{if $n=p_{1}p_{2}\\cdots p_{r}$, where the $p_{i}$ are distinct %primes.}\\end{cases}

In other words, $\\mu(n)=0$ if $n$ is not a square-free integer, while $\\mu(n)=(-1)^{r}$ if $n$ is square-free with $r$ prime factors. The function $\\mu$ is a multiplicative function, and obeys the identity

\\sum_{d|n}\\mu(d)=\\begin{cases}1&\\text{if $n=1$}\\\\0&\\text{if $n>1$}\\end{cases}

where $d$ runs through the positive divisors 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