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