# 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