Definition:Möbius Function

Definition

Let $n \in \Z_{>0}$, that is, a strictly positive integer.

The Möbius function is the function $\mu: \Z_{>0} \to \Z_{>0}$ defined as:

$\map \mu n = \begin{cases}

1 & : n = 1 \\ 0 & : \exists p \in \mathbb P: p^2 \divides n\\ \paren {-1}^k & : n = p_1 p_2 \ldots p_k: p_i \in \mathbb P \end{cases}$

That is:

$\map \mu n = 1$ if $n = 1$

$\map \mu n = 0$ if $n$ has any divisor which is the square of a prime, that is $n$ is not square-free

$\map \mu n = \paren {-1}^k$ if $n$ has $k$ distinct prime divisors.

Möbius function is also seen rendered as Moebius function in environments where rendering the umlaut is inconvenient.

This entry was named for August Ferdinand Möbius.