Möbius function

Definition

The Möbius function $\mu$ is an arithmetic function. For each positive integer $n$, it is given by $$\mu(n) = \begin{cases} (-1)^{\omega(n)} & \text{if } n \text{ is squarefree}, \\ 0 & \text{otherwise}. \end{cases}$$ Here $\omega(n)$ is the number of distinct prime factors of $n$. The word squarefree has the meaning of the definition of a squarefree integer.

Remarks

So $\mu(1) = 1$, $\mu(p) = -1$ for each prime $p$, and $\mu(n) = 0$ when $p^2 \mid n$ for some prime $p$. For example, $\mu(6) = 1$, $\mu(30) = -1$ and $\mu(12) = 0$. The function $\mu$ is multiplicative, and it is the inverse of the constant function 1 for the Dirichlet convolution.

Explanations

L1 human

The Möbius function is $0$ on numbers with a repeated prime factor. On the other numbers, it is $1$ for an even number of prime factors and $-1$ for an odd number. It is the sign rule of inclusion and exclusion, written as a function of $n$.