Liouville function

Definition

The Liouville function $\lambda$ is the arithmetic function given by $\lambda(n) = (-1)^{\Omega(n)}$ for each positive integer $n$. Here $\Omega(n)$ is the number of prime factors of n counted with multiplicity.

Remarks

For example, $\lambda(1) = 1$, $\lambda(12) = -1$ and $\lambda(36) = 1$. The function $\lambda$ is completely multiplicative, since $\Omega$ is completely additive. The sum $\sum_{d \mid n} \lambda(d)$ is $1$ if $n$ is a perfect square, and $0$ otherwise. On squarefree integers, $\lambda$ agrees with the Möbius function.

Explanations

L1 human

The Liouville function is $1$ when $n$ has an even number of prime factors, counted with repeats, and $-1$ when the number is odd. Unlike the Möbius function, it is never $0$.