---
tag: 08RQAPXF1T
kind: definition
name: Liouville function
msc: [11A25]
defines: 07NN4VNYWM
uses: [08420828EY]
sources:
  - url: https://en.wikipedia.org/wiki/Liouville_function
    locator: Definition
---

## 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|08420828EY]].

## Remarks

For example, $\lambda(1) = 1$, $\lambda(12) = -1$ and $\lambda(36) = 1$.
The function $\lambda$ is [[completely multiplicative|0GS27TKF1V]], 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|0C8QENHYZW]].

## 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 $0$.