In number theory, the Liouville function, named after French mathematician Joseph Liouville and denoted ${\displaystyle \lambda (n)}$, is an important arithmetic function. Its value is ${\displaystyle 1}$ if ${\displaystyle n}$ is the product of an even number of prime numbers, and ${\displaystyle -1}$ if it is the product of an odd number of prime numbers.

By the fundamental theorem of arithmetic, any positive integer ${\displaystyle n}$ can be represented uniquely as a product of powers of primes:

where ${\displaystyle p_{1},\dots ,p_{k}}$ are primes and the exponents ${\displaystyle a_{1},\dots ,a_{k}}$ are positive integers. The prime omega function ${\displaystyle \Omega (n)}$ counts the number of primes in the factorization of ${\displaystyle n}$ with multiplicity:

Thus, the Liouville function is defined by

(sequence A008836 in the OEIS).

Since ${\displaystyle \Omega (n)}$ is completely additive; i.e., ${\displaystyle \Omega (ab)=\Omega (a)+\Omega (b)}$, then ${\displaystyle \lambda (n)}$ is completely multiplicative. Since ${\displaystyle 1}$ has no prime factors, ${\displaystyle \Omega (1)=0}$, so ${\displaystyle \lambda (1)=1}$.

${\displaystyle \lambda (n)}$ is also related to the Möbius function ${\displaystyle \mu (n)}$: if we write ${\displaystyle n}$ as ${\displaystyle n=a^{2}b}$, where ${\displaystyle b}$ is squarefree, then

The sum of the Liouville function over the divisors of ${\displaystyle n}$ is the characteristic function of the squares: