Liouville function
The
Liouville function, denoted by λ(
n) and named after
Joseph Liouville, is an important
function in
number theory.
If n is a positive integer, then λ(n) is defined as:
- λ(n) = (-1)Ω(n),
where Ω(
n) is the number of
prime factors of
n, counted with multiplicity. (
SIDN A008836).
λ is completely multiplicative since Ω(n) is additive. We have Ω(1)=0 and therefore λ(1)=1. The Lioville function satisfies the identity:
- Σd|n λ(d) = 1 if n is a perfect square, and:
- Σd|n λ(d) = 0 otherwise.
The Liouville function is related to the
Riemann zeta function by the formula