Liouville function

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 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=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:

Σ_d|n λ(d) = 1 if n is a perfect square, and:

Σ_d|n λ(d) = 0 otherwise.

The Liouville function is related to the Riemann zeta function by the formula

<math>\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(s)}{n^s}</math>