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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