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:

where Ω(n) is the number of prime factors of n, counted with multiplicity (sequence A008836 in OEIS).

λ is completely multiplicative since Ω(n) is completely additive. The number one has no prime factors, so Ω(1) = 0 and therefore λ(1) = 1. The Liouville function satisfies the identity:

$\sum_{d|n}\lambda(d) = \begin{cases} 1 & \text{if }n\text{ is a perfect square,} \\ 0 & \text{otherwise.} \end{cases}$

The Liouville function's Dirichlet inverse is the absolute value of the Mobius function.

Read more about Liouville Function: Series, Conjectures

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