Completely multiplicative arithmetic function λ(n)=(-1)^{Ω(n)}, where Ω(n) is the total number of prime factors of n counted with multiplicity; takes values ±1 and extends multiplicativity to prime powers by λ(p^k)=(-1)^k.

Principle

Encode parity of the total prime factor count multiplicatively so that the value on a product equals the product of the values, making λ sensitive only to the parity of Ω(n).

Demonstration

Compute λ(12). Factor 12=2^2·3 so Ω(12)=3 and λ(12)=(-1)^3=-1. For a prime p, λ(p)=-1 and λ(p^2)=+1.

Misapplication

Using λ(n) as if it counted distinct primes (ω(n)) rather than prime factors with multiplicity leads to incorrect parity predictions and breaks multiplicativity on powers.

Consequence

As a completely multiplicative ±1 sequence, λ(n) is used in identities and transforms (Dirichlet series) that probe parity cancellations in sums over integers and in conjectures relating its partial sums to randomness or sign patterns.

Reversal

The complementary notion is the Möbius function μ(n), which vanishes for non-squarefree n and is multiplicative but not completely multiplicative; μ and λ agree on squarefree integers but differ on prime powers.

Boundary

Defined on positive integers only; information about distribution of λ(n) concerns sign patterns and partial sums rather than magnitudes; not suitable for counting distinct prime factors or for functions that require nonmultiplicative behavior.

Semantic Tension

Confused with the Möbius function μ because both take values ±1 on squarefree inputs; tension arises when multiplicativity on prime powers (λ) versus vanishing on nonsquarefree integers (μ) is ignored.

Synthesis

The Liouville function is the completely multiplicative ±1 arithmetic function determined by the parity of total prime factors; it provides a multiplicative encoding of factor-count parity whose partial sums measure global cancellation properties in number theory.