##  [Liouville Function](/liouville-function-0) 

 Definition

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

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

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

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

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

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

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

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

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.