Liouville function

The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product of an even number of prime numbers, and −1 if it is the product of an odd number of primes.

Explicitly, the fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: n = p₁^a₁ ⋯ p_k^a_k, where p₁ < p₂ < ... < p_k are primes and the a_j are positive integers. (1 is given by the empty product.) The prime omega functions count the number of primes, with (Ω) or without (ω) multiplicity:

${\displaystyle \omega (n)=k,}$

${\displaystyle \Omega (n)=a_{1}+a_{2}+\cdots +a_{k}.}$

λ(n) is defined by the formula

${\displaystyle \lambda (n)=(-1)^{\Omega (n)}}$

(sequence A008836 in the OEIS).

λ is completely multiplicative since Ω(n) is completely additive, i.e.: Ω(ab) = Ω(a) + Ω(b). Since 1 has no prime factors, Ω(1) = 0, so λ(1) = 1.

It is related to the Möbius function μ(n). Write n as n = a²b, where b is squarefree, i.e., ω(b) = Ω(b). Then

${\displaystyle \lambda (n)=\mu (b).}$

The sum of the Liouville function over the divisors of n is the characteristic function of the squares:

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

Möbius inversion of this formula yields

${\displaystyle \lambda (n)=\sum _{d^{2}|n}\mu \left({\frac {n}{d^{2}}}\right).}$

The Dirichlet inverse of Liouville function is the absolute value of the Möbius function, λ^–1(n) = |μ(n)| = μ²(n), the characteristic function of the squarefree integers. We also have that λ(n) = μ²(n).