Liouville function information

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: $n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}$ 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:

ω(n) = k,

Ω(n) = a₁ + a₂ + ... + a_k.

λ(n) is defined by the formula

\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

\lambda (n)=\mu (b).

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

\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

\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, $\lambda ^{-1}(n)=|\mu (n)|=\mu ^{2}(n),$ the characteristic function of the squarefree integers. We also have that $\lambda (n)\mu (n)=\mu ^{2}(n)$ .

Liouville function information

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