Liouville function

The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory. 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 the OEIS). If n is squarefree, i.e., if n = p 1 p 2 ⋯ p k {displaystyle n=p_{1}p_{2}cdots p_{k}} where p i {displaystyle p_{i}} is prime for all i and where p i ≠ p j ∀ i ≠ j {displaystyle p_{i} eq p_{j}forall i eq j} , then we have the following alternate formula for the function expressed in terms of the Möbius function and the distinct prime factor counting function ω ( n ) {displaystyle omega (n)} : λ is completely multiplicative since Ω(n) is completely additive, i.e.: Ω(ab) = Ω(a) + Ω(b). The number 1 has no prime factors, so Ω(1) = 0 and therefore λ(1) = 1. The Liouville function satisfies the identity: The Dirichlet inverse of Liouville function is the absolute value of the Möbius function, λ − 1 ( n ) = | μ ( n ) | = μ 2 ( n ) , {displaystyle lambda ^{-1}(n)=|mu (n)|=mu ^{2}(n),} which is equivalently the characteristic function of the squarefree integers. We also have that λ ( n ) μ ( n ) = μ 2 ( n ) {displaystyle lambda (n)mu (n)=mu ^{2}(n)} , and that for all natural numbers n: The Dirichlet series for the Liouville function is related to the Riemann zeta function by The Lambert series for the Liouville function is where ϑ 3 ( q ) {displaystyle vartheta _{3}(q)} is the Jacobi theta function. The Pólya conjecture is a conjecture made by George Pólya in 1919. Defining