Liouville function

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 in the OEIS).

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 Liouville function's Dirichlet inverse is the absolute value of the Möbius function.

The Dirichlet series for the Liouville function is related to the Riemann zeta function by

The Lambert series for the Liouville function is

where ${\displaystyle \vartheta _{3}(q)}$ is the Jacobi theta function.

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