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:

λ(n) = (-1)^Ω(n),

where Ω(n) is the number of prime factors of n, counted with multiplicity. (SIDN A008836).

λ is completely multiplicative since Ω(n) is additive. We have Ω(1)=0 and therefore λ(1)=1. The Lioville function satisfies the identity:

Σ_d|n λ(d) = 1 if n is a perfect square, and:

Σ_d|n λ(d) = 0 otherwise.

The Liouville function is related to the Riemann zeta function by the formula

${\frac {\zeta (2s)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}$

Polya conjectured that $L(n)=\sum _{k=1}^{n}\lambda (k)\leq 0$ for n>1. This turned out to be false, n=906150257 being a counterexample. It is not known as to whether L(n) changes sign infinitely often.

Also, if we define, $M(n)=\sum _{k=1}^{n}{\frac {\lambda (k)}{k}}$, the fact that $L(n)\geq 0$ is equivalent to the Riemann hypothesis.