Chemistry - 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:

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

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 Liouville function satisfies the identity:

$\Sigma_{(d|n)}\lambda(d)=1\,\!$ if n is a perfect square, and:

$\Sigma_{(d|n)}\lambda(d)=0\,\!$ 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 $M(n) \geq 0$ is equivalent to the Riemann hypothesis.

Categories: Number theory