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.

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