Liouville function
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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:
- <math>\lambda(n) = (-1)^{\Omega(n)}\,\! <math>,
where Ω(n) is the number of prime factors of n, counted with multiplicity. (SIDN A008836 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=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:
- <math>\Sigma_{(d|n)}\lambda(d)=1\,\! <math> if n is a perfect square, and:
- <math>\Sigma_{(d|n)}\lambda(d)=0\,\! <math> otherwise.
The Liouville function is related to the Riemann zeta function by the formula
- <math>\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}<math>
Polya conjectured that <math>L(n) = \sum_{k=1}^n \lambda(k) \leq 0 <math> 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, <math>M(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}<math>, the fact that <math>M(n) \geq 0 <math> is equivalent to the Riemann hypothesis.sv:Liouvilles lambda-funktion