Encyclopedia > Liouville function

  Article Content

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 (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 Lioville function satisfies the identity:

Σd|n λ(d) = 1 if n is a perfect square, and:
Σd|n λ(d) = 0 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(s)}{n^s}</math>



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Sanskrit language

... to that of Latin in Western Europe. It was (and still is) a language of religious ritual and scholarship, and it had locally varied spoken forms (Prakrits) such as ...

 
 
 
This page was created in 52.4 ms