Statements (31)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
arithmetic function
|
| gptkbp:citation |
https://en.wikipedia.org/wiki/Liouville_function
|
| gptkbp:codomain |
{-1, 1}
|
| gptkbp:completely_multiplicative |
true
|
| gptkbp:defines |
λ(n) = (-1)^{Ω(n)}
|
| gptkbp:domain |
positive integers
|
| gptkbp:hasConjecture |
Pólya conjecture (disproved using Liouville function)
related to Riemann Hypothesis |
| https://www.w3.org/2000/01/rdf-schema#label |
Liouville function
|
| gptkbp:multiplicative |
true
|
| gptkbp:namedAfter |
gptkb:Joseph_Liouville
|
| gptkbp:property |
λ(n) = 1 if n has even number of prime factors (with multiplicity)
λ(p^k) = (-1)^k for any prime p and integer k ≥ 1 λ(n) = -1 if n has odd number of prime factors (with multiplicity) λ(mn) = λ(m)λ(n) for all m, n λ(p) = -1 for any prime p |
| gptkbp:relatedTo |
gptkb:Möbius_function
gptkb:Riemann_zeta_function |
| gptkbp:sequence |
gptkb:OEIS_A008836
|
| gptkbp:sum |
Liouville summatory function
|
| gptkbp:symbol |
λ(n)
|
| gptkbp:usedIn |
number theory
analytic number theory |
| gptkbp:value_for_n=1 |
1
|
| gptkbp:value_for_n=12 |
-1
|
| gptkbp:value_for_n=2 |
-1
|
| gptkbp:value_for_n=4 |
1
|
| gptkbp:value_for_n=6 |
1
|
| gptkbp:Ω(n) |
number of prime factors of n counted with multiplicity
|
| gptkbp:bfsParent |
gptkb:Liouville_function_Dirichlet_series
|
| gptkbp:bfsLayer |
5
|