Liouville function
E637296
The Liouville function is a completely multiplicative arithmetic function that assigns values based on the parity of the total number of prime factors of an integer, playing a key role in analytic number theory and the study of prime distribution.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Liouville function in number theory | 1 |
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
arithmetic function
ⓘ
completely multiplicative function ⓘ |
| application |
construction of counterexamples to the Pólya conjecture
ⓘ
criteria equivalent to the Riemann hypothesis via bounds on its summatory function ⓘ |
| classification | completely multiplicative {−1,1}-valued function ⓘ |
| codomain | {-1, 1} ⓘ |
| definition | λ(n) = (-1)^{Ω(n)} where Ω(n) is the total number of prime factors of n counted with multiplicity ⓘ |
| DirichletSeries | ∑_{n=1}^{∞} λ(n)n^{-s} = ζ(2s) / ζ(s) for Re(s) > 1 ⓘ |
| domain | positive integers ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| generatingFunction | Dirichlet generating function is ζ(2s)/ζ(s) ⓘ |
| growthProperty | |λ(n)| = 1 for all positive integers n ⓘ |
| historicalNote | introduced by Joseph Liouville in the 19th century ⓘ |
| identity |
∑_{d^2|n} λ(d) = 1 for all n
ⓘ
∑_{d|n} λ(d) = 1 if n is a perfect square and 0 otherwise ⓘ |
| inverseRelation | Dirichlet inverse of λ(n) is given by the characteristic function of squares ⓘ |
| namedAfter | Joseph Liouville NERFINISHED ⓘ |
| orthogonalityProperty | exhibits cancellation in many average sums over n ⓘ |
| parityInterpretation |
λ(n) = -1 if n has an odd total number of prime factors counted with multiplicity
ⓘ
λ(n) = 1 if n has an even total number of prime factors counted with multiplicity ⓘ |
| property |
completely multiplicative: λ(mn) = λ(m)λ(n) for all positive integers m,n
ⓘ
multiplicative with respect to Dirichlet convolution ⓘ summatory function L(x) is conjectured to have strong cancellation properties ⓘ values are completely determined by the prime factorization of n ⓘ |
| range | {-1, 1} ⓘ |
| relatedConcept |
Dirichlet series
NERFINISHED
ⓘ
Möbius function ⓘ Riemann zeta function NERFINISHED ⓘ prime number theorem ⓘ Ω(n) (total number of prime factors with multiplicity) ⓘ |
| relatedConjecture |
Pólya conjecture (disproved)
NERFINISHED
ⓘ
Riemann hypothesis NERFINISHED ⓘ |
| relatedObject |
Liouville’s summatory function L(x)
NERFINISHED
ⓘ
characteristic function of squares via Dirichlet inversion ⓘ |
| summatoryFunction | L(x) = ∑_{n ≤ x} λ(n) ⓘ |
| symbol | λ(n) ⓘ |
| usedIn |
analytic number theory
ⓘ
study of prime distribution ⓘ study of sign changes in arithmetic functions ⓘ |
| valueAt |
λ(1) = 1
ⓘ
λ(p) = -1 for any prime p ⓘ λ(p^k) = (-1)^k for any prime p and integer k ≥ 1 ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Liouville function in number theory