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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Liouville function canonical | 2 |
| Liouville function in number theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030728 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Liouville function Context triple: [Multiplicative Number Theory, usesConcept, Liouville function]
-
A.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Ramanujan tau function
The Ramanujan tau function is a multiplicative arithmetic function arising from the Fourier coefficients of a modular discriminant form, central to the study of modular forms and number theory.
-
D.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
-
E.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Liouville function Target entity description: 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.
-
A.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Ramanujan tau function
The Ramanujan tau function is a multiplicative arithmetic function arising from the Fourier coefficients of a modular discriminant form, central to the study of modular forms and number theory.
-
D.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
-
E.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Liouville function Description of subject: 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.
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