Liouville function

E637296

arithmetic function completely multiplicative function

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.

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All labels observed (3)

Label	Occurrences
Liouville function canonical	2
Liouville function in number theory	1
Liouville’s summatory function L(x)	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 ⓘ
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 ⓘ Möbius function ⓘ Riemann zeta function ⓘ prime number theorem ⓘ Ω(n) (total number of prime factors with multiplicity) ⓘ
relatedConjecture	Pólya conjecture (disproved) ⓘ linked to: Pólya’s conjecture Riemann hypothesis ⓘ
relatedObject	Liouville’s summatory function L(x) ⓘ linked to: Liouville function 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

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Multiplicative Number Theory → usesConcept → Liouville function ⓘ

Joseph Liouville → notableWork → Liouville function in number theory ⓘ

linked to: Liouville function

Joseph Liouville → hasEponym → Liouville function ⓘ

Liouville function → relatedObject → Liouville’s summatory function L(x) ⓘ