Möbius function
E865102
The Möbius function is a multiplicative arithmetic function in number theory that assigns values based on the prime factorization of integers and plays a central role in inversion formulas and the study of prime distribution.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Möbius function canonical | 2 |
| Möbius function μ(n) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462058 — 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: Möbius function Context triple: [Selberg sieve, usesConcept, Möbius function]
-
A.
Liouville 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.
-
B.
Dirichlet convolution
Dirichlet convolution is a binary operation on arithmetic functions that combines them via summation over divisors and plays a central role in multiplicative number theory and Dirichlet series.
-
C.
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.
-
D.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
E.
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.
- 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: Möbius function Target entity description: The Möbius function is a multiplicative arithmetic function in number theory that assigns values based on the prime factorization of integers and plays a central role in inversion formulas and the study of prime distribution.
-
A.
Liouville 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.
-
B.
Dirichlet convolution
Dirichlet convolution is a binary operation on arithmetic functions that combines them via summation over divisors and plays a central role in multiplicative number theory and Dirichlet series.
-
C.
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.
-
D.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
E.
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.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf | arithmetic function ⓘ |
| alternativeName | Möbius μ-function ⓘ |
| appearsIn |
Mertens conjecture
NERFINISHED
ⓘ
equivalent criteria for the Riemann hypothesis ⓘ |
| classification | multiplicative arithmetic function ⓘ |
| codomain | {-1,0,1} ⓘ |
| definition |
μ(1) = 1
ⓘ
μ(n) = (-1)^k if n is the product of k distinct primes ⓘ μ(n) = 0 if n is divisible by the square of a prime ⓘ |
| DirichletSeries | ∑_{n≥1} μ(n)n^{-s} = 1/ζ(s) for Re(s) > 1 ⓘ |
| domain | positive integers ⓘ |
| field | number theory ⓘ |
| generalizationOf | Möbius functions on posets ⓘ |
| inspired | Möbius inversion in combinatorics ⓘ |
| introducedIn | 19th century ⓘ |
| inverseUnderDirichletConvolution | constant function 1 ⓘ |
| namedAfter | August Ferdinand Möbius NERFINISHED ⓘ |
| namedInLanguage | German: Möbiussche Funktion ⓘ |
| property |
average order is 0 in various senses
ⓘ
is a completely multiplicative function on square-free integers ⓘ multiplicative over coprime arguments ⓘ values are completely determined by prime factorization of n ⓘ μ(mn) = μ(m)μ(n) if gcd(m,n) = 1 ⓘ μ(n) = 0 if and only if n is not square-free ⓘ μ(n) ∈ {-1,0,1} for all positive integers n ⓘ μ(n) ≠ 0 if and only if n is square-free ⓘ μ(p) = -1 for any prime p ⓘ μ(p^k) = 0 for any prime p and integer k ≥ 2 ⓘ μ(pq) = 1 for distinct primes p and q ⓘ |
| relatedTo |
Dirichlet convolution
NERFINISHED
ⓘ
Mertens function NERFINISHED ⓘ Möbius inversion formula NERFINISHED ⓘ Riemann zeta function NERFINISHED ⓘ square-free integers ⓘ |
| satisfies |
μ * 1 = ε, where ε is the identity for Dirichlet convolution
ⓘ
∑_{d|n} μ(d) = 0 if n > 1 ⓘ ∑_{d|n} μ(d) = 1 if n = 1 ⓘ |
| summatoryFunction | Mertens function M(n) = ∑_{k≤n} μ(k) ⓘ |
| symbol | μ(n) ⓘ |
| usedFor |
Dirichlet series identities
ⓘ
Möbius inversion formula NERFINISHED ⓘ analysis of the Riemann zeta function ⓘ inversion of Dirichlet convolutions ⓘ recovering arithmetic functions from summatory functions ⓘ study of prime distribution ⓘ |
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: Möbius function Description of subject: The Möbius function is a multiplicative arithmetic function in number theory that assigns values based on the prime factorization of integers and plays a central role in inversion formulas 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:
Möbius function μ(n)