Möbius function

E865102

arithmetic function

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.

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All labels observed (3)

Label	Occurrences
Möbius function canonical	2
Möbius function μ	1
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 ⓘ 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 ⓘ
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 ⓘ Mertens function ⓘ Möbius inversion formula ⓘ Riemann zeta function ⓘ 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 ⓘ 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

Referenced by (4)

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

Selberg sieve → usesConcept → Möbius function ⓘ

Jordan’s totient functions → relatedConcept → Möbius function μ(n) ⓘ

linked to: Möbius function

Ramanujan’s sum → relatedTo → Möbius function ⓘ

Dirichlet convolution → keyFunction → Möbius function μ ⓘ