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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Möbius function μ(n) | 1 |
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 ⓘ |
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)