Statements (30)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:arithmetic_function
|
| gptkbp:codomain |
{-1, 0, 1}
|
| gptkbp:defines |
μ(n) = (-1)^k if n is a product of k distinct primes
μ(1) = 1 μ(n) = 0 if n has a squared prime factor |
| gptkbp:designer |
constant function 1 under Dirichlet convolution
|
| gptkbp:domain |
positive integers
|
| gptkbp:field |
number theory
|
| gptkbp:introducedIn |
1832
|
| gptkbp:multiplicative |
true
|
| gptkbp:namedAfter |
gptkb:August_Ferdinand_Möbius
|
| gptkbp:property |
μ(n) = 0 if n is not square-free
μ(n) alternates sign with number of prime factors |
| gptkbp:relatedTo |
gptkb:Riemann_zeta_function
prime numbers square-free numbers |
| gptkbp:sequence |
gptkb:A008683
|
| gptkbp:symbol |
μ(n)
|
| gptkbp:usedIn |
gptkb:Möbius_inversion_formula
gptkb:Dirichlet_convolution inclusion-exclusion principle |
| gptkbp:valueAt |
μ(2) = -1
μ(3) = -1 μ(4) = 0 μ(6) = 1 |
| gptkbp:bfsParent |
gptkb:Euler's_totient_function
gptkb:Number_theory gptkb:Sarnak_conjecture |
| gptkbp:bfsLayer |
6
|
| https://www.w3.org/2000/01/rdf-schema#label |
Möbius function
|