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gptkbp:instanceOf
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arithmetic function
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gptkbp:codomain
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{-1, 0, 1}
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gptkbp:defines
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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
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gptkbp:designer
|
constant function 1 under Dirichlet convolution
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gptkbp:domain
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positive integers
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gptkbp:field
|
number theory
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https://www.w3.org/2000/01/rdf-schema#label
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Möbius function
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gptkbp:introducedIn
|
1832
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gptkbp:multiplicative
|
true
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gptkbp:namedAfter
|
gptkb:August_Ferdinand_Möbius
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gptkbp:property
|
μ(n) = 0 if n is not square-free
μ(n) alternates sign with number of prime factors
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gptkbp:relatedTo
|
gptkb:Riemann_zeta_function
prime numbers
square-free numbers
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gptkbp:sequence
|
gptkb:A008683
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gptkbp:symbol
|
μ(n)
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gptkbp:usedIn
|
gptkb:Möbius_inversion_formula
gptkb:Dirichlet_convolution
inclusion-exclusion principle
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gptkbp:valueAt
|
μ(2) = -1
μ(3) = -1
μ(4) = 0
μ(6) = 1
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gptkbp:bfsParent
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gptkb:Möbius_function_Dirichlet_series
gptkb:Prime_zeta_function
gptkb:Number_theory
gptkb:Sarnak_conjecture
gptkb:Analytic_number_theory
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gptkbp:bfsLayer
|
5
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