MoebiusMu

The Möbius function: 0 if n has a squared prime factor, else (-1)^(number of prime factors).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaMöbius function↗MathWorldMoebiusFunction↗WikidataQ205243↗Encyclopedia of MathematicsMöbius_function↗DLMF27.2.E12↗FungrimMoebiusMu↗OEISA008683↗Wolfram LanguageMoebiusMu!↗

MoebiusMu(n)the Möbius function $\mu(n)$.

SageMathmoebius(n)✓↗SymPymobius(n)↗Oscarmoebius_mu(n)Juliamoebius_mu(n)

Typed signatures

Type	Package	Applies
(integer) -> integer	compute-engine	always
(number | quadratic_integer, any*) -> integer	number-theoryreplaces compute-engine	always

Domain: Number theory

Details

• $\mu(n)=0$ if $n$ has a squared prime factor; otherwise $\mu(n)=(-1)^{\omega(n)}$ for $\omega(n)$ distinct prime factors.
• $\mu(1)=1$ by convention, the empty product.
• Underlies Möbius inversion: $\sum_{d\mid n}\mu(d)=0$ for every $n>1$.
• $n$ is squarefree exactly when $\mu(n)\neq0$. See IsSquareFree.
• $\mu$ is only defined on positive integers; compute-engine leaves $n\le0$ unevaluated.

Examples

1 more kept as data, checked against the oracles

Basic #

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In$\mathrm{MoebiusMu}(1)$Out$1$

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In$\mathrm{MoebiusMu}(11)$Out$-1$

12 = 2² · 3 is not squarefree

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In$\mathrm{MoebiusMu}(12)$Out$0$

10 = 2·5, a product of an even number of distinct primes

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In$\mathrm{MoebiusMu}(10)$Out$1$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\mathrm{MoebiusMu}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack1, -1, -1, 0\bigr\rbrack$

Gaussian integers: $5+6i$ is a Gaussian prime (norm 61)

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In$\mathrm{MoebiusMu}(5+6i)$Out$-1$

Listable: threads over a list

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In$\mathrm{MoebiusMu}(\bigl\lbrack4, 10, 17, 20\bigr\rbrack)$Out$\bigl\lbrack0, 1, -1, 0\bigr\rbrack$

Listable over a Range

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In$\mathrm{MoebiusMu}(1..20)$Out$\bigl\lbrack1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0\bigr\rbrack$

Applications #

$15=3\cdot5$ is squarefree, so $\mu\ne0$

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In$\mathrm{MoebiusMu}(15)≠0$Out$\mathrm{True}$

$\sum_{d\mid n}\mu(d)=1$ only when $n=1$

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In$\mathrm{DivisorSum}(1, \operatorname{\_1}↦\mathrm{MoebiusMu}(\operatorname{\_1}))$Out$1$

Properties #

$\sum_{d\mid n}\mu(d)=0$ for $n>1$, the identity behind Möbius inversion

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In$\mathrm{MoebiusMu}(1)+\mathrm{MoebiusMu}(2)+\mathrm{MoebiusMu}(3)+\mathrm{MoebiusMu}(4)+\mathrm{MoebiusMu}(6)+\mathrm{MoebiusMu}(12)$Out$0$

Multiplicative over coprime factors, $253=11\cdot23$

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In$\mathrm{MoebiusMu}(253)=\mathrm{MoebiusMu}(11)\mathrm{MoebiusMu}(23)$Out$\mathrm{True}$

Squarefree with an even number of primes gives $1$

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In$\mathrm{MoebiusMu}(5\,187)$Out$1$

Squarefree with an odd number of primes gives $-1$

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In$\mathrm{MoebiusMu}(32\,538)$Out$-1$

Divisible by a square ($1440=2^5\cdot3^2\cdot5$), so $0$

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In$\mathrm{MoebiusMu}(1\,440)$Out$0$

A composite prime power has $\mu=0$

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In$\bigl\lbrack\mathrm{IsPrimePower}(121)∧\mathrm{IsComposite}(121), \mathrm{MoebiusMu}(121)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, 0\bigr\rbrack$

Every prime has $\mu=-1$

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In$\bigl\lbrack\mathrm{IsPrime}(11), \mathrm{MoebiusMu}(11)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, -1\bigr\rbrack$

A single prime gives $-1$

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In$\mathrm{MoebiusMu}(2)$Out$-1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

Another prime, again $-1$

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In$\mathrm{MoebiusMu}(3)$Out$-1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

$4=2^2$ is not squarefree

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In$\mathrm{MoebiusMu}(4)$Out$0$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

Prime, so $-1$

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In$\mathrm{MoebiusMu}(5)$Out$-1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

Two distinct primes give $+1$

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In$\mathrm{MoebiusMu}(6)$Out$1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

One prime factor, an odd count: $\mu(7) = -1$

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In$\mathrm{MoebiusMu}(7)$Out$-1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

A cube contains a square factor, so $0$

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In$\mathrm{MoebiusMu}(8)$Out$0$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

$9=3^2$ gives $0$

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In$\mathrm{MoebiusMu}(9)$Out$0$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

$\mu(p) = -1$ for every prime $p$

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In$\mathrm{MoebiusMu}(13)$Out$-1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

$14=2\cdot7$, an even number of primes

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In$\mathrm{MoebiusMu}(14)$Out$1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

$15=3\cdot5$ gives $+1$

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In$\mathrm{MoebiusMu}(15)$Out$1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

$2^4$ is not squarefree

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In$\mathrm{MoebiusMu}(16)$Out$0$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

17 is prime, so its Möbius value is $-1$

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In$\mathrm{MoebiusMu}(17)$Out$-1$

Known value · Wolfram MathematicalFunctionData, MoebiusMu

Possible issues #

$\mu$ is only defined for positive integers; compute-engine leaves $n=0$ unevaluated

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In$\mathrm{MoebiusMu}(0)$Out$\mathrm{MoebiusMu}(0)$

About

Examples	32
Checked against	Wolfram Language 30 agree2 differSageMath 27 agree1 unevaluatedSymPy 26 agree2 errormathlib4 22 agree1 unevaluated5 errorOscar 5 agree1 differ4 errorJulia 26 agree2 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsDLMFFungrimOEISWolfram LanguageSageMathSymPyOscarJulia

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedoscarpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

See also: FactorInteger, IsSquareFree