LiouvilleLambda

The Liouville function $\lambda(n)=(-1)^{\Omega(n)}$.

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LiouvilleLambda(n)$(-1)^{\Omega(n)}$

Juliaenumeratio_liouville(n)

Typed signatures

Type	Package	Applies
(integer) -> integer	number-theory	always

Domain: Number theory

Details

• Completely multiplicative: $\lambda(mn)=\lambda(m)\lambda(n)$ for every $m,n$, not just coprime ones — unlike MoebiusMu.
• $\sum_{d\mid n}\lambda(d)$ is 1 when $n$ is a perfect square and 0 otherwise.

Plot

Examples

Basic #

$\Omega(12)=3$

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In$\mathrm{LiouvilleLambda}(12)$Out$-1$

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

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

$8=2^3$ has three prime factors with multiplicity, so $-1$

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In$\mathrm{LiouvilleLambda}(8)$Out$-1$

$9=3^2$ has an even count of prime factors, so $+1$

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In$\mathrm{LiouvilleLambda}(9)$Out$1$

Scope #

Listable

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In$\mathrm{LiouvilleLambda}(1..10)$Out$\bigl\lbrack1, -1, -1, 1, -1, 1, -1, -1, 1, 1\bigr\rbrack$

Big integer input is factored exactly

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In$\mathrm{LiouvilleLambda}(1+10^{30})$Out$-1$

Threads over a list, giving the signs of $(-1)^{\Omega(n)}$

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In$\mathrm{LiouvilleLambda}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack)$Out$\bigl\lbrack1, -1, -1, 1, -1, 1\bigr\rbrack$

Properties #

$\lambda(n)=(-1)^{\Omega(n)}$. See PrimeOmega

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In$\mathrm{LiouvilleLambda}(360)=(-1)^{\mathrm{PrimeOmega}(360)}$Out$\mathrm{True}$

Completely multiplicative, even for non-coprime factors ($162=6\cdot27$)

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In$\mathrm{LiouvilleLambda}(162)=\mathrm{LiouvilleLambda}(6)\mathrm{LiouvilleLambda}(27)$Out$\mathrm{True}$

$60=2^2\cdot3\cdot5$ has four prime factors counted with multiplicity

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In$\mathrm{LiouvilleLambda}(60)$Out$1$

Three distinct primes give $-1$

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In$\mathrm{LiouvilleLambda}(30)$Out$-1$

$120=2^3\cdot3\cdot5$ has five prime factors with multiplicity

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In$\mathrm{LiouvilleLambda}(120)$Out$-1$

A prime square always gives $+1$

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In$\mathrm{LiouvilleLambda}(25)$Out$1$

Every prime has value $-1$

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

Known value · Wolfram MathematicalFunctionData, LiouvilleLambda

A second prime, same sign

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

Known value · Wolfram MathematicalFunctionData, LiouvilleLambda

$4=2\cdot2$, two factors with multiplicity

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In$\mathrm{LiouvilleLambda}(4)$Out$1$

Known value · Wolfram MathematicalFunctionData, LiouvilleLambda

Two distinct primes multiply the signs to $+1$

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

Known value · Wolfram MathematicalFunctionData, LiouvilleLambda

Over the divisors of a perfect square the sum is $1$

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In$\mathrm{DivisorSum}(64, \operatorname{\_1}↦\mathrm{LiouvilleLambda}(\operatorname{\_1}))$Out$1$

For a non-square the divisor sum is $0$

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In$\mathrm{DivisorSum}(7, \operatorname{\_1}↦\mathrm{LiouvilleLambda}(\operatorname{\_1}))$Out$0$

About

Examples	20
Checked against	Wolfram Language 20 agreeSageMath 17 agreeSymPy 17 agreemathlib4 16 agree2 errorJulia 17 agree
Crosswalk	FungrimFungrim entryWolfram LanguageJulia

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

ArithmeticFunction.liouville, (-1)^Omega(n), with lambda(0) = 0 by definition; natural n only.

See also: MoebiusMu, PrimeOmega