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单词 MultiplicativeFunction
释义

multiplicative function


In number theoryMathworldPlanetmath, a multiplicative functionMathworldPlanetmath is an arithmetic functionMathworldPlanetmath f: such that f(1)=1 and, for all a,b with gcd(a,b)=1, we have f(ab)=f(a)f(b).

An arithmetic function f(n) is said to be completely multiplicative if f(1)=1 and f(ab)=f(a)f(b) holds for all positive integers a and b, when they are not relatively prime. In this case, the functionMathworldPlanetmath is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of monoids and, because of the fundamental theorem of arithmeticMathworldPlanetmath, is completely determined by its restrictionPlanetmathPlanetmathPlanetmath (http://planetmath.org/Restriction) to prime numbersMathworldPlanetmath. Every completely multiplicative function is multiplicative.

Outside of number theory, the multiplicative is usually used for all functions with the property f(ab)=f(a)f(b) for all argumentsMathworldPlanetmath a and b. This entry discusses number theoretic multiplicative functions.

Examples

Examples of multiplicative functions include many important functions in number theory, such as:

  • φ(n): the Euler totient function (also denoted ϕ(n)), counting the totativesMathworldPlanetmath of n;

  • μ(n): the Möbius function, which determines the parity of the prime factors of n if n is squarefreeMathworldPlanetmath;

  • τ(n): the divisor functionDlmfDlmfMathworldPlanetmath (also denoted d(n)), counting the positive divisorsMathworldPlanetmathPlanetmath of n;

  • σ(n): the sum of divisors function (also denoted σ1(n)), summing the positive divisors of n;

  • σk(n): the sum of the k-th powers of all the positive divisors of n for any complex number k (typically a natural numberMathworldPlanetmath);

  • id(n): the identity functionMathworldPlanetmath, defined by id(n)=n;

  • idk(n): the power functionsDlmfDlmfPlanetmath, defined by idk(n)=nk for any complex number k (typically a natural number);

  • 1(n): the constant functionMathworldPlanetmath, defined by 1(n)=1;

  • ε(n): the convolution identity function, defined by:

    ε(n)=d|nμ(d)={1if n=10if n1

    where d runs through the positive divisors of n.

Properties

A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if n is a productPlanetmathPlanetmath of powers of distinct prime numbers, say n=paqb, then f(n)=f(pa)f(qb). This property of multiplicative functions significantly reduces the need for computation, as in the following examples for n=144=2432:

σ(144)=σ1(144)=σ1(24)σ1(32)=(11+21+41+81+161)(11+31+91)=3113=403
σ2(144)=σ2(24)σ2(32)=(12+22+42+82+162)(12+32+92)=34191=31031
σ3(144)=σ3(24)σ3(32)=(13+23+43+83+163)(13+33+93)=4681757=3543517

Similarly, we have:

τ(144)=τ(24)τ(32)=(4+1)(2+1)=53=15
φ(144)=φ(24)φ(32)=23(2-1)31(3-1)=8132=48

Convolution

Recall that, if f and g are two arithmetic functions, one defines a new arithmetic function f*g, the Dirichlet convolution (or simply convolution) of f and g, by

(f*g)(n)=d|nf(d)g(nd),

where the sum extends over all positive divisors d of n. Some general properties of this operationMathworldPlanetmath with respect to multiplicative functions include (here the argument n is omitted in all functions):

  • If both f and g are multiplicative, then so is f*g (proven here (http://planetmath.org/ElementaryResultsAboutMultiplicativeFunctionsAndConvolution));

  • f*g=g*f (proven here (http://planetmath.org/ArithmeticFunctionsFormARing));

  • (f*g)*h=f*(g*h) (proven here (http://planetmath.org/ArithmeticFunctionsFormARing));

  • f*ε=ε*f=f (proven here (http://planetmath.org/ArithmeticFunctionsFormARing));

  • If f is multiplicative, there exists a multiplicative function g such that f*g=ε (proven here (http://planetmath.org/ElementaryResultsAboutMultiplicativeFunctionsAndConvolution)). In other words, every multiplicative function has a convolution inverse that is also multiplicative.

This shows that, with respect to convolution, the multiplicative functions form an abelian groupMathworldPlanetmath with identity elementMathworldPlanetmath ε. among the multiplicative functions discussed above include:

  • μ*1=ε (the Möbius inversionMathworldPlanetmathPlanetmath formulaMathworldPlanetmathPlanetmath)

  • 1*1=τ

  • id*1=σ

  • idk*1=σk

  • ϕ*1=id

Given a completely multiplicative function f, its convolution inverse is fμ. See this entry (http://planetmath.org/FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction) for a proof.

Titlemultiplicative function
Canonical nameMultiplicativeFunction
Date of creation2013-03-22 12:47:00
Last modified on2013-03-22 12:47:00
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id50
AuthorWkbj79 (1863)
Entry typeDefinition
Classificationmsc 11A25
Related topicEulerProduct
Related topicConvolutionInversesForArithmeticFunctions
Related topicPropertyOfCompletelyMultiplicativeFunctions
Related topicDivisorSum
Related topicAdditiveFunction
Related topicProofThatEulerPhiIsAMultiplicativeFunction
Related topicDivisorSumOfAnArithmeticFunction
Definesmultiplicative
Definescompletely multiplicative
Definescompletely multiplicative function
Definesconvolution identity function
Definesconvolution inverse
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