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


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


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


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 (;

  • f*g=g*f (proven here (;

  • (f*g)*h=f*(g*h) (proven here (;

  • f*ε=ε*f=f (proven here (;

  • If f is multiplicative, there exists a multiplicative function g such that f*g=ε (proven here ( 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 ( for a proof.

Title multiplicative function
Canonical name MultiplicativeFunction
Date of creation 2013-03-22 12:47:00
Last modified on 2013-03-22 12:47:00
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 50
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11A25
Related topic EulerProduct
Related topic ConvolutionInversesForArithmeticFunctions
Related topic PropertyOfCompletelyMultiplicativeFunctions
Related topic DivisorSum
Related topic AdditiveFunction
Related topic ProofThatEulerPhiIsAMultiplicativeFunction
Related topic DivisorSumOfAnArithmeticFunction
Defines multiplicative
Defines completely multiplicative
Defines completely multiplicative function
Defines convolution identity function
Defines convolution inverse