# multiplicative function

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 function  is a homomorphism        of monoids and, because of the fundamental theorem of arithmetic  , is completely determined by its restriction   (http://planetmath.org/Restriction) to prime numbers  . 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 arguments  $a$ and $b$. This entry discusses number theoretic multiplicative functions.

## Examples

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

## 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 product  of powers of distinct prime numbers, say $n=p^{a}q^{b}\cdots$, then $f(n)=f(p^{a})f(q^{b})\cdots$. This property of multiplicative functions significantly reduces the need for computation, as in the following examples for $n=144=2^{4}\cdot 3^{2}$:

 $\displaystyle\sigma(144)$ $\displaystyle=$ $\displaystyle\sigma_{1}(144)=\sigma_{1}(2^{4})\sigma_{1}(3^{2})=(1^{1}+2^{1}+4% ^{1}+8^{1}+16^{1})(1^{1}+3^{1}+9^{1})=31\cdot 13=403$ $\displaystyle\sigma_{2}(144)$ $\displaystyle=$ $\displaystyle\sigma_{2}(2^{4})\sigma_{2}(3^{2})=(1^{2}+2^{2}+4^{2}+8^{2}+16^{2% })(1^{2}+3^{2}+9^{2})=341\cdot 91=31031$ $\displaystyle\sigma_{3}(144)$ $\displaystyle=$ $\displaystyle\sigma_{3}(2^{4})\sigma_{3}(3^{2})=(1^{3}+2^{3}+4^{3}+8^{3}+16^{3% })(1^{3}+3^{3}+9^{3})=4681\cdot 757=3543517$

Similarly, we have:

 $\displaystyle\tau(144)$ $\displaystyle=$ $\displaystyle\tau(2^{4})\tau(3^{2})=(4+1)(2+1)=5\cdot 3=15$ $\displaystyle\varphi(144)$ $\displaystyle=$ $\displaystyle\varphi(2^{4})\varphi(3^{2})=2^{3}(2-1)3^{1}(3-1)=8\cdot 1\cdot 3% \cdot 2=48$

## 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

 $(f*g)(n)=\sum_{d|n}f(d)g\left({n\over d}\right),$

where the sum extends over all positive divisors $d$ of $n$. Some general properties of this operation  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));

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

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

• $1*1=\tau$

• $\hbox{id}*1=\sigma$

• $\hbox{id}^{k}*1=\sigma^{k}$

• $\phi*1=\hbox{id}$

Given a completely multiplicative function $f$, its convolution inverse is $f\mu$. See this entry (http://planetmath.org/FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction) 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