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multiplicative function (Definition)

In number theory, a multiplicative function is an arithmetic function $ f \colon \mathbb{N} \to \mathbb{C}$ such that $ f(1)=1$ and, for all $ a,b \in \mathbb{N}$ 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$, even 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 to prime numbers. Every completely multiplicative function is multiplicative.

Outside of number theory, the term 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^aq^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

$\displaystyle (f*g)(n) = \sum_{d\vert 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);
  • $ f*g = g*f$ (proven here);
  • $ (f*g)*h = f*(g*h)$ (proven here);
  • $ f*\varepsilon = \varepsilon *f = f$ (proven here);
  • If $ f$ is multiplicative, there exists a multiplicative function $ g$ such that $ f*g=\varepsilon$ (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 group with identity element $ \varepsilon$. Relations among the multiplicative functions discussed above include:

Given a completely multiplicative function $ f$, its convolution inverse is $ f\mu$. See this entry for a proof.



"multiplicative function" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
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See Also: Euler product, convolution inverses for arithmetic functions, pointwise multiplication of a completely multiplicative function distibutes over convolution, divisor sum, additive function, proof that Euler $\varphi$ function is multiplicative, divisor sum of an arithmetic function

Also defines:  multiplicative, completely multiplicative, completely multiplicative function, convolution identity function, convolution inverse
Keywords:  number theory, arithmetic function

Attachments:
non-multiplicative function (Example) by Mathprof
proof that Euler $\varphi$ function is multiplicative (Proof) by Wkbj79
criterion for a multiplicative function to be completely multiplicative (Theorem) by Wkbj79
elementary results about multiplicative functions and convolution (Theorem) by Wkbj79
composition of multiplicative functions (Theorem) by Wkbj79
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Cross-references: Möbius inversion, identity element, abelian group, operation, Dirichlet convolution, product, consequence, constant function, power functions, identity function, natural number, complex number, summing, sum of divisors function, divisors, divisor function, squarefree, prime factors, parity, Möbius function, totatives, Euler totient function, number, arguments, prime numbers, fundamental theorem of arithmetic, monoids, homomorphism, function, relatively prime, integers, positive, arithmetic function, number theory
There are 89 references to this entry.

This is version 47 of multiplicative function, born on 2002-06-12, modified 2008-03-03.
Object id is 3097, canonical name is MultiplicativeFunction.
Accessed 22885 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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the tau function by Wkbj79 on 2003-03-10 03:42:41
Overall, I think that this entry is very informative and interesting. One comment that I'd like to make is that I have only seen the function that counts the positive divisors of a positive integer as the tau function. Again, a very good entry. :-)
[ reply | up ]
totient by Larry Hammick on 2003-01-28 00:57:14
Nice article. Maybe this is the object in which we should define "totient". Anyway, I wondered for years why Sylvester coined that strange word in 1882.

An arithmetic function f is called a "totient" if g*f=h for some two completely multiplicative functions g and h, where * is the convolution product.

Maybe the word "totient" is a hybrid of "total" (from "totally multiplicative") and "quotient", since we can indeed write f=h/g.
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What are the units in the monoid? by NeuRet on 2002-06-15 17:35:54
Why not mention, in passing, that the monoid of functions under Dirichlet convolution contains a submonoid, in fact a group, namely the functions that take at 1 a value different from 0 ?
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