# Multiplicative function

Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.

In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then

f(ab) = f(a) f(b).

An arithmetic function f(n) is said to be completely multiplicative (or totally 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 coprime.

## Examples

Some multiplicative functions are defined to make formulas easier to write:

• 1(n): the constant function, defined by 1(n) = 1 (completely multiplicative)
• Idk(n): the power functions, defined by Idk(n) = nk for any complex number k (completely multiplicative). As special cases we have
• Id0(n) = 1(n) and
• Id1(n) = Id(n).

Other examples of multiplicative functions include many functions of importance in number theory, such as:

An example of a non-multiplicative function is the arithmetic function r2(n) - the number of representations of n as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed. For example:

1 = 12 + 02 = (-1)2 + 02 = 02 + 12 = 02 + (-1)2

and therefore r2(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, r2(n)/4 is multiplicative.

In the On-Line Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult".

See arithmetic function for some other examples of non-multiplicative functions.

## 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 primes, say n = pa qb ..., 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 = 24 · 32:

d(144) = $\sigma$ 0(144) = $\sigma$ 0(24)$\sigma$ 0(32) = (10 + 20 + 40 + 80 + 160)(10 + 30 + 90) = 5 · 3 = 15,
$\sigma$ (144) = $\sigma$ 1(144) = $\sigma$ 1(24)$\sigma$ 1(32) = (11 + 21 + 41 + 81 + 161)(11 + 31 + 91) = 31 · 13 = 403,
$\sigma$ *(144) = $\sigma$ *(24)$\sigma$ *(32) = (11 + 161)(11 + 91) = 17 · 10 = 170.

Similarly, we have:

$\varphi$ (144)=$\varphi$ (24)$\varphi$ (32) = 8 · 6 = 48

In general, if f(n) is a multiplicative function and a, b are any two positive integers, then

f(a) · f(b) = f(gcd(a,b)) · f(lcm(a,b)).

Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

## Convolution

If f and g are two multiplicative functions, one defines a new multiplicative function f * g, the Dirichlet convolution of f and g, by

$(f\,*\,g)(n)=\sum _{d|n}f(d)\,g\left({\frac {n}{d}}\right)$ where the sum extends over all positive divisors d of n. With this operation, the set of all multiplicative functions turns into an abelian group; the identity element is $\epsilon$ . Convolution is commutative, associative, and distributive over addition.

Relations among the multiplicative functions discussed above include:

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

### Dirichlet series for some multiplicative functions

More examples are shown in the article on Dirichlet series.

## Multiplicative function over Fq[X]

Let A=Fq[X], the polynomial ring over the finite field with q elements. A is principal ideal domain and therefore A is unique factorization domain.

### Zeta function and Dirichlet series in Fq[X]

Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A). Its corresponding Dirichlet series define to be

The polynomial zeta function is then

Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Euler product):

Where the product runs over all monic irreducible polynomials P.

For example, the product representation of the zeta function is as for the integers: $\zeta _{A}(s)=\prod _{P}(1-|P|^{-s})^{-1}$ .

Unlike the classical zeta function, $\zeta _{A}(s)$ is a simple rational function:

In a similar way, If ƒ and g are two polynomial arithmetic functions, one defines ƒ * g, the Dirichlet convolution of ƒ and g, by

{\begin{aligned}(f*g)(m)&=\sum _{d\,\mid \,m}f(m)g\left({\frac {m}{d}}\right)\\&=\sum _{ab\,=\,f}f(a)g(b)\end{aligned}} where the sum extends over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m. The identity $D_{h}D_{g}=D_{h*g}$ still holds.