# 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 a 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)
• Id(n): identity function, defined by Id(n) = n (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).
• ε(n): the function defined by ε(n) = 1 if n = 1 and 0 otherwise, sometimes called multiplication unit for Dirichlet convolution or simply the unit function (completely multiplicative). Sometimes written as u(n), but not to be confused with μ(n) .
• 1C(n), the indicator function of the set CZ, for certain sets C. The indicator function 1C(n) is multiplicative precisely when the set C has the following property for any coprime numbers a and b: the product ab is in C if and only if the numbers a and b are both themselves in C. This is the case if C is the set of squares, cubes, or k-th powers, or if C is the set of square-free numbers.

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

• gcd(n,k): the greatest common divisor of n and k, as a function of n, where k is a fixed integer.
• $\varphi$ (n): Euler's totient function $\varphi$ , counting the positive integers coprime to (but not bigger than) n
• μ(n): the Möbius function, the parity (−1 for odd, +1 for even) of the number of prime factors of square-free numbers; 0 if n is not square-free
• σk(n): the divisor function, which is the sum of the k-th powers of all the positive divisors of n (where k may be any complex number). Special cases we have
• σ0(n) = d(n) the number of positive divisors of n,
• σ1(n) = σ(n), the sum of all the positive divisors of n.
• a(n): the number of non-isomorphic abelian groups of order n.
• λ(n): the Liouville function, λ(n) = (−1)Ω(n) where Ω(n) is the total number of primes (counted with multiplicity) dividing n. (completely multiplicative).
• γ(n), defined by γ(n) = (−1)ω(n), where the additive function ω(n) is the number of distinct primes dividing n.
• τ(n): the Ramanujan tau function.
• All Dirichlet characters are completely multiplicative functions. For example

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) = σ0(144) = σ0(24)σ0(32) = (10 + 20 + 40 + 80 + 160)(10 + 30 + 90) = 5 · 3 = 15,
σ(144) = σ1(144) = σ1(24)σ1(32) = (11 + 21 + 41 + 81 + 161)(11 + 31 + 91) = 31 · 13 = 403,
σ*(144) = σ*(24)σ*(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 ε. Convolution is commutative, associative, and distributive over addition.

Relations among the multiplicative functions discussed above include:

• μ * 1 = ε (the Möbius inversion formula)
• (μ Idk) * Idk = ε (generalized Möbius inversion)
• $\varphi$  * 1 = Id
• d = 1 * 1
• σ = Id * 1 = $\varphi$  * d
• σk = Idk * 1
• Id = $\varphi$  * 1 = σ * μ
• Idk = σk * μ

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

The Dirichlet convolution of two multiplicative functions is again multiplicative. A proof of this fact is given by the following expansion for relatively prime $a,b\in \mathbb {Z} ^{+}$ :

$(f\ast g)(ab)=\sum _{d|ab}f(d)g\left({\frac {ab}{d}}\right)=\sum _{d_{1}|a}\sum _{d_{2}|b}f(d_{1}d_{2})g\left({\frac {ab}{d_{1}d_{2}}}\right)=\sum _{d_{1}|a}f(d_{1})g\left({\frac {a}{d_{1}}}\right)\times \sum _{d_{2}|b}f(d_{2})g\left({\frac {b}{d_{2}}}\right)=(f\ast g)(a)\cdot (f\ast g)(b).$

### Dirichlet series for some multiplicative functions

• $\sum _{n\geq 1}{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}$
• $\sum _{n\geq 1}{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}$
• $\sum _{n\geq 1}{\frac {d(n)^{2}}{n^{s}}}={\frac {\zeta (s)^{4}}{\zeta (2s)}}$
• $\sum _{n\geq 1}{\frac {2^{\omega (n)}}{n^{s}}}={\frac {\zeta (s)^{2}}{\zeta (2s)}}$

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 a principal ideal domain and therefore A is a unique factorization domain.

A complex-valued function $\lambda$  on A is called multiplicative if $\lambda (fg)=\lambda (f)\lambda (g)$  whenever f and g are relatively prime.

### 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 is defined to be

$D_{h}(s)=\sum _{f{\text{ monic}}}h(f)|f|^{-s},$

where for $g\in A,$  set $|g|=q^{\deg(g)}$  if $g\neq 0,$  and $|g|=0$  otherwise.

The polynomial zeta function is then

$\zeta _{A}(s)=\sum _{f{\text{ monic}}}|f|^{-s}.$

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

$D_{h}(s)=\prod _{P}\left(\sum _{n\mathop {=} 0}^{\infty }h(P^{n})|P|^{-sn}\right),$

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:

$\zeta _{A}(s)=\sum _{f}|f|^{-s}=\sum _{n}\sum _{\deg(f)=n}q^{-sn}=\sum _{n}(q^{n-sn})=(1-q^{1-s})^{-1}.$

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

{\begin{aligned}(f*g)(m)&=\sum _{d\mid m}f(d)g\left({\frac {m}{d}}\right)\\&=\sum _{ab=m}f(a)g(b),\end{aligned}}

where the sum is 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.