# Multiplicative function

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 $f(ab)=f(a)f(b)$ whenever a and b are coprime.
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)
* Id$_k$(n): the power functions, defined by Id$_k$(n) = n$^k$ for any complex number k (completely multiplicative). As special cases we have
** Id$_0$(n) = 1(n) and
** Id$_1$(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) .
* 1$_C$(n), the indicator function of the set C ⊂ Z, for certain sets C. The indicator function 1$_C$(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. There are also other sets (not closed under multiplication) that give rise to such functions, such as 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
* σ$_k$(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
** σ$_0$(n) = d(n) the number of positive divisors of n,
** σ$_1$(n) = σ(n), the sum of all the positive divisors of n.
* The sum of the k-th powers of the unitary divisors is denoted by σ*$_k$(n):
    $\sigma _{k}^{*}(n)=\sum _{d\,\mid \,n \atop \gcd(d,\,n/d)=1}\!\!d^{k}.$
* 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
** (n/p), the Legendre symbol, considered as a function of n where p is a fixed prime number.
An example of a non-multiplicative function is the arithmetic function r$_2$(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:
and therefore r$_2$(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, r$_2$(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 = p$^a$ q$^b$ ..., then f(n) = f(p$^a$) f(q$^b$) ...
This property of multiplicative functions significantly reduces the need for computation, as in the following examples for n = 144 = 2$^4$ · 3$^2$: $d(144)=\sigma _{0}(144)=\sigma _{0}(2^{4})\,\sigma _{0}(3^{2})=(1^{0}+2^{0}+4^{0}+8^{0}+16^{0})(1^{0}+3^{0}+9^{0})=5\cdot 3=15$ $\sigma (144)=\sigma _{1}(144)=\sigma _{1}(2^{4})\,\sigma _{1}(3^{2})=(1^{1}+2^{1}+4^{1}+8^{1}+16^{1})(1^{1}+3^{1}+9^{1})=31\cdot 13=403$ $\sigma ^{*}(144)=\sigma ^{*}(2^{4})\,\sigma ^{*}(3^{2})=(1^{1}+16^{1})(1^{1}+9^{1})=17\cdot 10=170$
Similarly, we have: $\varphi (144)=\varphi (2^{4})\,\varphi (3^{2})=8\cdot 6=48$
In general, if f(n) is a multiplicative function and a, b are any two positive integers, then
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:
* $\mu *1=\varepsilon$ (the Möbius inversion formula)
* $(\mu \operatorname {Id} _{k})*\operatorname {Id} _{k}=\varepsilon$ (generalized Möbius inversion)
* $\varphi *1=\operatorname {Id}$
* $d=1*1$
* $\sigma =\operatorname {Id} *1=\varphi *d$
* $\sigma _{k}=\operatorname {Id} _{k}*1$
* $\operatorname {Id} =\varphi *1=\sigma *\mu$
* $\operatorname {Id} _{k}=\sigma _{k}*\mu$
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 $a,b\in \mathbb {Z} ^{+}$: ${\begin{aligned}(f\ast g)(ab)&=\sum _{d|ab}f(d)g\left({\frac {ab}{d}}\right)\\&=\sum _{d_{1}|a}\sum _{d_{2}|b}f(d_{1}d_{2})g\left({\frac {ab}{d_{1}d_{2}}}\right)\\&=\sum _{d_{1}|a}f(d_{1})g\left({\frac {a}{d_{1}}}\right)\times \sum _{d_{2}|b}f(d_{2})g\left({\frac {b}{d_{2}}}\right)\\&=(f\ast g)(a)\cdot (f\ast g)(b).\end{aligned}}$

### 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.

## Rational arithmetical functions

An arithmetical function f is said to be a rational arithmetical function of order $(r,s)$ if there exists completely multiplicative functions g$_1$,...,g$_r$, h$_1$,...,h$_s$ such that $f=g_{1}\ast \cdots \ast g_{r}\ast h_{1}^{-1}\ast \cdots \ast h_{s}^{-1},$ where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order $(1,1)$ are known as totient functions, and rational arithmetical functions of order $(2,0)$ are known as quadratic functions or specially multiplicative functions. Euler's function $\varphi (n)$ is a totient function, and the divisor function $\sigma _{k}(n)$ is a quadratic function. Completely multiplicative functions are rational arithmetical functions of order $(1,0)$. Liouville's function $\lambda (n)$ is completely multiplicative. The Möbius function $\mu (n)$ is a rational arithmetical function of order $(0,1)$. By convention, the identity element $\varepsilon$ under the Dirichlet convolution is a rational arithmetical function of order $(0,0)$.
All rational arithmetical functions are multiplicative. A multiplicative function f is a rational arithmetical function of order $(r,s)$ if and only if its Bell series is of the form ${\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}={\frac {(1-h_{1}(p)x)(1-h_{2}(p)x)\cdots (1-h_{s}(p)x)}{(1-g_{1}(p)x)(1-g_{2}(p)x)\cdots (1-g_{r}(p)x)}}}$ for all prime numbers $p$.
The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).

## Busche-Ramanujan identities

A multiplicative function $f$ is said to be specially multiplicative if there is a completely multiplicative function $f_{A}$ such that
    $f(m)f(n)=\sum _{d\mid (m,n)}f(mn/d^{2})f_{A}(d)$
for all positive integers $m$ and $n$, or equivalently
    $f(mn)=\sum _{d\mid (m,n)}f(m/d)f(n/d)\mu (d)f_{A}(d)$
for all positive integers $m$ and $n$, where $\mu$ is the Möbius function. These are known as Busche-Ramanujan identities. In 1906, E. Busche stated the identity
    $\sigma _{k}(m)\sigma _{k}(n)=\sum _{d\mid (m,n)}\sigma _{k}(mn/d^{2})d^{k},$
and, in 1915, S. Ramanujan gave the inverse form
    $\sigma _{k}(mn)=\sum _{d\mid (m,n)}\sigma _{k}(m/d)\sigma _{k}(n/d)\mu (d)d^{k}$
for $k=0$. S. Chowla gave the inverse form for general $k$ in 1929, see P. J. McCarthy (1986). The study of Busche-Ramanujan identities begun from an attempt to better understand the special cases given by Busche and Ramanujan.
It is known that quadratic functions $f=g_{1}\ast g_{2}$ satisfy the Busche-Ramanujan identities with $f_{A}=g_{1}g_{2}$. Quadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see R. Vaidyanathaswamy (1931).

## Multiplicative function over F$_q$[X]

Let A = F$_q$[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 F$_q$[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.

## Multivariate

Multivariate functions can be constructed using multiplicative model estimators. Where a matrix function of A is defined as $D_{N}=N^{2}\times N(N+1)/2$
a sum can be distributed across the product$y_{t}=\sum (t/T)^{1/2}u_{t}=\sum (t/T)^{1/2}G_{t}^{1/2}\epsilon _{t}$
For the efficient estimation of Σ(.), the following two nonparametric regressions can be considered: ${\tilde {y}}_{t}^{2}={\frac {y_{t}^{2}}{g_{t}}}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(\epsilon _{t}^{2}-1),$
and $y_{t}^{2}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(g_{t}\epsilon _{t}^{2}-1).$
Thus it gives an estimate value of $L_{t}(\tau ;u)=\sum _{t=1}^{T}K_{h}(u-t/T){\begin{bmatrix}ln\tau +{\frac {y_{t}^{2}}{g_{t}\tau }}\end{bmatrix}}$
with a local likelihood function for $y_{t}^{2}$ with known $g_{t}$ and unknown $\sigma ^{2}(t/T)$.

## Generalizations

An arithmetical function $f$ is quasimultiplicative if there exists a nonzero constant $c$ such that $c\,f(mn)=f(m)f(n)$ for all positive integers $m,n$ with $(m,n)=1$. This concept originates by Lahiri (1972).
An arithmetical function $f$ is semimultiplicative if there exists a nonzero constant $c$, a positive integer $a$ and a multiplicative function $f_{m}$ such that $f(n)=cf_{m}(n/a)$ for all positive integers $n$ (under the convention that $f_{m}(x)=0$ if $x$ is not a positive integer.) This concept is due to David Rearick (1966).
An arithmetical function $f$ is Selberg multiplicative if for each prime $p$ there exists a function $f_{p}$ on nonnegative integers with $f_{p}(0)=1$ for all but finitely many primes $p$ such that $f(n)=\prod _{p}f_{p}(\nu _{p}(n))$ for all positive integers $n$, where $\nu _{p}(n)$ is the exponent of $p$ in the canonical factorization of $n$. See Selberg (1977).
It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide. They both satisfy the arithmetical identity $f(m)f(n)=f((m,n))f([m,n])$ for all positive integers $m,n$. See Haukkanen (2012).
It is well known and easy to see that multiplicative functions are quasimultiplicative functions with $c=1$ and quasimultiplicative functions are semimultiplicative functions with $a=1$.