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)
 * 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 C ⊂ Z, 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.
 * 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) = (&minus;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 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:

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: $$d(144) = \sigma_0(144) = \sigma_0(2^4) \, \sigma_0(3^2) = (1^0 + 2^0 + 4^0 + 8^0 + 16^0)(1^0 + 3^0 + 9^0) = 5 \cdot 3 = 15$$ $$\sigma(144) = \sigma_1(144) = \sigma_1(2^4) \, \sigma_1(3^2) = (1^1 + 2^1 + 4^1 + 8^1 + 16^1)(1^1 + 3^1 + 9^1) = 31 \cdot 13 = 403$$ $$\sigma^*(144) = \sigma^*(2^4) \, \sigma^*(3^2) = (1^1 + 16^1)(1^1 + 9^1) = 17 \cdot 10 = 170$$

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 $$a,b \in \mathbb{Z}^{+}$$: $$\begin{align} (f \ast g)(ab) & = \sum_{d|ab} f(d) g\left(\frac{ab}{d}\right) \\ &= \sum_{d_1|a} \sum_{d_2|b} f(d_1d_2) g\left(\frac{ab}{d_1d_2}\right) \\ &= \sum_{d_1|a} f(d_1) g\left(\frac{a}{d_1}\right) \times \sum_{d_2|b} f(d_2) g\left(\frac{b}{d_2}\right) \\ &= (f \ast g)(a) \cdot (f \ast g)(b). \end{align} $$

Dirichlet series for some multiplicative functions
More examples are shown in the article on Dirichlet series.
 * $$\sum_{n\ge 1} \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}$$
 * $$\sum_{n\ge 1} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}$$
 * $$\sum_{n\ge 1} \frac{d(n)^2}{n^s} = \frac{\zeta(s)^4}{\zeta(2s)}$$
 * $$\sum_{n\ge 1} \frac{2^{\omega(n)}}{n^s} = \frac{\zeta(s)^2}{\zeta(2s)}$$

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 g1,...,gr, h1,...,hs 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 $$f_A=g_1g_2$$. In fact, 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 $$g\ne 0,$$ 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{align} (f*g)(m) &= \sum_{d \mid m} f(d)g\left(\frac{m}{d}\right) \\ &= \sum_{ab = m}f(a)g(b), \end{align} $$ where the sum is over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m. The identity $$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: $$\tilde{y}^2_t = \frac{y^2_t}{g_t} = \sigma^2(t/T) + \sigma^2(t/T)(\epsilon^2_t - 1),$$

and $$y^2_t = \sigma^2(t/T) + \sigma^2(t/T)(g_t\epsilon^2_t - 1).$$

Thus it gives an estimate value of $$L_t(\tau;u) = \sum_{t=1}^T K_h(u - t/T)\begin{bmatrix} ln\tau + \frac{y^2_t}{g_t\tau} \end{bmatrix}$$

with a local likelihood function for $$y^2_t$$ 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 $$ f(n)=c f_m(n/a) $$ 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 $$n$$.

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 $$a=1$$.