Multiplicative function

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 = 2⁴ · 3²: $${\displaystyle d(144)={\sigma }_0(144)={\sigma }_0(2^4){\sigma }_0(3^2)=(1^0+2^0+4^0+8^0+{16}^0)(1^0+3^0+9^0)=5\cdot 3=15}$$ $${\displaystyle \sigma (144)={\sigma }_1(144)={\sigma }_1(2^4){\sigma }_1(3^2)=(1^1+2^1+4^1+8^1+{16}^1)(1^1+3^1+9^1)=31\cdot 13=403}$$ $${\displaystyle {\sigma }^{\ast }(144)={\sigma }^{\ast }(2^4){\sigma }^{\ast }(3^2)=(1^1+{16}^1)(1^1+9^1)=17\cdot 10=170}$$

Similarly, we have: $${\displaystyle \phi (144)=\phi (2^4)\phi (3^2)=8\cdot 6=48}$$

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)).

The product of two multiplicative functions is also multiplicative.

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 ${\displaystyle f\ast g}$, the Dirichlet convolution of f and g, by $${\displaystyle (f\ast g)(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right)}$$ 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:

• ${\displaystyle \mu \ast 1=\epsilon }$ (the Möbius inversion formula)
• ${\displaystyle (\mu {\mathrm{Id}}_k)\ast {\mathrm{Id}}_k=\epsilon }$ (generalized Möbius inversion)
• ${\displaystyle \phi \ast 1=\mathrm{Id}}$
• ${\displaystyle d=1\ast 1}$
• ${\displaystyle \sigma =\mathrm{Id}\ast 1=\phi \ast d}$
• ${\displaystyle {\sigma }_k={\mathrm{Id}}_k\ast 1}$
• ${\displaystyle \mathrm{Id}=\phi \ast 1=\sigma \ast \mu }$
• ${\displaystyle {\mathrm{Id}}_k={\sigma }_k\ast \mu }$

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 ${\displaystyle a,b\in {\mathbb{Z}}^{+}}$: $${\displaystyle \begin{array}{cc}(f\ast g)(ab)& =\sum _{d|ab}f(d)g\left(\frac{ab}{d}\right)\\ & =\sum _{d_1|a}\sum _{d_2|b}f(d_1{d}_2)g\left(\frac{ab}{d_1{d}_2}\right)\\ & =\sum _{d_1|a}f(d_1)g\left(\frac{a}{d_1}\right)\times \sum _{d_2|b}f(d_2)g\left(\frac{b}{d_2}\right)\\ & =(f\ast g)(a)\cdot (f\ast g)(b).\end{array}}$$

Dirichlet series for some multiplicative functions

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

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 ${\displaystyle (r,s)}$ if there exists completely multiplicative functions g₁,...,g_r, h₁,...,h_s such that $${\displaystyle f=g_1\ast \cdots \ast g_r\ast h_1^{-1}\ast \cdots \ast h_s^{-1},}$$ where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order ${\displaystyle (1,1)}$ are known as totient functions, and rational arithmetical functions of order ${\displaystyle (2,0)}$ are known as quadratic functions or specially multiplicative functions. Euler's function ${\displaystyle \phi (n)}$ is a totient function, and the divisor function ${\displaystyle {\sigma }_k(n)}$ is a quadratic function. Completely multiplicative functions are rational arithmetical functions of order ${\displaystyle (1,0)}$. Liouville's function ${\displaystyle \lambda (n)}$ is completely multiplicative. The Möbius function ${\displaystyle \mu (n)}$ is a rational arithmetical function of order ${\displaystyle (0,1)}$. By convention, the identity element ${\displaystyle \epsilon }$ under the Dirichlet convolution is a rational arithmetical function of order ${\displaystyle (0,0)}$.

All rational arithmetical functions are multiplicative. A multiplicative function f is a rational arithmetical function of order ${\displaystyle (r,s)}$ if and only if its Bell series is of the form $${\displaystyle {\displaystyle f_p(x)=\sum _{n=0}^{\mathrm{\infty }}f(p^n)x^n=\frac{(1-h_1(p)x)(1-h_2(p)x)\cdots (1-h_s(p)x)}{(1-g_1(p)x)(1-g_2(p)x)\cdots (1-g_r(p)x)}}}$$ for all prime numbers ${\displaystyle p}$.

The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).

Busche-Ramanujan identities

A multiplicative function ${\displaystyle f}$ is said to be specially multiplicative if there is a completely multiplicative function ${\displaystyle f_A}$ such that

${\displaystyle f(m)f(n)=\sum _{d\mid (m,n)}f(mn/d^2)f_A(d)}$

for all positive integers ${\displaystyle m}$ and ${\displaystyle n}$, or equivalently

${\displaystyle f(mn)=\sum _{d\mid (m,n)}f(m/d)f(n/d)\mu (d)f_A(d)}$

for all positive integers ${\displaystyle m}$ and ${\displaystyle n}$, where ${\displaystyle \mu }$ is the Möbius function. These are known as Busche-Ramanujan identities. In 1906, E. Busche stated the identity

${\displaystyle {\sigma }_k(m){\sigma }_k(n)=\sum _{d\mid (m,n)}{\sigma }_k(mn/d^2)d^k,}$

and, in 1915, S. Ramanujan gave the inverse form

${\displaystyle {\sigma }_k(mn)=\sum _{d\mid (m,n)}{\sigma }_k(m/d){\sigma }_k(n/d)\mu (d)d^k}$

for ${\displaystyle k=0}$. S. Chowla gave the inverse form for general ${\displaystyle k}$ 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 ${\displaystyle f=g_1\ast g_2}$ satisfy the Busche-Ramanujan identities with ${\displaystyle f_A=g_1{g}_2}$. 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 Fq[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 ${\displaystyle \lambda }$ on A is called multiplicative if ${\displaystyle \lambda (fg)=\lambda (f)\lambda (g)}$ 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

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

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

The polynomial zeta function is then

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

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

$${\displaystyle D_h(s)=\prod _P\left(\sum _{n=0}^{\mathrm{\infty }}h(P^n)|P{|}^{-sn}\right),}$$

where the product runs over all monic irreducible polynomials P. For example, the product representation of the zeta function is as for the integers:

$${\displaystyle {\zeta }_A(s)=\prod _P(1-|P{|}^{-s}{)}^{-1}.}$$

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

$${\displaystyle {\zeta }_A(s)=\sum _f|f{|}^{-s}=\sum _n\sum _{\mathrm{deg}(f)=n}{q}^{-sn}=\sum _n(q^{n-sn})=(1-q^{1-s}{)}^{-1}.}$$

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

$${\displaystyle \begin{array}{cc}(f\ast g)(m)& =\sum _{d\mid m}f(d)g\left(\frac{m}{d}\right)\\ & =\sum _{ab=m}f(a)g(b),\end{array}}$$

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 ${\displaystyle D_h{D}_g=D_{h\ast g}}$ still holds.

Multivariate

Multivariate functions can be constructed using multiplicative model estimators. Where a matrix function of A is defined as $${\displaystyle D_N=N^2\times N(N+1)/2}$$

a sum can be distributed across the product$${\displaystyle y_t=\sum (t/T{)}^{1/2}{u}_t=\sum (t/T{)}^{1/2}{G}_t^{1/2}{ϵ}_t}$$

For the efficient estimation of Σ(.), the following two nonparametric regressions can be considered: $${\displaystyle {\tilde{y}}_t^2=\frac{y_t^2}{g_t}={\sigma }^2(t/T)+{\sigma }^2(t/T)({ϵ}_t^2-1),}$$

and $${\displaystyle y_t^2={\sigma }^2(t/T)+{\sigma }^2(t/T)(g_t{ϵ}_t^2-1).}$$

Thus it gives an estimate value of $${\displaystyle L_t(\tau ;u)=\sum _{t=1}^T{K}_h(u-t/T)\left[\begin{array}{c}ln\tau +\frac{y_t^2}{g_t\tau }\end{array}\right]}$$

with a local likelihood function for ${\displaystyle y_t^2}$ with known ${\displaystyle g_t}$ and unknown ${\displaystyle {\sigma }^2(t/T)}$.

Generalizations

An arithmetical function ${\displaystyle f}$ is quasimultiplicative if there exists a nonzero constant ${\displaystyle c}$ such that ${\displaystyle cf(mn)=f(m)f(n)}$ for all positive integers ${\displaystyle m,n}$ with ${\displaystyle (m,n)=1}$. This concept originates by Lahiri (1972).

An arithmetical function ${\displaystyle f}$ is semimultiplicative if there exists a nonzero constant ${\displaystyle c}$, a positive integer ${\displaystyle a}$ and a multiplicative function ${\displaystyle f_m}$ such that ${\displaystyle f(n)=c{f}_m(n/a)}$ for all positive integers ${\displaystyle n}$ (under the convention that ${\displaystyle f_m(x)=0}$ if ${\displaystyle x}$ is not a positive integer.) This concept is due to David Rearick (1966).

An arithmetical function ${\displaystyle f}$ is Selberg multiplicative if for each prime ${\displaystyle p}$ there exists a function ${\displaystyle f_p}$ on nonnegative integers with ${\displaystyle f_p(0)=1}$ for all but finitely many primes ${\displaystyle p}$ such that ${\displaystyle f(n)=\prod _p{f}_p({\nu }_p(n))}$ for all positive integers ${\displaystyle n}$, where ${\displaystyle {\nu }_p(n)}$ is the exponent of ${\displaystyle p}$ in the canonical factorization of ${\displaystyle n}$. See Selberg (1977).

It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide. They both satisfy the arithmetical identity ${\displaystyle f(m)f(n)=f((m,n))f([m,n])}$ for all positive integers ${\displaystyle m,n}$. See Haukkanen (2012).

It is well known and easy to see that multiplicative functions are quasimultiplicative functions with ${\displaystyle c=1}$ and quasimultiplicative functions are semimultiplicative functions with ${\displaystyle a=1}$.