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

${\displaystyle f(ab)=f(a)f(b)}$
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 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.
• ${\displaystyle \varphi (n)}$: Euler's totient function ${\displaystyle \varphi }$, 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):
${\displaystyle \sigma _{k}^{*}(n)=\sum _{d\,\mid \,n \atop \gcd(d,\,n/d)=1}\!\!d^{k}.}$
• 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

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:

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

Similarly, we have:

${\displaystyle \varphi (144)=\varphi (2^{4})\,\varphi (3^{2})=8\cdot 6=48}$

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

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 ${\displaystyle f*g}$, the Dirichlet convolution of f and g, by

${\displaystyle (f\,*\,g)(n)=\sum _{d|n}f(d)\,g\left({\frac {n}{d}}\right)}$
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:

• ${\displaystyle \mu *1=\varepsilon }$ (the Möbius inversion formula)
• ${\displaystyle (\mu \operatorname {Id} _{k})*\operatorname {Id} _{k}=\varepsilon }$ (generalized Möbius inversion)
• ${\displaystyle \varphi *1=\operatorname {Id} }$
• ${\displaystyle d=1*1}$
• ${\displaystyle \sigma =\operatorname {Id} *1=\varphi *d}$
• ${\displaystyle \sigma _{k}=\operatorname {Id} _{k}*1}$
• ${\displaystyle \operatorname {Id} =\varphi *1=\sigma *\mu }$
• ${\displaystyle \operatorname {Id} _{k}=\sigma _{k}*\mu }$

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 ${\displaystyle a,b\in \mathbb {Z} ^{+}}$:

{\displaystyle {\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

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

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 ${\displaystyle (r,s)}$ if there exists completely multiplicative functions g1,...,gr, h1,...,hs 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 ${\displaystyle \varphi (n)}$ 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 ${\displaystyle \varepsilon }$ 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

${\displaystyle {\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}={\frac {(1-h_{1}(p)x)(1-h_{2}(p)x)\cdots (1-h_{s}(p)x)}{(1-g_{1}(p)x)(1-g_{2}(p)x)\cdots (1-g_{r}(p)x)}}}}$
for all prime numbers ${\displaystyle p}$.

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

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 ${\displaystyle \lambda }$ 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 ${\displaystyle |g|=q^{\deg(g)}}$ if ${\displaystyle g\neq 0,}$ 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):

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

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

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

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

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

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{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 ${\displaystyle D_{h}D_{g}=D_{h*g}}$ still holds.

Multivariate

Multivariate functions can be constructed using multiplicative model estimators. Where a matrix function of A is defined as

${\displaystyle D_{N}=N^{2}\times N(N+1)/2}$

a sum can be distributed across the product

${\displaystyle y_{t}=\sum (t/T)^{1/2}u_{t}=\sum (t/T)^{1/2}G_{t}^{1/2}\epsilon _{t}}$

For the efficient estimation of Σ(.), the following two nonparametric regressions can be considered:

${\displaystyle {\tilde {y}}_{t}^{2}={\frac {y_{t}^{2}}{g_{t}}}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(\epsilon _{t}^{2}-1),}$

and

${\displaystyle y_{t}^{2}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(g_{t}\epsilon _{t}^{2}-1).}$

Thus it gives an estimate value of

${\displaystyle L_{t}(\tau ;u)=\sum _{t=1}^{T}K_{h}(u-t/T){\begin{bmatrix}ln\tau +{\frac {y_{t}^{2}}{g_{t}\tau }}\end{bmatrix}}}$

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