Multiplicative function

In number theory, a multiplicative function is an arithmetic function ${\displaystyle f}${\displaystyle f} of a positive integer ${\displaystyle n}${\displaystyle n} with the property that ${\displaystyle f(1)=1}${\displaystyle f(1)=1} and $${\displaystyle f(ab)=f(a)f(b)}$${\displaystyle f(ab)=f(a)f(b)} whenever ${\displaystyle a}${\displaystyle a} and ${\displaystyle b}${\displaystyle b} are coprime.

An arithmetic function is said to be completely multiplicative (or totally multiplicative) if ${\displaystyle f(1)=1}${\displaystyle f(1)=1} and ${\displaystyle f(ab)=f(a)f(b)}${\displaystyle f(ab)=f(a)f(b)} holds for all positive integers ${\displaystyle a}${\displaystyle a} and ${\displaystyle b}${\displaystyle b}, even when they are not coprime.

Some multiplicative functions are defined to make formulas easier to write:

• ${\displaystyle 1(n)}${\displaystyle 1(n)}: the constant function defined by ${\displaystyle 1(n)=1}${\displaystyle 1(n)=1}

• ${\displaystyle \operatorname {Id} (n)}${\displaystyle \operatorname {Id} (n)}: the identity function, defined by ${\displaystyle \operatorname {Id} (n)=n}${\displaystyle \operatorname {Id} (n)=n}

• ${\displaystyle \operatorname {Id} _{k}(n)}${\displaystyle \operatorname {Id} _{k}(n)}: the power functions, defined by ${\displaystyle \operatorname {Id} _{k}(n)=n^{k}}${\displaystyle \operatorname {Id} _{k}(n)=n^{k}} for any complex number ${\displaystyle k}${\displaystyle k}. As special cases we have
  • ${\displaystyle \operatorname {Id} _{0}(n)=1(n)}${\displaystyle \operatorname {Id} _{0}(n)=1(n)}, and
  • ${\displaystyle \operatorname {Id} _{1}(n)=\operatorname {Id} (n)}${\displaystyle \operatorname {Id} _{1}(n)=\operatorname {Id} (n)}.

• ${\displaystyle \varepsilon (n)}${\displaystyle \varepsilon (n)}: the function defined by ${\displaystyle \varepsilon (n)=1}${\displaystyle \varepsilon (n)=1} if ${\displaystyle n=1}${\displaystyle n=1} and ${\displaystyle 0}${\displaystyle 0} otherwise; this is the unit function, so called because it is the multiplicative identity for Dirichlet convolution. Sometimes written as ${\displaystyle u(n)}${\displaystyle u(n)}; not to be confused with ${\displaystyle \mu (n)}${\displaystyle \mu (n)}.

• ${\displaystyle \lambda (n)}${\displaystyle \lambda (n)}: the Liouville function, ${\displaystyle \lambda (n)=(-1)^{\Omega (n)}}${\displaystyle \lambda (n)=(-1)^{\Omega (n)}}, where ${\displaystyle \Omega (n)}${\displaystyle \Omega (n)} is the total number of primes (counted with multiplicity) dividing ${\displaystyle n}${\displaystyle n}

The above functions are all completely multiplicative.

• ${\displaystyle 1_{C}(n)}${\displaystyle 1_{C}(n)}: the indicator function of the set ${\displaystyle C\subseteq \mathbb {Z} }${\displaystyle C\subseteq \mathbb {Z} }. This function is multiplicative precisely when ${\displaystyle C}${\displaystyle C} is closed under multiplication of coprime elements. 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:

• ${\displaystyle \gcd(n,k)}${\displaystyle \gcd(n,k)}: the greatest common divisor of ${\displaystyle n}${\displaystyle n} and ${\displaystyle k}${\displaystyle k}, as a function of ${\displaystyle n}${\displaystyle n}, where ${\displaystyle k}${\displaystyle k} is a fixed integer

• ${\displaystyle \varphi (n)}${\displaystyle \varphi (n)}: Euler's totient function, which counts the positive integers coprime to (but not bigger than) ${\displaystyle n}${\displaystyle n}

• ${\displaystyle \mu (n)}${\displaystyle \mu (n)}: the Möbius function, the parity (${\displaystyle -1}${\displaystyle -1} for odd, ${\displaystyle +1}${\displaystyle +1} for even) of the number of prime factors of square-free numbers; ${\displaystyle 0}${\displaystyle 0} if ${\displaystyle n}${\displaystyle n} is not square-free

• ${\displaystyle \sigma _{k}(n)}${\displaystyle \sigma _{k}(n)}: the divisor function, which is the sum of the ${\displaystyle k}${\displaystyle k}-th powers of all the positive divisors of ${\displaystyle n}${\displaystyle n} (where ${\displaystyle k}${\displaystyle k} may be any complex number). As special cases we have
  • ${\displaystyle \sigma _{0}(n)=d(n)}${\displaystyle \sigma _{0}(n)=d(n)}, the number of positive divisors of ${\displaystyle n}${\displaystyle n},
  • ${\displaystyle \sigma _{1}(n)=\sigma (n)}${\displaystyle \sigma _{1}(n)=\sigma (n)}, the sum of all the positive divisors of ${\displaystyle n}${\displaystyle n}.

• ${\displaystyle \sigma _{k}^{*}(n)}${\displaystyle \sigma _{k}^{*}(n)}: the sum of the ${\displaystyle k}${\displaystyle k}-th powers of all unitary divisors of ${\displaystyle n}${\displaystyle n}

${\displaystyle \sigma _{k}^{*}(n),円=\!\!\sum _{d,円\mid ,円n \atop \gcd(d,,円n/d)=1}\!\!\!d^{k}}${\displaystyle \sigma _{k}^{*}(n),円=\!\!\sum _{d,円\mid ,円n \atop \gcd(d,,円n/d)=1}\!\!\!d^{k}}

• ${\displaystyle \operatorname {rad} (n)}${\displaystyle \operatorname {rad} (n)}: the radical of ${\displaystyle n}${\displaystyle n}, which is the product of the distinct prime factors of ${\displaystyle n}${\displaystyle n}.

• ${\displaystyle a(n)}${\displaystyle a(n)}: the number of non-isomorphic abelian groups of order ${\displaystyle n}${\displaystyle n}

• ${\displaystyle \gamma (n)}${\displaystyle \gamma (n)}, defined by ${\displaystyle \gamma (n)=(-1)^{\omega (n)}}${\displaystyle \gamma (n)=(-1)^{\omega (n)}}, where the additive function ${\displaystyle \omega (n)}${\displaystyle \omega (n)} is the number of distinct primes dividing ${\displaystyle n}${\displaystyle n}
• ${\displaystyle \tau (n)}${\displaystyle \tau (n)}: the Ramanujan tau function
• All Dirichlet characters are completely multiplicative functions, for example
  • ${\displaystyle (n/p)}${\displaystyle (n/p)}, the Legendre symbol, considered as a function of ${\displaystyle n}${\displaystyle n} where ${\displaystyle p}${\displaystyle p} is a fixed prime number

An example of a non-multiplicative function is the arithmetic function ${\displaystyle r_{2}(n)}${\displaystyle r_{2}(n)}, the number of representations of ${\displaystyle n}${\displaystyle 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 ${\displaystyle r_{2}(1)=4\neq 1}${\displaystyle r_{2}(1)=4\neq 1}. This shows that the function is not multiplicative. However, ${\displaystyle r_{2}(n)/4}${\displaystyle 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".^[1]

See arithmetic function for some other examples of non-multiplicative functions.

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

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

If f and g are two multiplicative functions, one defines a new multiplicative function ${\displaystyle f*g}${\displaystyle f*g}, the Dirichlet convolution of f and g, by $${\displaystyle (f,円*,円g)(n)=\sum _{d|n}f(d),円g\left({\frac {n}{d}}\right)}$${\displaystyle (f,円*,円g)(n)=\sum _{d|n}f(d),円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:

• ${\displaystyle \mu *1=\varepsilon }${\displaystyle \mu *1=\varepsilon } (the Möbius inversion formula)
• ${\displaystyle (\mu \operatorname {Id} _{k})*\operatorname {Id} _{k}=\varepsilon }${\displaystyle (\mu \operatorname {Id} _{k})*\operatorname {Id} _{k}=\varepsilon } (generalized Möbius inversion)
• ${\displaystyle \varphi *1=\operatorname {Id} }${\displaystyle \varphi *1=\operatorname {Id} }
• ${\displaystyle d=1*1}${\displaystyle d=1*1}
• ${\displaystyle \sigma =\operatorname {Id} *1=\varphi *d}${\displaystyle \sigma =\operatorname {Id} *1=\varphi *d}
• ${\displaystyle \sigma _{k}=\operatorname {Id} _{k}*1}${\displaystyle \sigma _{k}=\operatorname {Id} _{k}*1}
• ${\displaystyle \operatorname {Id} =\varphi *1=\sigma *\mu }${\displaystyle \operatorname {Id} =\varphi *1=\sigma *\mu }
• ${\displaystyle \operatorname {Id} _{k}=\sigma _{k}*\mu }${\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 a,b\in \mathbb {Z} ^{+}}: $${\displaystyle {\begin{aligned}(f\ast g)(ab)&=\sum _{d|ab}f(d)g\left({\frac {ab}{d}}\right)\\&=\sum _{d_{1}|a}\sum _{d_{2}|b}f(d_{1}d_{2})g\left({\frac {ab}{d_{1}d_{2}}}\right)\\&=\sum _{d_{1}|a}f(d_{1})g\left({\frac {a}{d_{1}}}\right)\times \sum _{d_{2}|b}f(d_{2})g\left({\frac {b}{d_{2}}}\right)\\&=(f\ast g)(a)\cdot (f\ast g)(b).\end{aligned}}}$${\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}}}

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

An arithmetical function f is said to be a rational arithmetical function of order ${\displaystyle (r,s)}${\displaystyle (r,s)} if there exists completely multiplicative functions g₁,...,g_r, h₁,...,h_s such that $${\displaystyle f=g_{1}\ast \cdots \ast g_{r}\ast h_{1}^{-1}\ast \cdots \ast h_{s}^{-1},}$${\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 ${\displaystyle (1,1)}${\displaystyle (1,1)} are known as totient functions, and rational arithmetical functions of order ${\displaystyle (2,0)}${\displaystyle (2,0)} are known as quadratic functions or specially multiplicative functions. Euler's function ${\displaystyle \varphi (n)}${\displaystyle \varphi (n)} is a totient function, and the divisor function ${\displaystyle \sigma _{k}(n)}${\displaystyle \sigma _{k}(n)} is a quadratic function. Completely multiplicative functions are rational arithmetical functions of order ${\displaystyle (1,0)}${\displaystyle (1,0)}. Liouville's function ${\displaystyle \lambda (n)}${\displaystyle \lambda (n)} is completely multiplicative. The Möbius function ${\displaystyle \mu (n)}${\displaystyle \mu (n)} is a rational arithmetical function of order ${\displaystyle (0,1)}${\displaystyle (0,1)}. By convention, the identity element ${\displaystyle \varepsilon }${\displaystyle \varepsilon } under the Dirichlet convolution is a rational arithmetical function of order ${\displaystyle (0,0)}${\displaystyle (0,0)}.

All rational arithmetical functions are multiplicative. A multiplicative function f is a rational arithmetical function of order ${\displaystyle (r,s)}${\displaystyle (r,s)} if and only if its Bell series is of the form $${\displaystyle {\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}={\frac {(1-h_{1}(p)x)(1-h_{2}(p)x)\cdots (1-h_{s}(p)x)}{(1-g_{1}(p)x)(1-g_{2}(p)x)\cdots (1-g_{r}(p)x)}}}}$${\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 ${\displaystyle p}${\displaystyle p}.

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

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

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

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

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

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

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

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

for ${\displaystyle k=0}${\displaystyle k=0}. S. Chowla gave the inverse form for general ${\displaystyle k}${\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 ${\displaystyle f=g_{1}\ast g_{2}}${\displaystyle f=g_{1}\ast g_{2}} satisfy the Busche-Ramanujan identities with ${\displaystyle f_{A}=g_{1}g_{2}}${\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).

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 ${\displaystyle \lambda }${\displaystyle \lambda } on A is called multiplicative if ${\displaystyle \lambda (fg)=\lambda (f)\lambda (g)}${\displaystyle \lambda (fg)=\lambda (f)\lambda (g)} whenever f and g are relatively prime.

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

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

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

The polynomial zeta function is then

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

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

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

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

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

$${\displaystyle {\begin{aligned}(f*g)(m)&=\sum _{d\mid m}f(d)g\left({\frac {m}{d}}\right)\\&=\sum _{ab=m}f(a)g(b),\end{aligned}}}$${\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 ${\displaystyle D_{h}D_{g}=D_{h*g}}${\displaystyle D_{h}D_{g}=D_{h*g}} still holds.

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

a sum can be distributed across the product$${\displaystyle y_{t}=\sum (t/T)^{1/2}u_{t}=\sum (t/T)^{1/2}G_{t}^{1/2}\epsilon _{t}}$${\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: $${\displaystyle {\tilde {y}}_{t}^{2}={\frac {y_{t}^{2}}{g_{t}}}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(\epsilon _{t}^{2}-1),}$${\displaystyle {\tilde {y}}_{t}^{2}={\frac {y_{t}^{2}}{g_{t}}}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(\epsilon _{t}^{2}-1),}

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

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

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

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

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

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

• Euler product
• Bell series
• Lambert series

• See chapter 2 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• P. J. McCarthy, Introduction to Arithmetical Functions, Universitext. New York: Springer-Verlag, 1986.
• Hafner, Christian M.; Linton, Oliver (2010). "Efficient estimation of a multivariate multiplicative volatility model" (PDF). Journal of Econometrics. 159 (1): 55–73. doi:10.1016/j.jeconom.201004007. S2CID 54812323.
• P. Haukkanen (2003). "Some characterizations of specially multiplicative functions". Int. J. Math. Math. Sci. 2003 (37): 2335–2344. doi:10.1155/S0161171203301139 .
• P. Haukkanen (2012). "Extensions of the class of multiplicative functions". East–West Journal of Mathematics. 14 (2): 101–113.
• DB Lahiri (1972). "Hypo-multiplicative number-theoretic functions". Aequationes Mathematicae. 8 (3): 316–317. doi:10.1007/BF01844515.

• D. Rearick (1966). "Semi-multiplicative functions". Duke Math. J. 33: 49–53. doi:10.1215/S0012-7094-66-03308-4.

• L. Tóth (2013). "Two generalizations of the Busche-Ramanujan identities". International Journal of Number Theory. 9 (5): 1301–1311. arXiv:1301.3331 . doi:10.1142/S1793042113500280.
• R. Vaidyanathaswamy (1931). "The theory of multiplicative arithmetic functions". Transactions of the American Mathematical Society. 33 (2): 579–662. doi:10.1090/S0002-9947-1931-1501607-1 .
• Ramanujan, S. (1916). "Some formulae in the analytic theory of numbers" (PDF). Messenger. 45: 81–84.

• E. Busche, Lösung einer Aufgabe über Teileranzahlen. Mitt. Math. Ges. Hamb. 4, 229--237 (1906)

• A. Selberg: Remarks on multiplicative functions. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), pp. 232–241, Springer, 1977.
• Mathar, Richard J. (2012). "Survey of Dirichlet series of multiplicative arithmetic functions". arXiv:1106.4038 [math.NT].

• Multiplicative function at PlanetMath.

• Multiplicative functions
• Number theory

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