Multiplicative function

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

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

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

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

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

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

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

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

The above functions are all completely multiplicative.

• 1 C ( n ) {\displaystyle 1_{C}(n)} : the indicator function of the set C ⊆ Z {\displaystyle C\subseteq \mathbb {Z} } . This function is multiplicative precisely when 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:

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

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

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

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

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

σ k ∗ ( n ) = ∑ d ∣ n gcd ( d , n / d ) = 1 d k {\displaystyle \sigma _{k}^{*}(n)\,=\!\!\sum _{d\,\mid \,n \atop \gcd(d,\,n/d)=1}\!\!\!d^{k}}

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

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

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

An example of a non-multiplicative function is the arithmetic function r 2 ( n ) {\displaystyle r_{2}(n)} , the number of representations of 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 r 2 ( 1 ) = 4 ≠ 1 {\displaystyle r_{2}(1)=4\neq 1} . This shows that the function is not multiplicative. However, 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: d ( 144 ) = σ 0 ( 144 ) = σ 0 ( 2 4 ) σ 0 ( 3 2 ) = ( 1 0 + 2 0 + 4 0 + 8 0 + 16 0 ) ( 1 0 + 3 0 + 9 0 ) = 5 ⋅ 3 = 15 {\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} σ ( 144 ) = σ 1 ( 144 ) = σ 1 ( 2 4 ) σ 1 ( 3 2 ) = ( 1 1 + 2 1 + 4 1 + 8 1 + 16 1 ) ( 1 1 + 3 1 + 9 1 ) = 31 ⋅ 13 = 403 {\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} σ ∗ ( 144 ) = σ ∗ ( 2 4 ) σ ∗ ( 3 2 ) = ( 1 1 + 16 1 ) ( 1 1 + 9 1 ) = 17 ⋅ 10 = 170 {\displaystyle \sigma ^{*}(144)=\sigma ^{*}(2^{4})\,\sigma ^{*}(3^{2})=(1^{1}+16^{1})(1^{1}+9^{1})=17\cdot 10=170}

Similarly, we have: φ ( 144 ) = φ ( 2 4 ) φ ( 3 2 ) = 8 ⋅ 6 = 48 {\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

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 f ∗ g {\displaystyle f*g} , the Dirichlet convolution of f and g, by ( f ∗ g ) ( n ) = ∑ d | n f ( d ) g ( n d ) {\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:

• μ ∗ 1 = ε {\displaystyle \mu *1=\varepsilon } (the Möbius inversion formula)
• ( μ Id k ) ∗ Id k = ε {\displaystyle (\mu \operatorname {Id} _{k})*\operatorname {Id} _{k}=\varepsilon } (generalized Möbius inversion)
• φ ∗ 1 = Id {\displaystyle \varphi *1=\operatorname {Id} }
• d = 1 ∗ 1 {\displaystyle d=1*1}
• σ = Id ∗ 1 = φ ∗ d {\displaystyle \sigma =\operatorname {Id} *1=\varphi *d}
• σ k = Id k ∗ 1 {\displaystyle \sigma _{k}=\operatorname {Id} _{k}*1}
• Id = φ ∗ 1 = σ ∗ μ {\displaystyle \operatorname {Id} =\varphi *1=\sigma *\mu }
• Id k = σ k ∗ μ {\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 a , b ∈ Z + {\displaystyle a,b\in \mathbb {Z} ^{+}} : ( f ∗ g ) ( a b ) = ∑ d | a b f ( d ) g ( a b d ) = ∑ d 1 | a ∑ d 2 | b f ( d 1 d 2 ) g ( a b d 1 d 2 ) = ∑ d 1 | a f ( d 1 ) g ( a d 1 ) × ∑ d 2 | b f ( d 2 ) g ( b d 2 ) = ( f ∗ g ) ( a ) ⋅ ( f ∗ g ) ( b ) . {\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}}}

• ∑ n ≥ 1 μ ( n ) n s = 1 ζ ( s ) {\displaystyle \sum _{n\geq 1}{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}}
• ∑ n ≥ 1 φ ( n ) n s = ζ ( s − 1 ) ζ ( s ) {\displaystyle \sum _{n\geq 1}{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}}
• ∑ n ≥ 1 d ( n ) 2 n s = ζ ( s ) 4 ζ ( 2 s ) {\displaystyle \sum _{n\geq 1}{\frac {d(n)^{2}}{n^{s}}}={\frac {\zeta (s)^{4}}{\zeta (2s)}}}
• ∑ n ≥ 1 2 ω ( n ) n s = ζ ( s ) 2 ζ ( 2 s ) {\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 ( r , s ) {\displaystyle (r,s)} if there exists completely multiplicative functions g₁,...,g_r, h₁,...,h_s such that f = g 1 ∗ ⋯ ∗ g r ∗ h 1 − 1 ∗ ⋯ ∗ 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 ( 1 , 1 ) {\displaystyle (1,1)} are known as totient functions, and rational arithmetical functions of order ( 2 , 0 ) {\displaystyle (2,0)} are known as quadratic functions or specially multiplicative functions. Euler's function φ ( n ) {\displaystyle \varphi (n)} is a totient function, and the divisor function σ k ( n ) {\displaystyle \sigma _{k}(n)} is a quadratic function. Completely multiplicative functions are rational arithmetical functions of order ( 1 , 0 ) {\displaystyle (1,0)} . Liouville's function λ ( n ) {\displaystyle \lambda (n)} is completely multiplicative. The Möbius function μ ( n ) {\displaystyle \mu (n)} is a rational arithmetical function of order ( 0 , 1 ) {\displaystyle (0,1)} . By convention, the identity element ε {\displaystyle \varepsilon } under the Dirichlet convolution is a rational arithmetical function of order ( 0 , 0 ) {\displaystyle (0,0)} .

All rational arithmetical functions are multiplicative. A multiplicative function f is a rational arithmetical function of order ( r , s ) {\displaystyle (r,s)} if and only if its Bell series is of the form f p ( x ) = ∑ n = 0 ∞ f ( p n ) x n = ( 1 − h 1 ( p ) x ) ( 1 − h 2 ( p ) x ) ⋯ ( 1 − h s ( p ) x ) ( 1 − g 1 ( p ) x ) ( 1 − g 2 ( p ) x ) ⋯ ( 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 p {\displaystyle p} .

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

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

f ( m ) f ( n ) = ∑ d ∣ ( m , n ) f ( m n / 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 m {\displaystyle m} and n {\displaystyle n} , or equivalently

f ( m n ) = ∑ d ∣ ( m , n ) f ( m / d ) f ( n / d ) μ ( 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 m {\displaystyle m} and n {\displaystyle n} , where μ {\displaystyle \mu } is the Möbius function. These are known as Busche-Ramanujan identities. In 1906, E. Busche stated the identity

σ k ( m ) σ k ( n ) = ∑ d ∣ ( m , n ) σ k ( m n / 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

σ k ( m n ) = ∑ d ∣ ( m , n ) σ k ( m / d ) σ k ( n / d ) μ ( 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 k = 0 {\displaystyle k=0} . S. Chowla gave the inverse form for general 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 f = g 1 ∗ g 2 {\displaystyle f=g_{1}\ast g_{2}} satisfy the Busche-Ramanujan identities with 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 λ {\displaystyle \lambda } on A is called multiplicative if λ ( f g ) = λ ( f ) λ ( 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

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

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

The polynomial zeta function is then

ζ A ( s ) = ∑ f  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):

D h ( s ) = ∏ P ( ∑ n = ⁡ 0 ∞ h ( P n ) | P | − s n ) , {\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:

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

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

ζ A ( s ) = ∑ f | f | − s = ∑ n ∑ deg ⁡ ( f ) = n q − s n = ∑ n ( q n − s n ) = ( 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

( f ∗ g ) ( m ) = ∑ d ∣ m f ( d ) g ( m d ) = ∑ a b = m f ( a ) g ( b ) , {\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 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 D N = N 2 × N ( N + 1 ) / 2 {\displaystyle D_{N}=N^{2}\times N(N+1)/2}

a sum can be distributed across the product y t = ∑ ( t / T ) 1 / 2 u t = ∑ ( t / T ) 1 / 2 G t 1 / 2 ϵ 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: y ~ t 2 = y t 2 g t = σ 2 ( t / T ) + σ 2 ( t / T ) ( ϵ 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 y t 2 = σ 2 ( t / T ) + σ 2 ( t / T ) ( g t ϵ 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 L t ( τ ; u ) = ∑ t = 1 T K h ( u − t / T ) [ l n τ + y t 2 g t τ ] {\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 y t 2 {\displaystyle y_{t}^{2}} with known g t {\displaystyle g_{t}} and unknown σ 2 ( t / T ) {\displaystyle \sigma ^{2}(t/T)} .

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

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

An arithmetical function f {\displaystyle f} is Selberg multiplicative if for each prime p {\displaystyle p} there exists a function f p {\displaystyle f_{p}} on nonnegative integers with f p ( 0 ) = 1 {\displaystyle f_{p}(0)=1} for all but finitely many primes p {\displaystyle p} such that f ( n ) = ∏ p f p ( ν p ( n ) ) {\displaystyle f(n)=\prod _{p}f_{p}(\nu _{p}(n))} for all positive integers n {\displaystyle n} , where ν p ( n ) {\displaystyle \nu _{p}(n)} is the exponent of p {\displaystyle p} in the canonical factorization of 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 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 m , n {\displaystyle m,n} . See Haukkanen (2012).

It is well known and easy to see that multiplicative functions are quasimultiplicative functions with c = 1 {\displaystyle c=1} and quasimultiplicative functions are semimultiplicative functions with 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.2010.04.007. 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.