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

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)
• Id_k(n): the power functions, defined by Id_k(n) = n^k 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) .
• 1_C(n), the indicator function of the set C ⊂ Z, for certain sets C. The indicator function 1_C(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
• σ_k(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
  • (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 = 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:

Similarly, we have:

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

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

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,...,g_r, h₁,...,h_s such that

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

for all prime numbers ${\displaystyle p}$.

The concept of a rational arithmetical function originates from 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 ${\displaystyle \lambda }$ 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 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

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

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

a sum can be distributed across the product

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

and

Thus it gives an estimate value of

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 ${\displaystyle c\,f(mn)=f(m)f(n)}$ for all positive integers ${\displaystyle m,n}$ with ${\displaystyle (m,n)=1}$.

It is easy to see that an arithmetical function ${\displaystyle f}$ not identically zero is quasimultiplicative if and only if ${\displaystyle f(1)\neq 0}$ and ${\displaystyle f(1)f(mn)=f(m)f(n)}$ for all ${\displaystyle m,n}$ with ${\displaystyle (m,n)=1}$. Then ${\displaystyle c=f(1)}$. Quasimultiplicative functions are multiplicative functions multiplied by a (nonzero) constant. An arithmetical function ${\displaystyle f}$ not identically zero is quasimultiplicative if and only if ${\displaystyle f(1)\neq 0}$ and ${\displaystyle f/f(1)}$ is multiplicative.

D. B. Lahiri introduced the concept of quasimultiplicative functions as a special case of hypomultiplicative functions.

The concept of a multiplicative function or a quasimultiplicative function is not satisfactory in the sense that compositions such as ${\displaystyle f(kn),f(k/n),f(n/k),f([k,n])}$ ${\displaystyle (k\neq 1)}$ preserve neither multiplicativity nor quasimultiplicativity. This has led to the concepts of semimultiplicative and Selberg multiplicative functions.

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

An arithmetical function ${\displaystyle f}$ not identically zero is semimultiplicative if and only if there exists a positive integer ${\displaystyle a}$ such that ${\displaystyle f(a)\neq 0}$ and ${\displaystyle f(ax)}$ is an arithmetical function (i.e. ${\displaystyle f(ax)=0}$ if ${\displaystyle x}$ is not a positive integer) and ${\displaystyle {\frac {f(an)}{f(a)}}}$ is multiplicative in ${\displaystyle n}$.

A nice characterization is as follows. An arithmetical function ${\displaystyle f}$ (not identically zero) is semimultiplicative if and only if ${\displaystyle f(m)f(n)=f((m,n))f([m,n])}$ for all positive integers ${\displaystyle m,n}$.

Semimultiplicative functions ${\displaystyle f}$ with ${\displaystyle a=1}$ and ${\displaystyle c=1}$ (i.e. ${\displaystyle f(1)=1}$) are multiplicative functions and semimultiplicative functions ${\displaystyle f}$ with ${\displaystyle a=1}$ (i.e. ${\displaystyle f(1)\neq 0}$) are quasimultiplicative functions.

Atle Selberg said on the concept of the usual multiplicative functions that ``I have never been very satisfied with this definition and would prefer to define a multiplicative function as follows". This leads to the following concept. An arithmetical function ${\displaystyle f}$ is Selberg multiplicative if for each prime ${\displaystyle p}$ there exists ${\displaystyle F_{p}}$ 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 ${\displaystyle n}$, where ${\displaystyle \nu _{p}(n)}$ is the exponent of ${\displaystyle p}$ in the canonical factorization of ${\displaystyle n}$.

Multiplicative functions ${\displaystyle f}$ are Selberg multiplicative with ${\displaystyle F_{p}(\nu _{p}(n))=f(p^{\nu _{p}(n)}).}$ Quasimultiplicative functions ${\displaystyle f}$ are Selberg multiplicative with Selberg factorization ${\displaystyle f(n)=f(1)\prod _{p}\left({\frac {f(p^{\nu _{p}(n)})}{f(1)}}\right)}$ provided that ${\displaystyle f(1)\neq 0}$. Finally, an arithmetical function is Selberg multiplicative if and only if it is semimultiplicative. In fact, a semimultiplicative function ${\displaystyle f}$ possesses a Selberg factorization as ${\displaystyle f(n)=f(a)\prod _{p}\left({\frac {f(ap^{\nu _{p}(n)-\nu _{p}(a)})}{f(a)}}\right).}$ Note that some authors define that the function identically zero is multiplicative and thus quasimultiplicative etc.

See also

• Euler product
• Bell series
• Lambert series

References

• 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
• 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 (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.

• D. Rearick (1966). "Semi-multiplicative functions". Duke Math. J. 33: 49–53.
• 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.

External links

• Multiplicative function at PlanetMath.