KNOWPIA
WELCOME TO KNOWPIA

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.

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- Id
_{0}(*n*) = 1(*n*) and - Id
_{1}(*n*) = Id(*n*).

- Id
*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. - : Euler's totient function , 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*σ*_{0}(*n*) =*d*(*n*) the number of positive divisors of*n*,*σ*_{1}(*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*):

*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 *r*_{2}(*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 = 1^{2} + 0^{2} = (−1)^{2} + 0^{2} = 0^{2} + 1^{2} = 0^{2} + (−1)^{2}

and therefore *r*_{2}(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, *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".

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 = 2^{4} · 3^{2}:

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.

If *f* and *g* are two multiplicative functions, one defines a new multiplicative function , 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:

- (the Möbius inversion formula)
- (generalized Möbius inversion)

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 :

More examples are shown in the article on Dirichlet series.

An arithmetical function *f* is said to be a rational arithmetical function of order if there exists completely multiplicative functions *g*_{1},...,*g*_{r},
*h*_{1},...,*h*_{s} such that
where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order are known as totient functions, and rational arithmetical functions of order are known as quadratic functions or specially multiplicative functions. Euler's function is a totient function, and the divisor function is a quadratic function.
Completely multiplicative functions are rational arithmetical functions of order . Liouville's function is completely multiplicative. The Möbius function is a rational arithmetical function of order .
By convention, the identity element under the Dirichlet convolution is a rational arithmetical function of order .

All rational arithmetical functions are multiplicative. A multiplicative function *f* is a rational arithmetical function of order if and only if its Bell series is of the form
for all prime numbers .

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

A multiplicative function is said to be specially multiplicative if there is a completely multiplicative function such that

for all positive integers and , or equivalently

for all positive integers and , where is the Möbius function. These are known as Busche-Ramanujan identities. In 1906, E. Busche stated the identity

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

for . S. Chowla gave the inverse form for general 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 satisfy the Busche-Ramanujan identities with . In fact, 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 on *A* is called **multiplicative** if 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

where for set if and otherwise.

The polynomial zeta function is then

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

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

Unlike the classical zeta function, is a simple rational function:

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

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 still holds.

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 with known and unknown .

An arithmetical function is quasimultiplicative if there exists a nonzero constant such that for all positive integers with . This concept originates by Lahiri (1972).

An arithmetical function is semimultiplicative if there exists a nonzero constant , a positive integer and a multiplicative function such that for all positive integers (under the convention that if is not a positive integer.) This concept is due to David Rearick (1966).

An arithmetical function is Selberg multiplicative if for each prime there exists a function on nonnegative integers with for all but finitely many primes such that for all positive integers , where is the exponent of in the canonical factorization of . See Selberg (1977).

It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide. They both satisfy the arithmetical identity for all positive integers . See Haukkanen (2012).

It is well known and easy to see that multiplicative functions are quasimultiplicative functions with and quasimultiplicative functions are semimultiplicative functions with .

- 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*.**37**: 2335–2344. - 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.

- L. Tóth (2013). "Two generalizations of the Busche-Ramanujan identities".
*International Journal of Number Theory*.**9**: 1301–1311. - 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. - S. Ramanujan, Some formulae in the analytic theory of numbers. Messenger 45 (1915), 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.

- Multiplicative function at PlanetMath.