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Wikipedia / Multiplicative function

In number theory, a multiplicative function is an arithmetic function of a positive integer are coprime.

An arithmetic function is said to be completely multiplicative (or totally multiplicative holds for all, even when they are not coprime.

Examples

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

. As special cases we have

The above functions are all completely multiplicative.

  • : the indicator function 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:

: the divisor function may be any complex number). As special cases we have

All Dirichlet characters are completely multiplicative functions, for example

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:

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

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

The product of two multiplicative functions is also multiplicative.

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, the Dirichlet convolution of f and g 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 Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

The Dirichlet convolution

Dirichlet series for some multiplicative functions

More examples are shown in the article on Dirichlet series.

Rational arithmetical functions

An arithmetical function f if there exists completely multiplicative functions g1,...,gr, h1,...,hs.

All rational arithmetical functions are multiplicative. A multiplicative function f.

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

Busche-Ramanujan identities

such that

, or equivalently

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

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.

. Quadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see R. Vaidyanathaswamy (1931).

Multiplicative function over Fq[X]

Let , the polynomial ring over the finite field with q elements. A is a principal ideal domain and therefore A is a unique factorization domain.

on A is called multiplicative 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

otherwise.

The polynomial zeta function is then

Similar to the situation in , 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 ofm, or equivalently over all pairs (a, b) of monic polynomials whose product is m still holds.

Multivariate

Multivariate functions can be constructed using multiplicative model estimators. Where a matrix function of

a sum can be distributed

For the efficient estimation of , the following two nonparametric regressions

with a local likelihood function.

Generalizations

. This concept originates by Lahiri (1972).

is not a positive integer.) This concept is due to David Rearick (1966).

. See Selberg (1977).

. See Haukkanen (2012).

.

See also

References

  • See chapter 2 of

  • P. J. McCarthy, Introduction to Arithmetical Functions, Universitext. New York: Springer-Verlag, 1986.

  • 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.

References