Idk: the power functions, defined by Idk = nk for any complex numberk. As special cases we have
* Id0 = 1 and
* Id1 = Id.
ε: the function defined by ε = 1 if n = 1 and 0 otherwise, sometimes called multiplication unit for Dirichlet convolution or simply the unit function. Sometimes written as u, but not to be confused with μ.
1C, the indicator function of the set C ⊂ Z, for certain sets C. The indicator function 1C is multiplicative precisely when the set C has the following property for any coprime numbersa and b: the product ab is in Cif 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:
An example of a non-multiplicative function is the arithmetic function r2 - 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 = 4 ≠ 1. This shows that the function is not multiplicative. However, r2/4 is multiplicative. In the On-Line Encyclopedia of Integer Sequences, 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 = paqb..., then f = ff... 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 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 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:
μ * 1 = ε
* Idk = ε
* 1 = Id
d = 1 * 1
σ = Id * 1 = * d
σk = Idk * 1
Id = * 1 = σ * μ
Idk = σk * μ
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 :
Let h be a polynomial arithmetic function. 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, every Dirichlet series of a multiplicative function h has a product representation : 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 of monic polynomials whose product is m. The identity still holds.
