In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then
An arithmetic function f(n) is said to be completely 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. In this case the function is a homomorphism of monoids and, because of the fundamental theorem of arithmetic, is completely determined by its restriction to the prime numbers. Every completely multiplicative function is multiplicative.
Outside number theory, the term multiplicative is usually used for functions with the property f(ab) = f(a) f(b) for all arguments a and b. This article discusses number theoretic multiplicative functions.
Examples of multiplicative functions include many functions of importance in number theory, such as:
An example of a nonmultiplicative 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:
and therefore r_{2}(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, r_{2}(n)/4 is multiplicative.
See arithmetic function for some examples of nonmultiplicative 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 two positive integers, then
If f and g are two multiplicative functions, one defines a new multiplicative function f * g, the Dirichlet convolution of f and g, by
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.
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