Completely multiplicative function

In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. Especially in number theory, a weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.

Definition

A completely multiplicative function (or totally multiplicative function) is an arithmetic function (that is, a function whose domain is the natural numbers), such that f(1) = 1 and f(ab) = f(a) f(b) holds for all positive integers a and b.

Without the requirement that f(1) = 1, one could still have f(1) = 0, but then f(a) = 0 for all positive integers a, so this is not a very strong restriction.

Examples

The easiest example of a multiplicative function is a monomial: For any particular positive integer n, define f(a) = aⁿ.

Properties

A completely multiplicative function is completely determined by its values at the 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 ...

There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function f multiplicative then is completely multiplicative if and only if the Dirichlet inverse is $\mu f$ where $\mu$ is the Mobius function.

See also

• Dirichlet character
• Liouville function