In mathematics, especially in the field of number theory, functions from the natural numbers which respect products are important, and are given the name completely multiplicative functions. 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 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.
See also
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