在数论中，积性函数是指一个定义域为正整数n 的算术函数f(n)，有如下性质：f(1) = 1，且当a 和b 互质时，f(ab) = f(a) f(b)。

若一个函数f(n) 有如下性质：f(1) = 1，且对两个随意正整数a 和b 而言，不只限这两数互质时，f(ab) = f(a)f(b) 都成立，则称此函数为完全积性函数。

在数论以外的其他数学领域中所谈到的积性函数通常是指完全积性函数。此条目则只讨论数论中的积性函数。

Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.

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 (or totally 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.