A multiplicative function is a function which commutes with multiplication. That is, and must be sets with multiplication such that for all , i.e. it preserves the multiplicative structure. A prominent special case of this would be a homomorphism between groups, which preserves the whole group structure (inverses and identity in addition to multiplication).
Most frequently, one deals with multiplicative functions . These functions appear frequently in number theory, especially in analytic number theory. In this case, one sometimes also defines weak multiplicative functions: a function is weak multiplicative if and only if for all pairs of relatively prime integers .
Let and be multiplicative in the number theoretic sense ("weak multiplicative"). Then the function of defined by is also multiplicative; the Mobius inversion formula relates these two quantities.
Examples in elementary number theory include the identity map, the number of divisors, the sum of divisors (and its generalization , the Euler phi function, the number of divisors (also denoted , $\mu( This article is a stub. Help us out by.