Multiplicative function

This is an AoPSWiki Word of the Week for March 28-April 5

A multiplicative function $f : S \to T$ is a function which commutes with multiplication. That is, $S$ and $T$ must be sets with multiplication such that $f(x\cdot y) = f(x) \cdot f(y)$ for all $x, y \in S$ , 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 $f : \mathbb{Z}_{>0} \to \mathbb{C}$ . These functions appear frequently in number theory, especially in analytic number theory. In this case, one sometimes also defines weak multiplicative functions: a function $f: \mathbb{Z}_{>0} \to \mathbb{C}$ is weak multiplicative if and only if $f(mn) = f(m)f(n)$ for all pairs of relatively prime integers $(m, n)$ .

Let $f(n)$ and $g(n)$ be multiplicative in the number theoretic sense ("weak multiplicative"). Then the function of $n$ defined by $\sum_{d|n} f(d) g(\frac{n}{d})$ is also multiplicative; the Mobius inversion formula relates these two quantities.

Examples in elementary number theory include the identity map, $d(n)$ the number of divisors, $\sigma(n)$ the sum of divisors, $\phi(n)$ the Euler phi function, $\tau(n)$ (I actually forget what this is), $\mu(n)$ the Mobius function, and bunches of other totally awesome stuff. This article is a stub. Help us out by expanding it.

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Multiplicative function