# Multiplicative function

A multiplicative function $f$ is a function such that $f(x\cdot y) = f(x) \cdot f(y)$ for all $x, y$. That is, $f$ commutes with multiplication. Multiplicative functions arise most commonly in the field of number theory, where an alternate definition is often used: a function from the positive integers to the complex numbers is said to be multiplicative if $f(m \cdot n) = f(m) \cdot f(n)$ for all relatively prime $m, n$.

The function $f$ defined on the real numbers by $f(x) = x^2$ is a simple example of a multiplicative function.

For a function $f : S \to T$ to be multiplicative, the domain $S$ and range $T$ must be sets with multiplication. Then $f(x\cdot y) = f(x) \cdot f(y)$ means that $f$ preserves the multiplicative structure. One prominent class of functions with this property are homomorphisms of groups (where the group operation is multiplication).

## Multiplicative functions in number theory

Multiplicative functions are of special importance in the field of analytic number theory. In this context, one works with multiplicative functions $f : \mathbb{Z}_{>0} \to \mathbb{C}$.

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 positive integers $(m, n)$. If actually $f(mn) = f(m)f(n)$ for all positive integers $m, n$, we say that $f$ is strongly multiplicative (but this notion is of less central importance).

Examples of multiplicative functions in elementary number theory include the identity map, the divisor function $d(n)$ that gives the number of divisors of the integer $n$, the sum of divisors function $\sigma(n)$ that gives the sum of divisors of the integer $n$ (and its generalization $\sigma_k(n) = \sum_{d|n}d^k$), Euler's totient function $\phi(n)$, and the Mobius function $\mu(n)$.

Let $f(n)$ be multiplicative in the number theoretic sense ("weak multiplicative"). Then the function $g(n)$ defined by $$g(n) = \sum_{d|n} f(d)$$ is also multiplicative. In this situation, the Mobius inversion formula allows us to write $f(n)$ in terms of $g(n)$.