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 $g(n)$.