Multiplicative function

Definition

Let $f$ be an arithmetic function: in other words, $f$ is a function from the set of natural numbers to a commutative unital ring $R$. We say that $f$ is multiplicative if it satisfies the following two conditions:

• $f(1)=1$.
• $f(mn)=f(m)f(n)$ for all pairs of relatively prime numbers $m,n\in \mathbb{N}$.

Relation with other properties

Stronger properties

• Completely multiplicative function

Incomparable properties

• Divisibility-preserving function