Completely multiplicative function

Definition

Suppose $f$ is an arithmetic function. In other words, $f$ is a function from the natural numbers to some commutative unital ring $R$. Then, $f$ is termed completely multiplicative if it is a monoid homomorphism from the multiplicative monoid of natural numbers to the multiplicative monoid of $R$. In other words, $f$ satisfies the following two conditions:

• $f(1)=1$.
• $f(mn)=f(m)f(n)$ for all natural numbers $m,n\in \mathbb{N}$. Here, the multiplication on the left happens in $\mathbb{N}$ while the multiplication on the right happens in $R$.

Relation with other properties

Weaker properties

• Multiplicative function
• Divisibility-preserving function