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# Multiplicative arithmetic functions

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Not to be confused with multiplicative functions, a term used in algebra.

In number theory, multiplicative arithmetic functions are arithmetic functions $a(n),\,n\,\in \,\mathbb {N} ^{+},$ such that

$m\perp n\Rightarrow a(mn)=a(m)a(n),\quad m,\,n\in \mathbb {N} ^{+},\,$ where $m\,\perp \,n$ means $m$ is coprime to $n$ . Obviously, $a(1)$ must be 1.

For example, Euler's totient function $\varphi (n)$ is multiplicative: note that $\varphi (3)=2$ and $\varphi (7)=6,$ thus $\varphi (21)=\varphi (3)\varphi (7)=2\times 6=12.$ However, $\varphi (24)\neq \phi (4)\varphi (6)$ as $\gcd(4,6)=2$ ; instead we should do $\varphi (24)=\varphi (3)\varphi (8)$ . Functions where the identity holds even when the numbers are not coprime are called completely multiplicative.

Erdős proved that if a function is multiplicative and increasing then there is some $\alpha \geq 0$ such that $a(n)=n^{\alpha }$ for $n\geq 1.$ In fact an analogous result holds for decreasing functions, so if a function is multiplicative and monotone then either it is $a(n)=n^{\alpha }$ for some $\alpha$ or $a(n)=0$ for $n>2.$ 