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Multiplicative arithmetic functions

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Not to be confused with multiplicative functions, a term used in algebra.[1]

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

${\displaystyle m\perp n\Rightarrow a(mn)=a(m)a(n),\quad m,\,n\in \mathbb {N} ^{+},\,}$

where ${\displaystyle \scriptstyle m\,\perp \,n}$ means ${\displaystyle \scriptstyle m}$ is coprime to ${\displaystyle \scriptstyle n}$. Obviously, ${\displaystyle \scriptstyle a(1)}$ must be 1.

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

Erdős proved[2] that if a function is multiplicative and increasing then there is some ${\displaystyle \scriptstyle \alpha \geq 0}$ such that ${\displaystyle \scriptstyle a(n)=n^{\alpha }}$ for ${\displaystyle \scriptstyle n\geq 1.}$[3] In fact an analogous result holds for decreasing functions,[4] so if a function is multiplicative and monotone then either it is ${\displaystyle \scriptstyle a(n)=n^{\alpha }}$ for some ${\displaystyle \scriptstyle \alpha }$ or ${\displaystyle \scriptstyle a(n)=0}$ for ${\displaystyle \scriptstyle n>2.}$

3. For ${\displaystyle n=1}$, it is trivial, so he meant ${\displaystyle n>1}$.