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 $\scriptstyle a(n),\,n\,\in \,\mathbb {N} ^{+},$ such that

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

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

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

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

See also

• Multiplicative arithmetic functions
• Completely multiplicative arithmetic functions
• Additive arithmetic functions
• Completely additive arithmetic functions