# §27.3 Multiplicative Properties

Except for $\mathop{\nu\/}\nolimits\!\left(n\right)$, $\mathop{\Lambda\/}\nolimits\!\left(n\right)$, $p_{n}$, and $\mathop{\pi\/}\nolimits\!\left(x\right)$, the functions in §27.2 are multiplicative, which means $f(1)=1$ and

 27.3.1 $f(mn)=f(m)f(n),$ $\left(m,n\right)=1$.

If $f$ is multiplicative, then the values $f(n)$ for $n>1$ are determined by the values at the prime powers. Specifically, if $n$ is factored as in (27.2.1), then

 27.3.2 $f(n)=\prod_{r=1}^{\mathop{\nu\/}\nolimits\!\left(n\right)}f(p^{a_{r}}_{r}).$

In particular,

 27.3.3 $\displaystyle\mathop{\phi\/}\nolimits\!\left(n\right)$ $\displaystyle=n\prod_{p\divides n}(1-p^{-1}),$ 27.3.4 $\displaystyle\mathop{J_{k}\/}\nolimits\!\left(n\right)$ $\displaystyle=n^{k}\prod_{p\divides n}(1-p^{-k}),$ 27.3.5 $\displaystyle\mathop{d\/}\nolimits\!\left(n\right)$ $\displaystyle=\prod_{r=1}^{\mathop{\nu\/}\nolimits\!\left(n\right)}(1+a_{r}),$ 27.3.6 $\displaystyle\mathop{\sigma_{\alpha}\/}\nolimits\!\left(n\right)$ $\displaystyle=\prod_{r=1}^{\mathop{\nu\/}\nolimits\!\left(n\right)}\frac{p^{% \alpha(1+a_{r})}_{r}-1}{p^{\alpha}_{r}-1},$ $\alpha\neq 0$.

Related multiplicative properties are

 27.3.7 $\mathop{\sigma_{\alpha}\/}\nolimits\!\left(m\right)\mathop{\sigma_{\alpha}\/}% \nolimits\!\left(n\right)=\sum_{d\divides\left(m,n\right)}d^{\alpha}\mathop{% \sigma_{\alpha}\/}\nolimits\left(\frac{mn}{d^{2}}\right),$
 27.3.8 $\mathop{\phi\/}\nolimits\!\left(m\right)\mathop{\phi\/}\nolimits\!\left(n% \right)=\mathop{\phi\/}\nolimits\!\left(mn\right)\mathop{\phi\/}\nolimits\!% \left(\left(m,n\right)\right)/\left(m,n\right).$

A function $f$ is completely multiplicative if $f(1)=1$ and

 27.3.9 $f(mn)=f(m)f(n),$ $m,n=1,2,\dots$.

Examples are $\left\lfloor 1/n\right\rfloor$ and $\mathop{\lambda\/}\nolimits\!\left(n\right)$, and the Dirichlet characters, defined in §27.8.

If $f$ is completely multiplicative, then (27.3.2) becomes

 27.3.10 $f(n)=\prod_{r=1}^{\mathop{\nu\/}\nolimits\!\left(n\right)}\left(f(p_{r})\right% )^{a_{r}}.$