# §27.6 Divisor Sums

Sums of number-theoretic functions extended over divisors are of special interest. For example,

 27.6.1 $\sum_{d\divides n}\mathop{\lambda\/}\nolimits\!\left(d\right)=\begin{cases}1,&% n\mbox{ is a square},\\ 0,&\mbox{otherwise}.\end{cases}$

If $f$ is multiplicative, then

 27.6.2 $\sum_{d\divides n}\mathop{\mu\/}\nolimits\!\left(d\right)f(d)=\prod_{p\divides n% }(1-f(p)),$ $n>1$.

Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. Examples include:

 27.6.3 $\displaystyle\sum_{d\divides n}|\mathop{\mu\/}\nolimits\!\left(d\right)|$ $\displaystyle=2^{\mathop{\nu\/}\nolimits\!\left(n\right)},$ 27.6.4 $\displaystyle\sum_{d^{2}\divides n}\mathop{\mu\/}\nolimits\!\left(d\right)$ $\displaystyle=|\mathop{\mu\/}\nolimits\!\left(n\right)|,$ 27.6.5 $\displaystyle\sum_{d\divides n}\frac{|\mathop{\mu\/}\nolimits\!\left(d\right)|% }{\mathop{\phi\/}\nolimits\!\left(d\right)}$ $\displaystyle=\frac{n}{\mathop{\phi\/}\nolimits\!\left(n\right)},$
 27.6.6 $\sum_{d\divides n}\mathop{\phi_{k}\/}\nolimits\!\left(d\right)\left(\frac{n}{d% }\right)^{k}=1^{k}+2^{k}+\dots+n^{k},$
 27.6.7 $\sum_{d\divides n}\mathop{\mu\/}\nolimits\!\left(d\right)\left(\frac{n}{d}% \right)^{k}=\mathop{J_{k}\/}\nolimits\!\left(n\right),$
 27.6.8 $\sum_{d\divides n}\mathop{J_{k}\/}\nolimits\!\left(d\right)=n^{k}.$