# §27.6 Divisor Sums

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

 27.6.1 $\sum_{d\mathbin{|}n}\mathop{\lambda\/}\nolimits\!\left(d\right)=\begin{cases}1% ,&n\mbox{ is a square},\\ 0,&\mbox{otherwise}.\end{cases}$ Symbols: $\mathop{\lambda\/}\nolimits\!\left(\NVar{n}\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer Permalink: http://dlmf.nist.gov/27.6.E1 Encodings: TeX, pMML, png See also: Annotations for 27.6

If $f$ is multiplicative, then

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

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

 27.6.3 $\displaystyle\sum_{d\mathbin{|}n}|\mathop{\mu\/}\nolimits\!\left(d\right)|$ $\displaystyle=2^{\mathop{\nu\/}\nolimits\!\left(n\right)},$ 27.6.4 $\displaystyle\sum_{d^{2}\mathbin{|}n}\mathop{\mu\/}\nolimits\!\left(d\right)$ $\displaystyle=|\mathop{\mu\/}\nolimits\!\left(n\right)|,$ Symbols: $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: Möbius function, $d$: positive integer and $n$: positive integer Permalink: http://dlmf.nist.gov/27.6.E4 Encodings: TeX, pMML, png See also: Annotations for 27.6 27.6.5 $\displaystyle\sum_{d\mathbin{|}n}\frac{|\mathop{\mu\/}\nolimits\!\left(d\right% )|}{\mathop{\phi\/}\nolimits\!\left(d\right)}$ $\displaystyle=\frac{n}{\mathop{\phi\/}\nolimits\!\left(n\right)},$
 27.6.6 $\sum_{d\mathbin{|}n}\mathop{\phi_{k}\/}\nolimits\!\left(d\right)\left(\frac{n}% {d}\right)^{k}=1^{k}+2^{k}+\dots+n^{k},$
 27.6.7 $\sum_{d\mathbin{|}n}\mathop{\mu\/}\nolimits\!\left(d\right)\left(\frac{n}{d}% \right)^{k}=\mathop{J_{k}\/}\nolimits\!\left(n\right),$
 27.6.8 $\sum_{d\mathbin{|}n}\mathop{J_{k}\/}\nolimits\!\left(d\right)=n^{k}.$