# §27.6 Divisor Sums

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

 27.6.1 $\sum_{d\mathbin{|}n}\lambda\left(d\right)=\begin{cases}1,&n\mbox{ is a square}% ,\\ 0,&\mbox{otherwise}.\end{cases}$ ⓘ Symbols: $\lambda\left(\NVar{n}\right)$: 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 and Ch.27

If $f$ is multiplicative, then

 27.6.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)f(d)=\prod_{p\mathbin{|}n}(1-f(p)),$ $n>1$. ⓘ Symbols: $\mu\left(\NVar{n}\right)$: Möbius function, $d$: positive integer, $n$: positive integer and $p,p_{1},\ldots$: prime numbers Permalink: http://dlmf.nist.gov/27.6.E2 Encodings: TeX, pMML, png See also: Annotations for §27.6 and Ch.27

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

 27.6.3 $\displaystyle\sum_{d\mathbin{|}n}|\mu\left(d\right)|$ $\displaystyle=2^{\nu\left(n\right)},$ 27.6.4 $\displaystyle\sum_{d^{2}\mathbin{|}n}\mu\left(d\right)$ $\displaystyle=|\mu\left(n\right)|,$ ⓘ Symbols: $\mu\left(\NVar{n}\right)$: 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 and Ch.27 27.6.5 $\displaystyle\sum_{d\mathbin{|}n}\frac{|\mu\left(d\right)|}{\phi\left(d\right)}$ $\displaystyle=\frac{n}{\phi\left(n\right)},$ ⓘ Symbols: $\phi\left(\NVar{n}\right)$: Euler’s totient, $\mu\left(\NVar{n}\right)$: Möbius function, $d$: positive integer and $n$: positive integer Permalink: http://dlmf.nist.gov/27.6.E5 Encodings: TeX, pMML, png See also: Annotations for §27.6 and Ch.27
 27.6.6 $\sum_{d\mathbin{|}n}\phi_{k}\left(d\right)\left(\frac{n}{d}\right)^{k}=1^{k}+2% ^{k}+\dots+n^{k},$
 27.6.7 $\sum_{d\mathbin{|}n}\mu\left(d\right)\left(\frac{n}{d}\right)^{k}=J_{k}\left(n% \right),$
 27.6.8 $\sum_{d\mathbin{|}n}J_{k}\left(d\right)=n^{k}.$ ⓘ Symbols: $J_{\NVar{k}}\left(\NVar{n}\right)$: Jordan’s function, $d$: positive integer, $k$: positive integer and $n$: positive integer Permalink: http://dlmf.nist.gov/27.6.E8 Encodings: TeX, pMML, png See also: Annotations for §27.6 and Ch.27