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27 Functions of Number Theory Multiplicative Number Theory 27.5 Inversion Formulas 27.7 Lambert Series as Generating Functions

§27.6 Divisor Sums

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Keywords:: divisor sums, number-theoretic functions
Notes:: See Apostol (1976, Chapter 2).
Permalink:: http://dlmf.nist.gov/27.6
See also:: Annotations for Ch.27

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

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

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Symbols:: $\lambda\left(\NVar{n}\right)$ : Liouville’s function, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §27.6 and Ch.27

If is multiplicative, then

27.6.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)f(d)=\prod_{p\mathbin{|}n}(1-f(p)),$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer, : 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 $\sum_{d\mathbin{|}n}|\mu\left(d\right)|=2^{\nu\left(n\right)},$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §27.6 and Ch.27

27.6.4 $\sum_{d^{2}\mathbin{|}n}\mu\left(d\right)=|\mu\left(n\right)|,$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer and : 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 $\sum_{d\mathbin{|}n}\frac{|\mu\left(d\right)|}{\phi\left(d\right)}=\frac{n}{% \phi\left(n\right)},$

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Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer and : 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 $\sum_{d\mathbin{|}n}\phi_{k}\left(d\right)\left(\frac{n}{d}\right)^{k}=1^{k}+2% ^{k}+\dots+n^{k},$

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Symbols:: $\phi_{\NVar{k}}\left(\NVar{n}\right)$ : sum of powers of integers relatively prime to a number, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §27.6 and Ch.27

27.6.7 $\sum_{d\mathbin{|}n}\mu\left(d\right)\left(\frac{n}{d}\right)^{k}=J_{k}\left(n% \right),$

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Symbols:: $J_{\NVar{k}}\left(\NVar{n}\right)$ : Jordan’s function, $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §27.6 and Ch.27

27.6.8 $\sum_{d\mathbin{|}n}J_{k}\left(d\right)=n^{k}.$

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Symbols:: $J_{\NVar{k}}\left(\NVar{n}\right)$ : Jordan’s function, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §27.6 and Ch.27

27.5 Inversion Formulas 27.7 Lambert Series as Generating Functions

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