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

§27.6 Divisor Sums

Notes:: See Apostol (1976, Chapter 2).
Keywords:: number-theoretic functions
Permalink:: http://dlmf.nist.gov/27.6

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

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

Symbols:: $\mathop{\lambda\/}\nolimits\!\left(n\right)$ : Liouville’s function, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E1
Encodings:: TeX, pMML, png

If is multiplicative, then

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

Symbols:: $\mathop{\mu\/}\nolimits\!\left(n\right)$ : 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

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

27.6.3 $\sum_{{d\divides n}}|\mathop{\mu\/}\nolimits\!\left(d\right)|=2^{{\mathop{\nu% \/}\nolimits\!\left(n\right)}},$

Symbols:: $\mathop{\mu\/}\nolimits\!\left(n\right)$ : Möbius function, $\mathop{\nu\/}\nolimits\!\left(n\right)$ : number of distinct primes dividing , : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E3
Encodings:: TeX, pMML, png

27.6.4 $\sum_{{d^{2}\divides n}}\mathop{\mu\/}\nolimits\!\left(d\right)=|\mathop{\mu\/% }\nolimits\!\left(n\right)|,$

Symbols:: $\mathop{\mu\/}\nolimits\!\left(n\right)$ : Möbius function, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E4
Encodings:: TeX, pMML, png

27.6.5 $\sum_{{d\divides n}}\frac{|\mathop{\mu\/}\nolimits\!\left(d\right)|}{\mathop{% \phi\/}\nolimits\!\left(d\right)}=\frac{n}{\mathop{\phi\/}\nolimits\!\left(n% \right)},$

Symbols:: $\mathop{\phi_{{k}}\/}\nolimits\!\left(n\right)$ : sum of powers of integers relatively prime to , $\mathop{\mu\/}\nolimits\!\left(n\right)$ : Möbius function, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E5
Encodings:: TeX, pMML, png

27.6.6 $\sum_{{d\divides n}}\mathop{\phi_{{k}}\/}\nolimits\!\left(d\right)\left(\frac{% n}{d}\right)^{k}=1^{k}+2^{k}+\dots+n^{k},$

Symbols:: $\mathop{\phi_{{k}}\/}\nolimits\!\left(n\right)$ : sum of powers of integers relatively prime to , : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E6
Encodings:: TeX, pMML, png

27.6.7 $\sum_{{d\divides n}}\mathop{\mu\/}\nolimits\!\left(d\right)\left(\frac{n}{d}% \right)^{k}=\mathop{J_{{k}}\/}\nolimits\!\left(n\right),$

Symbols:: $\mathop{J_{{k}}\/}\nolimits\!\left(n\right)$ : Jordan’s function, $\mathop{\mu\/}\nolimits\!\left(n\right)$ : Möbius function, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E7
Encodings:: TeX, pMML, png

27.6.8 $\sum_{{d\divides n}}\mathop{J_{{k}}\/}\nolimits\!\left(d\right)=n^{k}.$

Symbols:: $\mathop{J_{{k}}\/}\nolimits\!\left(n\right)$ : Jordan’s function, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E8
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 27.5 Inversion Formulas 27.7 Lambert Series as Generating Functions