# divisor sums

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##### 1: 27.6 Divisor Sums
###### §27.6 DivisorSums
Sums of number-theoretic functions extended over divisors are of special interest. … Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
##### 2: 27.10 Periodic Number-Theoretic Functions
It can also be expressed in terms of the Möbius function as a divisor sum:
27.10.5 $c_{k}\left(n\right)=\sum_{d\mathbin{|}\left(n,k\right)}d\mu\left(\frac{k}{d}% \right).$
27.10.6 $s_{k}(n)=\sum_{d\mathbin{|}\left(n,k\right)}f(d)g\left(\frac{k}{d}\right)$
27.10.8 $a_{k}(m)=\sum_{d\mathbin{|}\left(m,k\right)}g(d)f\left(\frac{k}{d}\right)\frac% {d}{k}.$
##### 3: 27.1 Special Notation
 $d,k,m,n$ positive integers (unless otherwise indicated). … sum, product taken over divisors of $n$. …
##### 4: 27.5 Inversion Formulas
which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: …
##### 5: 27.21 Tables
Glaisher (1940) contains four tables: Table I tabulates, for all $n\leq 10^{4}$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\phi\left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma(n)$ of these divisors. …
##### 6: 27.3 Multiplicative Properties
27.3.7 $\sigma_{\alpha}\left(m\right)\sigma_{\alpha}\left(n\right)=\sum_{d\mathbin{|}% \left(m,n\right)}d^{\alpha}\sigma_{\alpha}\left(\frac{mn}{d^{2}}\right),$
##### 7: 27.2 Functions
27.2.6 $\phi_{k}\left(n\right)=\sum_{\left(m,n\right)=1}m^{k},$
27.2.9 $d\left(n\right)=\sum_{d\mathbin{|}n}1$
27.2.10 $\sigma_{\alpha}\left(n\right)=\sum_{d\mathbin{|}n}d^{\alpha},$
is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. … Table 27.2.2 tabulates the Euler totient function $\phi\left(n\right)$, the divisor function $d\left(n\right)$ ($=\sigma_{0}(n)$), and the sum of the divisors $\sigma(n)$ ($=\sigma_{1}(n)$), for $n=1(1)52$. …
##### 8: 27.7 Lambert Series as Generating Functions
27.7.5 $\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sigma_{% \alpha}\left(n\right)x^{n},$
##### 9: 27.11 Asymptotic Formulas: Partial Sums
###### §27.11 Asymptotic Formulas: Partial Sums
27.11.2 $\sum_{n\leq x}d\left(n\right)=x\ln x+(2\gamma-1)x+O\left(\sqrt{x}\right),$
27.11.3 $\sum_{n\leq x}\frac{d\left(n\right)}{n}=\frac{1}{2}(\ln x)^{2}+2\gamma\ln x+O% \left(1\right),$
27.11.4 $\sum_{n\leq x}\sigma_{1}\left(n\right)=\frac{{\pi}^{2}}{12}x^{2}+O\left(x\ln x% \right).$
27.11.5 $\sum_{n\leq x}\sigma_{\alpha}\left(n\right)=\frac{\zeta\left(\alpha+1\right)}{% \alpha+1}x^{\alpha+1}+O\left(x^{\beta}\right),$ $\alpha>0$, $\alpha\neq 1$, $\beta=\max(1,\alpha)$.
##### 10: 27.4 Euler Products and Dirichlet Series
27.4.11 $\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)% \zeta\left(s-\alpha\right),$ $\Re s>\max(1,1+\Re\alpha)$,