# divisor function

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## 1—10 of 26 matching pages

##### 1: 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. …Table III lists all solutions $n\leq 10^{4}$ of the equation $d\left(n\right)=m$, and Table IV lists all solutions $n$ of the equation $\sigma(n)=m$ for all $m\leq 10^{4}$. …6 lists $\phi\left(n\right),d\left(n\right)$, and $\sigma(n)$ for $n\leq 1000$; Table 24. …
##### 2: 27.2 Functions
27.2.9 $d\left(n\right)=\sum_{d\mathbin{|}n}1$
It is the special case $k=2$ of the function $d_{k}\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …Note that $\sigma_{0}\left(n\right)=d\left(n\right)$. … 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 $n=1(1)52$. …
##### 3: 27.3 Multiplicative Properties
27.3.5 $d\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}(1+a_{r}),$
##### 4: 27.4 Euler Products and Dirichlet Series
27.4.11 $\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)% \zeta\left(s-\alpha\right),$ $\Re s>\max(1,1+\Re\alpha)$,
##### 5: 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 $\sum_{n\leq x}\sigma_{\alpha}\left(n\right)=\frac{\zeta\left(\alpha+1\right)}{% \alpha+1}x^{\alpha+1}+O\left(x^{\beta}\right),$ $\alpha>0$, $\alpha\neq 1$, $\beta=\max(1,\alpha)$.
##### 6: 27.6 Divisor Sums
###### §27.6 Divisor Sums
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. …
##### 7: 27.14 Unrestricted Partitions
27.14.7 $np\left(n\right)=\sum_{k=1}^{n}\sigma_{1}\left(k\right)p\left(n-k\right),$
27.14.10 $A_{k}(n)=\sum_{\begin{subarray}{c}h=1\\ \left(h,k\right)=1\end{subarray}}^{k}\exp\left(\pi\mathrm{i}s(h,k)-2\pi\mathrm% {i}n\frac{h}{k}\right),$
27.14.19 $\tau\left(m\right)\tau\left(n\right)=\sum_{d\mathbin{|}\left(m,n\right)}d^{11}% \tau\left(\frac{mn}{d^{2}}\right),$ $m,n=1,2,\dots$.
##### 8: 27.10 Periodic Number-Theoretic Functions
Examples are the Dirichlet characters (mod $k$) and the greatest common divisor $\left(n,k\right)$ regarded as a function of $n$. … 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 $a_{k}(m)=\sum_{d\mathbin{|}\left(m,k\right)}g(d)f\left(\frac{k}{d}\right)\frac% {d}{k}.$
##### 9: 27.5 Inversion Formulas
###### §27.5 Inversion Formulas
The multiplicative functions are a subgroup of this group. Generating functions yield many relations connecting number-theoretic functions. …which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: …