divisor function

SearchAdvancedHelp

(0.001 seconds)

1—10 of 26 matching pages

1: 27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^4$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma (n)$ of these divisors. …Table III lists all solutions $n\le {10}^4$ of the equation $d\left(n\right)=m$, and Table IV lists all solutions $n$ of the equation $\sigma (n)=m$ for all $m\le {10}^4$. …6 lists $\varphi \left(n\right),d\left(n\right)$, and $\sigma (n)$ for $n\le 1000$; Table 24. …

2: 27.2 Functions

… ►

27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

ⓘ

Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►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 $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. … ►

Table 27.2.2: Functions related to division.

► ►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…

►

3: 27.3 Multiplicative Properties

… ►

27.3.1 $$f(mn)=f(m)f(n),$$ $\left(m,n\right)=1$.

ⓘ

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer, $n$: positive integer and $f(n)$: multiplicative function

Permalink:

http://dlmf.nist.gov/27.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

… ►

27.3.5 $$d\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}(1+a_r),$$

ⓘ

Symbols:

$d_k\left(n\right)$: divisor function, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a_r$: positive exponent

Permalink:

http://dlmf.nist.gov/27.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

… ►

27.3.8 $$\varphi \left(m\right)\varphi \left(n\right)=\varphi \left(mn\right)\varphi \left(\left(m,n\right)\right)/\left(m,n\right).$$

ⓘ

Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

…

4: 27.4 Euler Products and Dirichlet Series

… ►

27.4.10 $$\sum _{n=1}^{\mathrm{\infty }}{d}_k\left(n\right)n^{-s}={(\zeta \left(s\right))}^k,$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $d_k\left(n\right)$: divisor function, $\mathrm{\Re }$: real part, $k$: positive integer and $n$: positive integer

Referenced by:

§27.4

Permalink:

http://dlmf.nist.gov/27.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

►

27.4.11 $$\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)n^{-s}=\zeta \left(s\right)\zeta \left(s-\alpha \right),$$ $\mathrm{\Re }s>\max (1,1+\mathrm{\Re }\alpha )$,

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

…

5: 27.11 Asymptotic Formulas: Partial Sums

… ►

27.11.2 $$\sum _{n\le x}d\left(n\right)=x\ln x+(2\gamma -1)x+O\left(\sqrt{x}\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 III

Referenced by:

§27.11, §27.11, Erratum (V1.1.8) for Section 27.11

Permalink:

http://dlmf.nist.gov/27.11.E2

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

… ►

27.11.3 $$\sum _{n\le x}\frac{d\left(n\right)}{n}=\frac{1}{2}{(\ln x)}^2+2\gamma \ln x+O\left(1\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E3

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

… ►

27.11.4 $$\sum _{n\le x}{\sigma }_1\left(n\right)=\frac{{\pi }^2}{12}{x}^2+O\left(x\ln x\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 III (in slightly different form)

Permalink:

http://dlmf.nist.gov/27.11.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

►

27.11.5 $$\sum _{n\le x}{\sigma }_{\alpha }\left(n\right)=\frac{\zeta \left(\alpha +1\right)}{\alpha +1}{x}^{\alpha +1}+O\left(x^{\beta }\right),$$ $\alpha >0$, $\alpha \ne 1$, $\beta =\max (1,\alpha )$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

…

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 $$np\left(n\right)=\sum _{k=1}^n{\sigma }_1\left(k\right)p\left(n-k\right),$$

ⓘ

Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer and $n$: positive integer

A&S Ref:

24.2.1 II.A (in slightly different form) 24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20, Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/27.14.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

The erroneous ${\sigma }_1\left(n\right)$ in
$np\left(n\right)={\sum }_{k=1}^n{\sigma }_1\left(n\right)p\left(n-k\right)$
was replaced.

Reported 2015-08-17 by Howard Cohl and Hannah Cohen

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►

27.14.10 $$A_k(n)=\sum _{\begin{array}{c}h=1\\ \left(h,k\right)=1\end{array}}^k\exp \left(\pi \mathrm{i}s(h,k)-2\pi \mathrm{i}n\frac{h}{k}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $k$: positive integer, $n$: positive integer, $A_k(n)$: coefficients and $s(h,k)$: Dedekind sum

A&S Ref:

24.2.1 I.C

Permalink:

http://dlmf.nist.gov/27.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iii), §27.14 and Ch.27

… ►

27.14.19 $$\tau \left(m\right)\tau \left(n\right)=\sum _{d|\left(m,n\right)}{d}^{11}\tau \left(\frac{mn}{d^2}\right),$$ $m,n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, $\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.14.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

… ►

27.14.20 $$\tau \left(n\right)\equiv {\sigma }_{11}\left(n\right)(mod691).$$

ⓘ

Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\equiv$: modular equivalence and $n$: positive integer

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

…

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 $$c_k\left(n\right)=\sum _{d|\left(n,k\right)}d\mu \left(\frac{k}{d}\right).$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $c_k\left(n\right)$: Ramanujan’s sum, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

… ►

27.10.6 $$s_k(n)=\sum _{d|\left(n,k\right)}f(d)g\left(\frac{k}{d}\right)$$

ⓘ

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

… ►

27.10.8 $$a_k(m)=\sum _{d|\left(m,k\right)}g(d)f\left(\frac{k}{d}\right)\frac{d}{k}.$$

ⓘ

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer, $m$: positive integer, $f(n)$: function, $g(m)$: periodic function and $a_k(m)$: coefficients

Permalink:

http://dlmf.nist.gov/27.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

…

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: … ►

10: 27.13 Functions

… ►

27.13.6 $${(\vartheta \left(x\right))}^2=1+4\sum _{n=1}^{\mathrm{\infty }}\left({\delta }_1(n)-{\delta }_3(n)\right)x^n,$$

ⓘ

Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $n$: positive integer, $x$: real number and ${\delta }_j(n)$: number of divisors

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

…