divisor sums

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11: 27.14 Unrestricted Partitions

… ►

27.14.7 $$np\left(n\right)=\sum _{k=1}^n{\sigma }_1\left(k\right)p\left(n-k\right),$$

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

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

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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

…

12: 26.10 Integer Partitions: Other Restrictions

… ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. … ►

26.10.18

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $h$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $A_k(n)$: coefficient and $f(h,k)$: function

Permalink:

http://dlmf.nist.gov/26.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(vi), §26.10 and Ch.26

…

13: 27.13 Functions

… ►

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

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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

… ►By similar methods Jacobi proved that $r_4\left(n\right)=8{\sigma }_1\left(n\right)$ if $n$ is odd, whereas, if $n$ is even, $r_4\left(n\right)=24$ times the sum of the odd divisors of $n$. …

14: 26.9 Integer Partitions: Restricted Number and Part Size

… ►where the inner sum is taken over all positive divisors of $t$ that are less than or equal to $k$. …

15: 26.12 Plane Partitions

… ►where ${\sigma }_2(j)$ is the sum of the squares of the divisors of $j$. …

16: 23.18 Modular Transformations

… ►

23.18.7 $$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$ $c>0$.

ⓘ

Symbols:

$\lfloor x\rfloor$: floor of $x$ , $\left(m,n\right)$: greatest common divisor (gcd), $c$: integer and $d$: integer

Referenced by:

§23.18, Erratum (V1.0.11) for Equation (23.18.7)

Permalink:

http://dlmf.nist.gov/23.18.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the sum ${\sum }_{r=1}^{c-1}$ was written with an additional constraint on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare

See also:

Annotations for §23.18, §23.18 and Ch.23

…

17: 24.4 Basic Properties

… ►

24.4.11 $$\sum _{\begin{array}{c}k=1\\ \left(k,m\right)=1\end{array}}^m{k}^n=\frac{1}{n+1}\sum _{j=1}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{j}\right)\left(\prod _{p|m}(1-p^{n-j})B_{n+1-j}\right)m^j.$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\left(m,n\right)$: greatest common divisor (gcd), $j$: integer, $k$: integer, $m$: integer, $n$: integer and $p$: prime

Referenced by:

§24.4(iii)

Permalink:

http://dlmf.nist.gov/24.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

…

18: 27.8 Dirichlet Characters

… ►

27.8.4 $$\chi \left(n\right)=0,$$ $\left(n,k\right)>1$.

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

… ►

27.8.5

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

… ►A Dirichlet character $\chi (modk)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor ), there exists an integer $a\equiv 1(modd)$, with $\left(a,k\right)=1$ and $\chi \left(a\right)\ne 1$. …A divisor $d$ of $k$ is called an induced modulus for $\chi$ if ►

27.8.7 $$\chi \left(a\right)=1\text{ for all }a\equiv 1\text{ (mod }d\text{)},$$ $\left(a,k\right)=1$.

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $\equiv$: modular equivalence, $d$: positive integer and $k$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

…

19: 25.15 Dirichlet $L$-functions

… ►

25.15.1 $$L(s,\chi )=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi (n)}{n^s},$$ $\mathrm{\Re }s>1$,

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Defines:

$L(s,\chi )$: Dirichlet $L$-function

Symbols:

$\mathrm{\Re }$: real part, $n$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, p. 249)

Referenced by:

(25.11.36), (25.11.36), §25.11(x), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.3 $$L(s,\chi )=k^{-s}\sum _{r=1}^{k-1}\chi (r)\zeta (s,\frac{r}{k}),$$

ⓘ

Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

finite sum

Source:

Apostol (1976, (8), p. 255)

Notes:

In the original source the upper-index of the finite sum is $k$. We use $k-1$ since $\chi (k)=0$.

Referenced by:

(25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►where ${\chi }_0$ is a primitive character (mod $d$) for some positive divisor $d$ of $k$ (§27.8). … ►

25.15.6 $$G(\chi )\equiv \sum _{r=1}^{k-1}\chi (r){\mathrm{e}}^{2\pi \mathrm{i}r/k}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

definition

Source:

Apostol (1976, p. 262)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.6)

Permalink:

http://dlmf.nist.gov/25.15.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.10

ⓘ

Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1976, Theorem 12.20, p. 268)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.10)

Permalink:

http://dlmf.nist.gov/25.15.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(ii), §25.15 and Ch.25

20: 23.20 Mathematical Applications

… ►It follows from the addition formula (23.10.1) that the points $P_j=P(z_j)$, $j=1,2,3$, have zero sum iff $z_1+z_2+z_3\in 𝕃$, so that addition of points on the curve $C$ corresponds to addition of parameters $z_j$ on the torus $ℂ/𝕃$; see McKean and Moll (1999, §§2.11, 2.14). … ►The addition law states that to find the sum of two points, take the third intersection with $C$ of the chord joining them (or the tangent if they coincide); then its reflection in the $x$-axis gives the required sum. … ►To determine $T$, we make use of the fact that if $(x,y)\in T$ then $y^2$ must be a divisor of $\mathrm{\Delta }$; hence there are only a finite number of possibilities for $y$. Values of $x$ are then found as integer solutions of $x^3+ax+b-y^2=0$ (in particular $x$ must be a divisor of $b-y^2$). …