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21: 25.6 Integer Arguments

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25.6.2 $$\zeta \left(2n\right)=\frac{{(2\pi )}^{2n}}{2(2n)!}\left|B_{2n}\right|,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: nonnegative integer

Sources:

Magnus et al. (1966, p. 19); Apostol (1976, (22), p. 266)

A&S Ref:

23.2.16

Referenced by:

(25.11.21)

Permalink:

http://dlmf.nist.gov/25.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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25.6.6 $$\zeta \left(2k+1\right)=\frac{{(-1)}^{k+1}{(2\pi )}^{2k+1}}{2(2k+1)!}{\int }_0^1{B}_{2k+1}\left(t\right)\cot \left(\pi t\right)dt,$$ $k=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral and $k$: nonnegative integer

Source:

Nörlund (1924, (81*), p. 66); with $\nu =k$

A&S Ref:

23.2.17

Permalink:

http://dlmf.nist.gov/25.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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25.6.11 $${\zeta }^{\prime }\left(0\right)=-\frac{1}{2}\ln \left(2\pi \right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln z$: principal branch of logarithm function

Source:

Apostol (1985a, p. 223)

A&S Ref:

23.2.13

Permalink:

http://dlmf.nist.gov/25.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(ii), §25.6 and Ch.25

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25.6.12 $${\zeta }^{\prime \prime }\left(0\right)=-\frac{1}{2}{(\ln \left(2\pi \right))}^2+\frac{1}{2}{\gamma }^2-\frac{1}{24}{\pi }^2+{\gamma }_1,$$

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln z$: principal branch of logarithm function

Source:

Apostol (1985a, pp. 226, 227, 229)

Referenced by:

Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(ii), §25.6 and Ch.25

…

22: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

23: William P. Reinhardt

… ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

24: 20.7 Identities

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20.7.17 $${\theta }_2\left(2z\right|2\tau )=A{\theta }_1\left(\frac{1}{4}\pi -z\right|\tau ){\theta }_1\left(\frac{1}{4}\pi +z\right|\tau ),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex, $\tau$: lattice parameter and $A$

Permalink:

http://dlmf.nist.gov/20.7.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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20.7.18 $${\theta }_3\left(2z\right|2\tau )=A{\theta }_3\left(\frac{1}{4}\pi -z\right|\tau ){\theta }_3\left(\frac{1}{4}\pi +z\right|\tau ),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex, $\tau$: lattice parameter and $A$

Permalink:

http://dlmf.nist.gov/20.7.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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20.7.20 $$B\equiv B(\tau )=1/\left({\theta }_3\left(0\right|\tau ){\theta }_4\left(0\right|\tau ){\theta }_3\left(\frac{1}{4}\pi \right|\tau )\right),$$

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Defines:

$B$ (locally)

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

… ►See Lawden (1989, pp. 19–20). … ►

20.7.34 $$\frac{{\theta }_1(z,q^2){\theta }_3(z,q^2)}{{\theta }_1(z,\mathrm{i}q)}=\frac{{\theta }_2(z,q^2){\theta }_4(z,q^2)}{{\theta }_2(z,\mathrm{i}q)}={\mathrm{i}}^{-1/4}\sqrt{\frac{{\theta }_2(0,q^2){\theta }_4(0,q^2)}{2}}.$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{i}$: imaginary unit, $z$: complex and $q$: nome

Referenced by:

§20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/20.7.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. See Walker (2012).

See also:

Annotations for §20.7(ix), §20.7 and Ch.20

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25: 20.10 Integrals

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_3\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_3\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.3 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}(1-{\theta }_4\left(0\right|\mathrm{i}{x}^2))dx=(1-2^{1-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>0$ because $1-{\theta }_4\left(0\right|\mathrm{i}{x}^2)=1-x^{-1}{\theta }_2\left(0\right|\mathrm{i}{x}^{-2})\sim 1$ as $x\to 0+$, which follows from (20.7.33) and (20.4.3).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

►

20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

…

26: 6.19 Tables

… ►

Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=[0,100]$, 8S.

… ►

Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $z{\mathrm{e}}^z{E}_1\left(z\right)$, $x=-19(1)20$, $y=0(1)20$, 6D; ${\mathrm{e}}^z{E}_1\left(z\right)$, $x=-4(.5)-2$, $y=0(.2)1$, 6D; $E_1\left(z\right)+\ln z$, $x=-2(.5)2.5$, $y=0(.2)1$, 6D.

►

Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_1\left(z\right)$, $\pm x=0.5,1,3,5,10,15,20,50,100$, $y=0(.5)1(1)5(5)30,50,100$, 8S.

27: Peter L. Walker

… ►

20 Theta Functions

…

28: Staff

… ►

William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

… ►

William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

…

29: 26.10 Integer Partitions: Other Restrictions

… ►

Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from $A_{j,k}$.

► ►►►

	$p(𝒟,n)$	$p(𝒟2,n)$	$p(𝒟2,\in T,n)$	$p({𝒟}^{\prime }3,n)$
…
$20$	$64$	$31$	$20$	$18$

►

… ►

26.10.16 $$p(𝒟,n)\sim \frac{{\mathrm{e}}^{\pi \sqrt{n/3}}}{{(768n^3)}^{1/4}},$$ $n\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ and $n$: nonnegative integer

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E16

Encodings:

pMML, png, TeX

See also:

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

… ►

26.10.17 $$p(𝒟,n)=\pi \sum _{k=1}^{\mathrm{\infty }}\frac{A_{2k-1}(n)}{(2k-1)\sqrt{24n+1}}{I}_1\left(\frac{\pi }{2k-1}\sqrt{\frac{24n+1}{72}}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer, $n$: nonnegative integer and $A_k(n)$: coefficient

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E17

Encodings:

pMML, png, TeX

See also:

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

… ►

26.10.18

ⓘ

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

…

30: 12.11 Zeros

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12.11.1 $$z_{a,s}={\mathrm{e}}^{\frac{3}{4}\pi \mathrm{i}}\sqrt{2{\tau }_s}\left(1-\frac{\mathrm{i}a{\lambda }_s}{2{\tau }_s}+\frac{2a^2{\lambda }_s^2-8a^2{\lambda }_s+4a^2+3}{16{\tau }_s^2}+O\left({\lambda }_s^3{\tau }_s^{-3}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter, ${\tau }_s$ and ${\lambda }_s$

Permalink:

http://dlmf.nist.gov/12.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(ii), §12.11 and Ch.12

… ►

12.11.2 $${\tau }_s=\left(2s+\frac{1}{2}-a\right)\pi +\mathrm{i}\ln \left({\pi }^{-\frac{1}{2}}{2}^{-a-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)\right),$$

ⓘ

Defines:

${\tau }_s$ (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $s$: nonnegative integer and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(ii), §12.11 and Ch.12

… ►

12.11.3 $${\lambda }_s=\ln {\tau }_s-\frac{1}{2}\pi \mathrm{i}.$$

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Defines:

${\lambda }_s$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $s$: nonnegative integer and ${\tau }_s$

Permalink:

http://dlmf.nist.gov/12.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(ii), §12.11 and Ch.12

… ►

12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

…