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11: 27.4 Euler Products and Dirichlet Series

… ►The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)): … ►The Riemann zeta function is the prototype of series of the form …The following examples have generating functions related to the zeta function: ►

27.4.5 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)n^{-s}=\frac{1}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

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

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.4.E5

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Annotations for §27.4 and Ch.27

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27.4.7 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)n^{-s}=\frac{\zeta \left(2s\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.4.E7

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Annotations for §27.4 and Ch.27

…

12: 25.19 Tables

… ►

Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta \left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta (s,a)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

13: 25.10 Zeros

… ►

§25.10(i) Distribution

… ►The functional equation (25.4.1) implies $\zeta \left(-2n\right)=0$ for $n=1,2,3,\mathrm{\dots }$. … … ►

25.10.1 $$Z(t)\equiv \exp \left(\mathrm{i}\vartheta (t)\right)\zeta \left(\frac{1}{2}+\mathrm{i}t\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\equiv$: equals by definition, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

definition

Source:

Titchmarsh (1986b, (4.17.2), p. 89)

Permalink:

http://dlmf.nist.gov/25.10.E1

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Annotations for §25.10(i), §25.10 and Ch.25

… ►

§25.10(ii) Riemann–Siegel Formula

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14: 25.8 Sums

§25.8 Sums

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25.8.1 $$\sum _{k=2}^{\mathrm{\infty }}\left(\zeta \left(k\right)-1\right)=1.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (1.4), p. 132)

Permalink:

http://dlmf.nist.gov/25.8.E1

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Annotations for §25.8 and Ch.25

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25.8.2 $$\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)}{(k+1)!}\left(\zeta \left(s+k\right)-1\right)=\mathrm{\Gamma }\left(s-1\right),$$ $s\ne 1,0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Landau (1953, (2), p. 273); with (5.2.4), (5.2.5)

Permalink:

http://dlmf.nist.gov/25.8.E2

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Annotations for §25.8 and Ch.25

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25.8.4 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}(\zeta \left(nk\right)-1)=\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(2-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $n$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (2.7), p. 135)

Permalink:

http://dlmf.nist.gov/25.8.E4

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Annotations for §25.8 and Ch.25

… ►

25.8.9 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Srivastava and Choi (2001, (467), p. 212)

Permalink:

http://dlmf.nist.gov/25.8.E9

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Annotations for §25.8 and Ch.25

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15: 25.4 Reflection Formulas

§25.4 Reflection Formulas

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25.4.1 $$\zeta \left(1-s\right)=2{(2\pi )}^{-s}\cos \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(s\right)\zeta \left(s\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (12), p. 259)

Referenced by:

§25.10(i), §5.17

Permalink:

http://dlmf.nist.gov/25.4.E1

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Annotations for §25.4 and Ch.25

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25.4.2 $$\zeta \left(s\right)=2{(2\pi )}^{s-1}\sin \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(1-s\right)\zeta \left(1-s\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (13), p. 259)

A&S Ref:

23.2.6

Permalink:

http://dlmf.nist.gov/25.4.E2

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Annotations for §25.4 and Ch.25

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25.4.4 $$\xi \left(s\right)=\frac{1}{2}s(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s\right){\pi }^{-s/2}\zeta \left(s\right).$$

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Defines:

$\xi \left(s\right)$: Riemann’s $\xi$-function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $s$: complex variable

Keywords:

definition

Source:

Apostol (1976, p. 260)

Permalink:

http://dlmf.nist.gov/25.4.E4

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Annotations for §25.4 and Ch.25

… ►

25.4.5 $${(-1)}^k{\zeta }^{(k)}\left(1-s\right)=\frac{2}{{(2\pi )}^s}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\left(\mathrm{\Re }\left(c^{k-m}\right)\cos \left(\frac{1}{2}\pi s\right)+\mathrm{\Im }\left(c^{k-m}\right)\sin \left(\frac{1}{2}\pi s\right)\right){\mathrm{\Gamma }}^{(r)}\left(s\right){\zeta }^{(m-r)}\left(s\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $\sin z$: sine function, $k$: nonnegative integer, $m$: nonnegative integer, $s$: complex variable and $c$

Keywords:

derivative

Source:

Apostol (1985a, (6), p. 224)

Permalink:

http://dlmf.nist.gov/25.4.E5

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Annotations for §25.4 and Ch.25

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16: 25.6 Integer Arguments

… ►

§25.6(i) Function Values

… ►

25.6.4 $$\zeta \left(-2n\right)=0,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, Theorem 12.16, p. 266)

A&S Ref:

23.2.14

Permalink:

http://dlmf.nist.gov/25.6.E4

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

… ►

§25.6(ii) Derivative Values

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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

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

… ►

§25.6(iii) Recursion Formulas

…

17: 25.2 Definition and Expansions

§25.2 Definition and Expansions

… ►When $\mathrm{\Re }s>1$, … ►

§25.2(ii) Other Infinite Series

… ►

§25.2(iii) Representations by the Euler–Maclaurin Formula

… ►

§25.2(iv) Infinite Products

…

18: 25.5 Integral Representations

§25.5 Integral Representations

… ►

25.5.8 $$\zeta \left(s\right)=\frac{1}{2(1-2^{-s})\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{\sinh x}dx,$$ $\mathrm{\Re }s>1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.12.6), p. 32)

Permalink:

http://dlmf.nist.gov/25.5.E8

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Annotations for §25.5(i), §25.5 and Ch.25

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25.5.9 $$\zeta \left(s\right)=\frac{2^{s-1}}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{x^s}{{(\sinh x)}^2}dx,$$ $\mathrm{\Re }s>1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.12.7), p. 32)

Permalink:

http://dlmf.nist.gov/25.5.E9

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Annotations for §25.5(i), §25.5 and Ch.25

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25.5.19 $$\zeta \left(m+s\right)={(-1)}^{m-1}\frac{\mathrm{\Gamma }\left(s\right)\sin \left(\pi s\right)}{\pi \mathrm{\Gamma }\left(m+s\right)}{\int }_0^{\mathrm{\infty }}{\psi }^{(m)}\left(1+x\right)x^{-s}dx,$$ $m=1,2,3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $m$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

de Bruijn (1937, p. 178)

Referenced by:

§25.5(ii)

Permalink:

http://dlmf.nist.gov/25.5.E19

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Annotations for §25.5(ii), §25.5 and Ch.25

►

§25.5(iii) Contour Integrals

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19: 25.21 Software

… ►

§25.21(ii) Zeta Functions for Real Arguments

… ►

§25.21(iii) Zeta Functions for Complex Arguments

… ►

§25.21(iv) Hurwitz Zeta Function

…

20: 25.16 Mathematical Applications

§25.16 Mathematical Applications

… ►which is related to the Riemann zeta function by … ►

§25.16(ii) Euler Sums

… ►For integer $s$ ($\ge 2$), $H\left(s\right)$ can be evaluated in terms of the zeta function: … ►which satisfies the reciprocity law …