Riemann zeta function

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11: 25.10 Zeros

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§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

Encodings:

pMML, png, TeX

See also:

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

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§25.10(ii) Riemann–Siegel Formula

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12: 25.6 Integer Arguments

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§25.6(i) Function Values

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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

Encodings:

pMML, png, TeX

See also:

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

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§25.6(ii) Derivative Values

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25.6.13 $${(-1)}^k{\zeta }^{(k)}\left(-2n\right)=\frac{2{(-1)}^n}{{(2\pi )}^{2n+1}}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\mathrm{\Im }\left(c^{k-m}\right){\mathrm{\Gamma }}^{(r)}\left(2n+1\right){\zeta }^{(m-r)}\left(2n+1\right),$$

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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, $\mathrm{\Im }$: imaginary part, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $c$

Source:

Apostol (1985a, p. 225)

Permalink:

http://dlmf.nist.gov/25.6.E13

Encodings:

pMML, png, TeX

See also:

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

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§25.6(iii) Recursion Formulas

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13: 25.5 Integral Representations

§25.5 Integral Representations

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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

Encodings:

pMML, png, TeX

See also:

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

►

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

Encodings:

pMML, png, TeX

See also:

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

Encodings:

pMML, png, TeX

See also:

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

►

§25.5(iii) Contour Integrals

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14: 25.2 Definition and Expansions

§25.2 Definition and Expansions

… ►When $\mathrm{\Re }s>1$, … ►

§25.2(ii) Other Infinite Series

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

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§25.2(iv) Infinite Products

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15: 25.19 Tables

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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.

16: 25.16 Mathematical Applications

§25.16 Mathematical Applications

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

25.16.2 $$\psi \left(x\right)=x-\frac{{\zeta }^{\prime }\left(0\right)}{\zeta \left(0\right)}-\sum _{\rho }\frac{x^{\rho }}{\rho }+o\left(1\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$\psi \left(x\right)$: Chebyshev $\psi$-function, $\zeta \left(s\right)$: Riemann zeta function, $o\left(x\right)$: order less than and $x$: real variable

Keywords:

asymptotic approximation

Source:

Apostol (2000, p. 9)

Permalink:

http://dlmf.nist.gov/25.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(i), §25.16 and Ch.25

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§25.16(ii) Euler Sums

… ►which satisfies the reciprocity law …

17: 20.10 Integrals

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

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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

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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

►Here $\zeta \left(s\right)$ again denotes the Riemann zeta function (§25.2). …

18: 25.9 Asymptotic Approximations

§25.9 Asymptotic Approximations

… ►

25.9.1 $$\zeta \left(\sigma +\mathrm{i}t\right)=\sum _{1\le n\le x}\frac{1}{n^s}+\chi (s)\sum _{1\le n\le y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma }\right)+O\left(y^{\sigma -1}{t}^{\frac{1}{2}-\sigma }\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable, $s$: complex variable and $\sigma$: fixed

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.12.4), p. 79)

Referenced by:

§25.9

Permalink:

http://dlmf.nist.gov/25.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.9 and Ch.25

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25.9.3 $$\zeta \left(\frac{1}{2}+\mathrm{i}t\right)=\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi \left(\frac{1}{2}+\mathrm{i}t\right)\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+O\left(t^{-1/4}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.17.1), p. 88)

Referenced by:

§25.10(ii)

Permalink:

http://dlmf.nist.gov/25.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.9 and Ch.25

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19: 5.16 Sums

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5.16.2 $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}{\psi }^{\prime }\left(k+1\right)=\zeta \left(3\right)=-\frac{1}{2}{\psi }^{\prime \prime }\left(1\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

…

20: 25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: ►

25.11.2 $$\zeta (s,1)=\zeta \left(s\right).$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function and $s$: complex variable

Keywords:

specialization

Source:

Apostol (1976, p. 255)

Proof sketch:

Derivable from (25.11.1) and (25.2.1).

Referenced by:

(25.11.10), (25.11.24), (25.12.12)

Permalink:

http://dlmf.nist.gov/25.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(i), §25.11 and Ch.25

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25.11.11 $$\zeta (s,\frac{1}{2})=(2^s-1)\zeta \left(s\right),$$ $s\ne 1$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function and $s$: complex variable

Keywords:

special value

Proof sketch:

Derivable using (25.11.1), (25.2.2).

Referenced by:

§25.11(iv)

Permalink:

http://dlmf.nist.gov/25.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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25.11.24 $$\sum _{r=1}^{k-1}{\zeta }^{\prime }(s,\frac{r}{k})=(k^s-1){\zeta }^{\prime }\left(s\right)+k^s\zeta \left(s\right)\ln k,$$ $s\ne 1$, $k=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $a$: real or complex parameter, $s$: complex variable and $k$: integer

Keywords:

derivative

Proof sketch:

Derivable from (25.11.15) with $a=1/k$, multiplying by $k^s$, differentiating with respect to $s$, and using (25.11.2).

Referenced by:

(25.11.21), §25.11(vi)

Permalink:

http://dlmf.nist.gov/25.11.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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25.11.41 $$\zeta (s,a+1)=\zeta \left(s\right)-s\zeta \left(s+1\right)a+O\left(a^2\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $a$: real or complex parameter and $s$: complex variable

Keywords:

asymptotic approximation

Source:

Apostol (1952, (15), p. 6)

Permalink:

http://dlmf.nist.gov/25.11.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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