Riemann zeta function

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21: Bibliography Y

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The calculation of the Riemann zeta function in the complex domain. USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.

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External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Computation of the derivatives of the Riemann zeta-function in the complex domain. USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.

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Notes:

Includes Fortran programs.

External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i), §25.21(iii).

Encodings:

BibTeX

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22: 5.7 Series Expansions

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5.7.2 $$(k-1)c_k=\gamma c_{k-1}-\zeta \left(2\right)c_{k-2}+\zeta \left(3\right)c_{k-3}-\mathrm{\cdots }+{(-1)}^k\zeta \left(k-1\right)c_1,$$ $k\ge 3$.

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $k$: nonnegative integer and $c_k$: coefficient

Referenced by:

§5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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5.7.3 $$\ln \mathrm{\Gamma }\left(1+z\right)=-\ln \left(1+z\right)+z(1-\gamma )+\sum _{k=2}^{\mathrm{\infty }}{(-1)}^k(\zeta \left(k\right)-1)\frac{z^k}{k},$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.33

Referenced by:

§5.21, §5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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5.7.4 $$\psi \left(1+z\right)=-\gamma +\sum _{k=2}^{\mathrm{\infty }}{(-1)}^k\zeta \left(k\right)z^{k-1},$$ ,

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.3.14

Permalink:

http://dlmf.nist.gov/5.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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5.7.5 $$\psi \left(1+z\right)=\frac{1}{2z}-\frac{\pi }{2}\cot \left(\pi z\right)+\frac{1}{z^2-1}+1-\gamma -\sum _{k=1}^{\mathrm{\infty }}(\zeta \left(2k+1\right)-1)z^{2k},$$ , $z\ne 0,\pm 1$.

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.3.15

Referenced by:

§5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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23: 25.12 Polylogarithms

… ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►Further properties include ►

25.12.12 $${\mathrm{Li}}_s\left(z\right)=\mathrm{\Gamma }\left(1-s\right){\left(\ln \frac{1}{z}\right)}^{s-1}+\sum _{n=0}^{\mathrm{\infty }}\zeta \left(s-n\right)\frac{{(\ln z)}^n}{n!},$$ $s\ne 1,2,3,\mathrm{\dots }$, ,

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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, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

infinite series, series representation

Source:

Erdélyi et al. (1953a, (1.11.8), p. 29); take $v=1$ and use (25.14.3), (25.11.2)

Permalink:

http://dlmf.nist.gov/25.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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24: 20.9 Relations to Other Functions

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§20.9(iii) Riemann Zeta Function

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25: 27.18 Methods of Computation: Primes

… ►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). …

26: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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$\zeta \left(s\right)$	Riemann zeta function; see §25.2(i).
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27: 5.15 Polygamma Functions

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5.15.2 $${\psi }^{(n)}\left(1\right)={(-1)}^{n+1}n!\zeta \left(n+1\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), ${\psi }^{(n)}\left(z\right)$: polygamma functions and $n$: nonnegative integer

A&S Ref:

6.4.2

Referenced by:

§5.15

Permalink:

http://dlmf.nist.gov/5.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

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5.15.3 $${\psi }^{(n)}\left(\frac{1}{2}\right)={(-1)}^{n+1}n!(2^{n+1}-1)\zeta \left(n+1\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), ${\psi }^{(n)}\left(z\right)$: polygamma functions and $n$: nonnegative integer

A&S Ref:

6.4.4

Permalink:

http://dlmf.nist.gov/5.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

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28: 26.12 Plane Partitions

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26.12.26 $$\mathrm{pp}\left(n\right)\sim \frac{{\left(\zeta \left(3\right)\right)}^{7/36}}{2^{11/36}{(3\pi )}^{1/2}{n}^{25/36}}\exp \left(3{\left(\zeta \left(3\right)\right)}^{1/3}{\left(\frac{1}{2}n\right)}^{2/3}+{\zeta }^{\prime }\left(-1\right)\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\sim$: asymptotic equality, $\exp z$: exponential function, $\mathrm{pp}\left(n\right)$: number of plane partitions of $n$ , $n$: nonnegative integer and $\pi$: plane partition

Referenced by:

§26.12(iv), Erratum (V1.0.5) for Equation (26.12.26)

Permalink:

http://dlmf.nist.gov/26.12.E26

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

The original statement of this equation, $\mathrm{pp}\left(n\right)\sim {\left(\frac{\zeta \left(3\right)}{2^{11}{n}^{25}}\right)}^{1/36}\exp \left(3{\left(\frac{\zeta \left(3\right)n^2}{4}\right)}^{1/3}+{\zeta }^{\prime }\left(-1\right)\right)$, has been corrected.

Reported 2011-09-05 by Suresh Govindarajan

See also:

Annotations for §26.12(iv), §26.12 and Ch.26

►where $\zeta$ is the Riemann $\zeta$-function (§25.2(i)). …

29: Bibliography K

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The Riemann Zeta-Function. de Gruyter Expositions in Mathematics, Vol. 5, Walter de Gruyter & Co., Berlin.

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Notes:

Translated from the Russian by Neal Koblitz

External Links:

ISBN 3-11-013170-6, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

Ch.25.

Encodings:

BibTeX

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The Riemann Zeta-Function and Quantum Chaology. In Quantum Chaos (Varenna, 1991), Proc. Internat. School of Phys. Enrico Fermi, CXIX, pp. 145–185.

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External Links:

MathReview (Dieter H. Mayer)

Referenced by:

§25.17.

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

Encodings:

BibTeX

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Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.22(ii).

Encodings:

BibTeX

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30: 24.17 Mathematical Applications

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§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …