Riemann zeta function

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31: 5.11 Asymptotic Expansions

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5.11.11 $$\left|R_K(z)\right|\le \frac{(1+\zeta \left(K\right))\mathrm{\Gamma }\left(K\right)}{2{(2\pi )}^{K+1}{\left|z\right|}^K}\left(1+\min (\sec \left(\mathrm{ph}z\right),2K^{\frac{1}{2}})\right),$$ $\left|\mathrm{ph}z\right|\le \frac{1}{2}\pi$,

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Defines:

$R_K(z)$: remainder (locally)

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, $\mathrm{ph}$: phase, $\sec z$: secant function and $z$: complex variable

Referenced by:

§5.11(ii)

Permalink:

http://dlmf.nist.gov/5.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(ii), §5.11 and Ch.5

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32: Bibliography I

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The Riemann Zeta-Function. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:

Reprinted by Dover, 2003.

External Links:

ISBN 0-471-80634-X, MathReview (S. W. Graham), ZentralBlatt

Referenced by:

(25.2.4), (25.2.5), §25.2(i), §25.2(ii), §25.2, Ch.25.

Encodings:

BibTeX

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33: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.7 $$C=\underset{n\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^{n}k\ln k-\left(\frac{1}{2}{n}^2+\frac{1}{2}n+\frac{1}{12}\right)\ln n+\frac{1}{4}{n}^2\right)=\frac{\gamma +\ln \left(2\pi \right)}{12}-\frac{{\zeta }^{\prime }\left(2\right)}{2{\pi }^2}=\frac{1}{12}-{\zeta }^{\prime }\left(-1\right),$$

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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, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $k$: nonnegative integer and $C$: log of Glaisher’s Constant

Referenced by:

§5.17

Permalink:

http://dlmf.nist.gov/5.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

►and ${\zeta }^{\prime }$ is the derivative of the zeta function (Chapter 25). …

34: Bibliography E

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Riemann’s Zeta Function. Academic Press, New York-London.

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Notes:

Pure and Applied Mathematics, Vol. 58. Reprinted by Dover Publications Inc., Mineola, NY, 2001.

External Links:

MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

Ch.25.

Encodings:

BibTeX

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An asymptotic expansion for the first derivative of the generalized Riemann zeta function. Math. Comp. 47 (175), pp. 347–350.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Audinarayana Moorthy), ZentralBlatt

Referenced by:

(25.11.44), (25.11.45), §25.11(xii), §25.11(xii).

Encodings:

BibTeX

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35: Bibliography

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Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.

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External Links:

MathReview

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:

ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

Encodings:

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Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.

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External Links:

ISSN 0025-5718, FullText, MathReview (Aleksandar Ivić), ZentralBlatt

Referenced by:

(25.11.19), (25.11.20), (25.11.6), §25.11(vi), §25.18(i), (25.4.5), (25.4.6), §25.4, (25.6.11), (25.6.12), (25.6.13), (25.6.14), §25.6(ii), §25.6.

Encodings:

BibTeX

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36: 24.4 Basic Properties

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§24.4(ix) Relations to Other Functions

►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

37: Bibliography V

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On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (174), pp. 667–681.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (Jürgen G. Hinz), ZentralBlatt

Referenced by:

§25.10(ii), §25.18(ii).

Encodings:

BibTeX

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38: 5.9 Integral Representations

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5.9.11 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)=-\gamma z-\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{\infty }\mathrm{i}}^{-c+\mathrm{\infty }\mathrm{i}}\frac{\pi z^{-s}}{s\sin \left(\pi s\right)}\zeta \left(-s\right)ds,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: 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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $s$: real or complex variable and $z$: complex variable

Referenced by:

§5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.9.E11

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+1\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.17 $$\psi \left(z+1\right)=-\gamma +\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{\infty }\mathrm{i}}^{-c+\mathrm{\infty }\mathrm{i}}\frac{\pi z^{-s-1}}{\sin \left(\pi s\right)}\zeta \left(-s\right)ds,$$

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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, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $s$: real or complex variable and $z$: complex variable

Referenced by:

§5.9(ii)

Permalink:

http://dlmf.nist.gov/5.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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39: Bibliography T

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The Theory of the Riemann Zeta-Function. 2nd edition, The Clarendon Press Oxford University Press, New York-Oxford.

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Notes:

Edited and with a preface by D. R. Heath-Brown

External Links:

ISBN 0-19-853369-1, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§20.10(i), §20.9(iii), (25.10.1), (25.10.2), (25.10.3), (25.10.4), §25.10(i), §25.10(ii), §25.10, (25.2.12), §25.2(iv), (25.5.13), (25.5.14), (25.5.5), §25.5(ii), (25.9.1), (25.9.2), (25.9.3), §25.9, Ch.25, §27.4.

Encodings:

BibTeX

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40: Bibliography P

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Chebyshev polynomial expansions of the Riemann zeta function. Math. Comp. 26 (120), pp. G1–G5.

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External Links:

ISSN 0025-5718, FullText, MathReview

Referenced by:

2nd item, 3rd item.

Encodings:

BibTeX

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