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21: 8.22 Mathematical Applications

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§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

… ►See Paris and Cang (1997). ►If ${\zeta }_x(s)$ denotes the incomplete Riemann zeta function defined by …so that ${lim}_{x\to \mathrm{\infty }}{\zeta }_x(s)=\zeta \left(s\right)$, then …For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

22: 21.4 Graphics

§21.4 Graphics

►Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\widehat{\theta }\left(𝐳\right|𝛀)$, with …This Riemann matrix originates from the Riemann surface represented by the algebraic curve ${\mu }^3-{\lambda }^7+2{\lambda }^3\mu =0$; compare §21.7(i). … ►For the scaled Riemann theta functions depicted in Figures 21.4.2–21.4.5 … ►

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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. … Magnify 3D Help

23: 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)): ►

27.4.3 $$\zeta \left(s\right)=\sum _{n=1}^{\mathrm{\infty }}{n}^{-s}=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

A&S Ref:

23.2.1 and 23.2.2

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http://dlmf.nist.gov/27.4.E3

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Annotations for §27.4 and Ch.27

►The Riemann zeta function is the prototype of series of the form … ►

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

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

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Annotations for §27.4 and Ch.27

… ►In (27.4.12) and (27.4.13) ${\zeta }^{\prime }\left(s\right)$ is the derivative of $\zeta \left(s\right)$.

24: 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.

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Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

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Periodic Orbits, Spectral Statistics, and the Riemann Zeros. In Supersymmetry and Trace Formulae: Chaos and Disorder, J. P. Keating, D. E. Khmelnitskii, and I. V. Lerner (Eds.), pp. 1–15.

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Referenced by:

§25.17.

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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).

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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:

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25: Bibliography D

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Integralen voor de $\zeta$-functie van Riemann. Mathematica (Zutphen) B5, pp. 170–180 (Dutch).

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Notes:

In Dutch.

External Links:

ZentralBlatt

Referenced by:

(25.5.15), (25.5.16), (25.5.17), (25.5.18), (25.5.19), §25.5(ii).

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

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Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

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External Links:

ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt

Referenced by:

3rd item.

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Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York.

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External Links:

ISBN 0-9658703-2-4; 0-8218-2695-6, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§18.38(ii), §18.38(ii).

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Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.22(ii).

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26: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

Encodings:

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

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Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

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Tables of the Riemann Zeta Function. Royal Society Mathematical Tables, Vol. 6, Cambridge University Press, New York.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§25.18(i), §25.18(ii).

Encodings:

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Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

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External Links:

ISSN 0002-9890, Document, Link, MathReview Entry

Referenced by:

§1.16(ix).

Encodings:

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27: 21.5 Modular Transformations

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§21.5(i) Riemann Theta Functions

… ►Equation (21.5.4) is the modular transformation property for Riemann theta functions. ►The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi (𝚪)$ is determinate: … ►

§21.5(ii) Riemann Theta Functions with Characteristics

… ►For explicit results in the case $g=1$, see §20.7(viii).

28: 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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25.8.9 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2.$$

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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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25.8.10 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4{\pi }^2}\zeta \left(3\right).$$

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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 $k$: nonnegative integer

Keywords:

infinite series

Source:

Ewell (1990, (1), p. 219)

Permalink:

http://dlmf.nist.gov/25.8.E10

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Annotations for §25.8 and Ch.25

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29: 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:

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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

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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.

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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.

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The Riemann Hypothesis and the Hamiltonian of a Quantum Mechanical System. In Number Theory and Dynamical Systems (York, 1987), M. M. Dodson and J. A. G. Vickers (Eds.), London Math. Soc. Lecture Note Ser., Vol. 134, pp. 153–172.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.17.

Encodings:

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30: 21.8 Abelian Functions

§21.8 Abelian Functions

… ►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …