relation to Riemann zeta function

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21: Software Index

… ►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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A cross index of mathematical software in use at NIST.

22: 26.12 Plane Partitions

… ►A plane partition is self-complementary if it is equal to its complement. … ►

§26.12(ii) Generating Functions

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§26.12(iii) Recurrence Relation

… ►where $\zeta$ is the Riemann $\zeta$-function (§25.2(i)). ►Addendum: To ten decimal places, …

23: 27.11 Asymptotic Formulas: Partial Sums

… ►The behavior of a number-theoretic function $f(n)$ for large $n$ is often difficult to determine because the function values can fluctuate considerably as $n$ increases. It is more fruitful to study partial sums and seek asymptotic formulas of the form … ►Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. … ►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3). The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.

24: Bibliography T

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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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External Links:

ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§13.8(iii), §13.8(iii).

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

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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.

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Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:

Document

Referenced by:

§31.17(ii).

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Combinatorial Group Theory, Riemann Surfaces and Differential Equations. In Contributions to Group Theory, Contemp. Math., Vol. 33, pp. 467–519.

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External Links:

MathReview (William Harvey), ZentralBlatt

Referenced by:

2nd item.

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25: Errata

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Specific source citations and proof metadata are now given for all equations in Chapter 25 Zeta and Related Functions.

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The Gegenbauer function $C_{\alpha }^{(\lambda )}\left(z\right)$, was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C_n^{(\lambda )}\left(z\right)$. In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

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The original constraint, $\mathrm{\Re }s>0$, was removed because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$, and therefore $\zeta \left(s\right)-{(s-1)}^{-1}$ is an entire function.

Suggested by John Harper.

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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26: Bibliography B

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

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1st item.

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The Riemann-Siegel expansion for the zeta function: High orders and remainders. Proc. Roy. Soc. London Ser. A 450, pp. 439–462.

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External Links:

ISSN 0962-8444, FullText, MathReview (Christian Grosche), ZentralBlatt

Referenced by:

§25.10(ii).

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

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Computational strategies for the Riemann zeta function. J. Comput. Appl. Math. 121 (1-2), pp. 247–296.

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ISSN 0377-0427, Document, MathReview (Cem Y. Yildirim), ZentralBlatt

Referenced by:

§27.18.

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Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt

Referenced by:

§25.18(i).

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27: Bibliography I

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Special Functions, $q$-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Mourad E.H. Ismail.

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:

ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

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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.

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From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.

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External Links:

ISBN 3-528-06355-6, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§32.2(vi), §32.7(vii).

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28: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

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Contribution no. 14. The Riemann zeta function. BIT 5, pp. 138–141.

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Referenced by:

§25.21(ii).

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Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.22(i).

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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External Links:

ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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29: Bibliography S

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Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.

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External Links:

ISSN 0019-5588, MathReview (Jean-Marie De Koninck), ZentralBlatt

Referenced by:

§24.14(ii).

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:

ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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Introduction to Riemann Surfaces. Addison-Wesley Publishing Company, Reading, Massachusetts.

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Notes:

A second edition was published in 1981 (Chelsea Publishing Co., New York, and AMS Chelsea Book Series, American Mathematical Society, Providence, Rhode Island)

External Links:

MathReview (L. Sario), ZentralBlatt (H. Wittich)

Referenced by:

§21.7(i).

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Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 0-7923-7054-6, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

(25.11.25), (25.11.8), §25.11, §25.11(iv), §25.11(x), (25.8.9), §25.8.

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Sums of certain series of the Riemann zeta function. J. Math. Anal. Appl. 134 (1), pp. 129–140.

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External Links:

ISSN 0022-247X, Document, MathReview (Aleksandar Ivić), ZentralBlatt

Referenced by:

(25.8.3), §25.8.

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30: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

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§19.25(iv) Theta Functions

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§19.25(vii) Hypergeometric Function

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