Riemann%20zeta%20function

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11: 6.16 Mathematical Applications

… ►It occurs with Fourier-series expansions of all piecewise continuous functions. … … ►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then ►

6.16.5 $$\mathrm{li}\left(x\right)-\pi (x)=O\left(\sqrt{x}\ln x\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $x$: real variable and $\pi (x)$: number of primes $\le x$

A&S Ref:

5.1.50

Permalink:

http://dlmf.nist.gov/6.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(ii), §6.16 and Ch.6

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

12: Bibliography P

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Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

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External Links:

ZentralBlatt

Referenced by:

§5.1, Notations F.

Encodings:

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An asymptotic representation for $\zeta (\frac{1}{2}+it)$ . Methods Appl. Anal. 4 (4), pp. 449–470.

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External Links:

ISSN 1073-2772, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.9, §8.22(ii).

Encodings:

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The Stokes phenomenon associated with the Hurwitz zeta function $\zeta (s,a)$ . Proc. Roy. Soc. London Ser. A 461, pp. 297–304.

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External Links:

ISSN 1364-5021, Document, MathReview (Marc Prevost), ZentralBlatt

Referenced by:

(25.11.43), §25.11(xii).

Encodings:

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

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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:

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13: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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§25.11(i) Definition

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

§25.11(ii) Graphics

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§25.11(vi) Derivatives

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14: 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).

Encodings:

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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).

Encodings:

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

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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).

Encodings:

BibTeX

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15: 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:

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Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.

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Notes:

A reprint was published in 1986.

External Links:

ISBN 90-6196-277-3, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§5.21, §6.18(iv), §7.22(v), §7.6(i), §7.7(i), §7.7(ii).

Encodings:

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A new estimate of the function $\zeta (1+it)$ . Izv. Akad. Nauk SSSR. Ser. Mat. 22, pp. 161–164 (Russian).

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External Links:

MathReview (H. Davenport), ZentralBlatt

Referenced by:

§27.12.

Encodings:

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

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Über die Riemann’sche Primzahlfunction. Math. Ann. 55, pp. 441–464 (German).

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External Links:

ISSN 0025-5831; 1432-1807/e, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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16: 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 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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26.12.26 $$\mathrm{pp}\left(n\right)\sim \frac{{\left(\zeta \left(3\right)\right)}^{7/36}}{2^{11/36}{(3\pi )}^{1/2}{n}^{25/36}}\exp \left(3{\left(\zeta \left(3\right)\right)}^{1/3}{\left(\frac{1}{2}n\right)}^{2/3}+{\zeta }^{\prime }\left(-1\right)\right)$$

Originally this equation was given incorrectly as

$$\mathrm{pp}\left(n\right)\sim {\left(\frac{\zeta \left(3\right)}{2^{11}{n}^{25}}\right)}^{1/36}\exp \left(3{\left(\frac{\zeta \left(3\right)n^2}{4}\right)}^{1/3}+{\zeta }^{\prime }\left(-1\right)\right).$$

Reported 2011-09-05 by Suresh Govindarajan.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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17: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

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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).

Encodings:

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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.

Encodings:

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

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18: 5.11 Asymptotic Expansions

… ►and … ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►where $\zeta \left(K\right)$ is as in Chapter 25. In the case $K=1$ the factor $1+\zeta \left(K\right)$ is replaced with 4. … ►

19: Bibliography

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Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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External Links:

ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt

Referenced by:

(25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.

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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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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

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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20: Bibliography C

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The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

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Chebyshev approximations for the Riemann zeta function. Math. Comp. 25 (115), pp. 537–547.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

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On some series representations of the Hurwitz zeta function. J. Comput. Appl. Math. 216 (1), pp. 297–305.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.11(iv), §25.11(iv).

Encodings:

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

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Zeros of the Hankel function of real order out of the principal Riemann sheet. J. Comput. Appl. Math. 37 (1-3), pp. 89–99.

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External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

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