incomplete Riemann zeta function

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1: 8.22 Mathematical Applications

… ►If ${\zeta }_x(s)$ denotes the incomplete Riemann zeta function defined by ►

8.22.2 $${\zeta }_x(s)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^x\frac{t^{s-1}}{{\mathrm{e}}^t-1}dt,$$ $\mathrm{\Re }s>1$,

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Defines:

${\zeta }_x(s)$: incomplete Riemann zeta function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/8.22.E2

Encodings:

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See also:

Annotations for §8.22(ii), §8.22 and Ch.8

►so that ${lim}_{x\to \mathrm{\infty }}{\zeta }_x(s)=\zeta \left(s\right)$, then ►

8.22.3 $${\zeta }_x(s)=\sum _{k=1}^{\mathrm{\infty }}{k}^{-s}P(s,kx),$$ $\mathrm{\Re }s>1$.

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Symbols:

$P(a,z)$: normalized incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $k$: nonnegative integer and ${\zeta }_x(s)$: incomplete Riemann zeta function

Permalink:

http://dlmf.nist.gov/8.22.E3

Encodings:

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See also:

Annotations for §8.22(ii), §8.22 and Ch.8

►For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

2: Bibliography K

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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:

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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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3: Bibliography D

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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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4: 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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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.12, §19.5.

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).

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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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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:

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5: Bibliography

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Theory of Incomplete Cylindrical Functions and Their Applications. Springer-Verlag, Berlin.

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Notes:

Translated from the Russian by H. E. Fettis, J. W. Goresh and D. A. Lee, Die Grundlehren der mathematischen Wissenschaften, Band 160

External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

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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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On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§8.10.

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.

Encodings:

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6: 5.11 Asymptotic Expansions

… ►and … ►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. … ►For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). … ►

7: Bibliography L

… ►

On incomplete Airy functions and their application to diffraction problems. Radio Sci. 4 (10), pp. 959–969.

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External Links:

ISSN 0048-6604, Document, MathReview (T. Erber)

Referenced by:

§9.14.

Encodings:

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More than one third of zeros of Riemann’s zeta-function are on $\sigma =\frac{1}{2}$ . Advances in Math. 13 (4), pp. 383–436.

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External Links:

Document, MathReview (H. L. Montgomery), ZentralBlatt

Referenced by:

§25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

Encodings:

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On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. Constr. Approx. 10 (1), pp. 15–30.

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External Links:

ISSN 0176-4276, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§8.13(i).

Encodings:

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Generalized Riemann $\zeta$-function regularization and Casimir energy for a piecewise uniform string. Phys. Rev. D 44 (2), pp. 560–562.

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External Links:

Document

Referenced by:

§24.18.

Encodings:

BibTeX

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Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

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External Links:

ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

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8: 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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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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26.12.26 $$\mathrm{pp}\left(n\right)\sim \frac{{\left(\zeta \left(3\right)\right)}^{7/36}}{2^{11/36}{(3\pi )}^{1/2}{n}^{25/36}}\exp \left(3{\left(\zeta \left(3\right)\right)}^{1/3}{\left(\frac{1}{2}n\right)}^{2/3}+{\zeta }^{\prime }\left(-1\right)\right)$$

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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9: 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

Referenced by:

1st item.

Encodings:

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On the Hurwitz zeta-function. Rocky Mountain J. Math. 2 (1), pp. 151–157.

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External Links:

MathReview (K. Thanigasalam), ZentralBlatt

Referenced by:

(25.11.26).

Encodings:

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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).

Encodings:

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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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External Links:

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

Referenced by:

§27.18.

Encodings:

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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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External Links:

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

Referenced by:

§25.18(i).

Encodings:

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10: Bibliography R

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Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:

MathReview (H. H. Martens), ZentralBlatt

Referenced by:

§20.11(iv), §20.12(ii).

Encodings:

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

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Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monats. Berlin Akad. November 1859, pp. 671–680.

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Notes:

Reprinted in Gesammelte Mathematische Werke, pp. 145–155, published by Dover Publications, New York, N.Y. 1953.

Referenced by:

§25.1, (25.2.1), §25.2(i).

Encodings:

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Elliptische Functionen. Teubner, Leipzig.

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Notes:

Based on notes of lectures by B. Riemann edited and published by H. Stahl. Reprinted by UMI Books on Demand (Ann Arbor, MI, USA).

External Links:

ZentralBlatt

Referenced by:

§20.12(ii).

Encodings:

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Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Inauguraldissertation, Göttingen.

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Referenced by:

§21.7(i).

Encodings:

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