Riemann identity

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21: 2.10 Sums and Sequences

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2.10.6 $$C=\frac{\gamma +\ln \left(2\pi \right)}{12}-\frac{{\zeta }^{\prime }\left(2\right)}{2{\pi }^2}=\frac{1}{12}-{\zeta }^{\prime }\left(-1\right),$$

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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 and $C$: constant

Permalink:

http://dlmf.nist.gov/2.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(i), §2.10(i), §2.10 and Ch.2

►where $\gamma$ is Euler’s constant (§5.2(ii)) and ${\zeta }^{\prime }$ is the derivative of the Riemann zeta function (§25.2(i)). … ►This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). … ►From the identities … ►Hence by the Riemann–Lebesgue lemma (§1.8(i)) …

22: 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.

Encodings:

BibTeX

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Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

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External Links:

ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt

Referenced by:

§22.9(iv), §22.9(v).

Encodings:

BibTeX

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Cyclic identities involving Jacobi elliptic functions. J. Math. Phys. 43 (7), pp. 3798–3806.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§22.9(i), §22.9(ii), §22.9(iii), §22.9(iv), §22.9(v).

Encodings:

BibTeX

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Dilogarithm identities. Progr. Theoret. Phys. Suppl. (118), pp. 61–142.

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External Links:

ISSN 0375-9687, FullText, MathReview (J. Browkin), ZentralBlatt

Referenced by:

§25.12(i).

Encodings:

BibTeX

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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:

BibTeX

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23: Bibliography

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$q$-identities of Auluck, Carlitz, and Rogers. Duke Math. J. 33 (3), pp. 575–581.

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External Links:

Document, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§17.9(iv).

Encodings:

BibTeX

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

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:

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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:

BibTeX

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24: Bibliography L

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Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.

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External Links:

ISSN 0001-8708, Document, MathReview (S. I. Gel’fand), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:

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

Referenced by:

§17.16.

Encodings:

BibTeX

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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:

BibTeX

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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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25: Errata

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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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Three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection.

Suggested by Tom Koornwinder.

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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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21.3.4 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶+{𝐦}_1}{𝜷+{𝐦}_2}\right]\left(𝐳\right|𝛀)={\mathrm{e}}^{2\pi \mathrm{i}𝜶\cdot {𝐦}_2}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$$

Originally the vector ${𝐦}_2$ on the right-hand side was given incorrectly as ${𝐦}_1$.

Reported 2012-08-27 by Klaas Vantournhout.

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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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