Riemann%20hypothesis

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31: 21.1 Special Notation

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$g,h$	positive integers.
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$𝛀$	$g\times g$ complex, symmetric matrix with $\mathrm{\Im }𝛀$ strictly positive definite, i.e., a Riemann matrix.
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$a\circ b$	intersection index of $a$ and $b$, two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$.
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… ►Uppercase boldface letters are $g\times g$ real or complex matrices. ►The main functions treated in this chapter are the Riemann theta functions $\theta \left(𝐳\right|𝛀)$, and the Riemann theta functions with characteristics $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$. ►The function $\mathrm{\Theta }(\mathit{\varphi }|𝐁)=\theta \left(\mathit{\varphi }/(2\pi \mathrm{i})\right|𝐁/(2\pi \mathrm{i}))$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).

32: 25.19 Tables

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Abramowitz and Stegun (1964) tabulates: $\zeta \left(n\right)$, $n=2,3,4,\mathrm{\dots }$, 20D (p. 811); ${\mathrm{Li}}_2\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta )$, $\theta ={15}^{\circ }(1^{\circ }){30}^{\circ }(2^{\circ }){90}^{\circ }(5^{\circ }){180}^{\circ }$, $f(\theta )+\theta \ln \theta$, $\theta =0(1^{\circ }){15}^{\circ }$, 6D (p. 1006). Here $f(\theta )$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta \left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta (s,a)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

33: 25.2 Definition and Expansions

§25.2 Definition and Expansions

… ►Elsewhere $\zeta \left(s\right)$ is defined by analytic continuation. … ►

§25.2(ii) Other Infinite Series

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§25.2(iii) Representations by the Euler–Maclaurin Formula

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§25.2(iv) Infinite Products

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34: 25.11 Hurwitz Zeta Function

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

25.11.2 $$\zeta (s,1)=\zeta \left(s\right).$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function and $s$: complex variable

Keywords:

specialization

Source:

Apostol (1976, p. 255)

Proof sketch:

Derivable from (25.11.1) and (25.2.1).

Referenced by:

(25.11.10), (25.11.24), (25.12.12)

Permalink:

http://dlmf.nist.gov/25.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(i), §25.11 and Ch.25

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Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

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25.11.11 $$\zeta (s,\frac{1}{2})=(2^s-1)\zeta \left(s\right),$$ $s\ne 1$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function and $s$: complex variable

Keywords:

special value

Proof sketch:

Derivable using (25.11.1), (25.2.2).

Referenced by:

§25.11(iv)

Permalink:

http://dlmf.nist.gov/25.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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25.11.41 $$\zeta (s,a+1)=\zeta \left(s\right)-s\zeta \left(s+1\right)a+O\left(a^2\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $a$: real or complex parameter and $s$: complex variable

Keywords:

asymptotic approximation

Source:

Apostol (1952, (15), p. 6)

Permalink:

http://dlmf.nist.gov/25.11.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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35: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►Such a solution is given in terms of a Riemann theta function with two phases. …

36: 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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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).

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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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).

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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37: 20 Theta Functions

Chapter 20 Theta Functions

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38: 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:

BibTeX

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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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:

BibTeX

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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:

BibTeX

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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:

BibTeX

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39: 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:

BibTeX

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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:

BibTeX

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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.

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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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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40: 5.16 Sums

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5.16.2 $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}{\psi }^{\prime }\left(k+1\right)=\zeta \left(3\right)=-\frac{1}{2}{\psi }^{\prime \prime }\left(1\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

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