Riemann identity

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11: 15.17 Mathematical Applications

… ►

§15.17(iv) Combinatorics

►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. … ►The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

12: 25.9 Asymptotic Approximations

§25.9 Asymptotic Approximations

… ►

25.9.1 $$\zeta \left(\sigma +\mathrm{i}t\right)=\sum _{1\le n\le x}\frac{1}{n^s}+\chi (s)\sum _{1\le n\le y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma }\right)+O\left(y^{\sigma -1}{t}^{\frac{1}{2}-\sigma }\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable, $s$: complex variable and $\sigma$: fixed

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.12.4), p. 79)

Referenced by:

§25.9

Permalink:

http://dlmf.nist.gov/25.9.E1

Encodings:

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See also:

Annotations for §25.9 and Ch.25

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25.9.2 $$\chi (s)\equiv {\pi }^{s-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}s\right)/\mathrm{\Gamma }\left(\frac{1}{2}s\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition and $s$: complex variable

Keywords:

definition

Source:

Titchmarsh (1986b, p. 78); with (5.5.5), (5.5.3)

Referenced by:

(25.10.2)

Permalink:

http://dlmf.nist.gov/25.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.9 and Ch.25

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25.9.3 $$\zeta \left(\frac{1}{2}+\mathrm{i}t\right)=\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi \left(\frac{1}{2}+\mathrm{i}t\right)\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+O\left(t^{-1/4}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.17.1), p. 88)

Referenced by:

§25.10(ii)

Permalink:

http://dlmf.nist.gov/25.9.E3

Encodings:

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See also:

Annotations for §25.9 and Ch.25

…

13: 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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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:

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See also:

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

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25.11.40 $$G\equiv \sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(2n+1)}^2}=\mathrm{0.91596\hspace{0.33em}55941\hspace{0.33em}772}\mathrm{\dots }.$$

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Symbols:

$\equiv$: equals by definition, $n$: nonnegative integer and $G$: Catalan’s constant

Keywords:

Catalan’s constant, definition

Source:

Adamchik and Srivastava (1998, (2.32), p. 139)

Notes:

For more digits see OEIS Sequence A006752; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/25.11.E40

Encodings:

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See also:

Annotations for §25.11(xi), §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:

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See also:

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

…

14: 25.5 Integral Representations

… ►

25.5.6 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-1$.

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Symbols:

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

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.5.1) by first replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x})$, using the identity ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$, and replacing ${({\mathrm{e}}^x-1)}^{-1}\mapsto ({({\mathrm{e}}^x-1)}^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1).

Referenced by:

(25.5.7)

Permalink:

http://dlmf.nist.gov/25.5.E6

Encodings:

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See also:

Annotations for §25.5(i), §25.5 and Ch.25

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15: 21.1 Special Notation

… ► ►►►►

$g,h$	positive integers.
…
$𝛀$	$g\times g$ complex, symmetric matrix with $\mathrm{\Im }𝛀$ strictly positive definite, i.e., a Riemann matrix.
…
${𝐈}_g$	$g\times g$ identity matrix.
…

… ►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).

16: Bibliography R

… ►

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

… ►

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

►

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

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Referenced by:

§21.7(i).

Encodings:

BibTeX

… ►

Combinatorial Identities. Robert E. Krieger Publishing Co., Huntington, NY.

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Notes:

Reprint of the 1968 original with corrections.

External Links:

ISBN 0-88275-829-2, MathReview

Referenced by:

§24.5(iii), §26.5(iii), §26.8(i), §26.8(iv), §26.8(v).

Encodings:

BibTeX

… ►

Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

…

17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►In this section we will only consider the special case $w(x)=1$, so $d\alpha (x)=dx$; in which case $L^2\left(X\right)\equiv L^2(X,dx)$. … ►Other choices of boundary conditions, identical for $f(x)$ and $g(x)$, and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of $T$. … ► It is to be noted that if any of the $\lambda \in 𝝈$ have degenerate sub-spaces, that is subspaces of orthogonal eigenfunctions with identical eigenvalues, that in the expansions below all such distinct eigenfunctions are to be included. … ► … ►What then is the condition on $q(x)$ to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if $q(x)\equiv 0$ then there is only a continuous spectrum. …

18: Bibliography B

… ►

Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

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External Links:

ISSN 0022-4715, Document, MathReview, ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

… ►

Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.

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External Links:

ISSN 1431-0643, FullText, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

… ►

Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.

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External Links:

ISSN 0305-4470, Document, MathReview (J. S. Joel), ZentralBlatt

Referenced by:

§36.9.

Encodings:

BibTeX

… ►

A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.

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External Links:

ISSN 0002-9947, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

… ►

The hypergeometric identities of Cayley, Orr, and Bailey. Proc. London Math. Soc. (2) 50, pp. 56–74.

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External Links:

Document, MathReview (N. A. Hall), ZentralBlatt

Referenced by:

§15.16.

Encodings:

BibTeX

…

19: 27.12 Asymptotic Formulas: Primes

… ► $\pi \left(x\right)-\mathrm{li}\left(x\right)$ changes sign infinitely often as $x\to \mathrm{\infty }$; see Littlewood (1914), Bays and Hudson (2000). ►The Riemann hypothesis (§25.10(i)) is equivalent to the statement that for every $x\ge 2657$, … ►For example, if $2^n\cancel{\equiv }2(modn)$, then $n$ is composite. … ►A Carmichael number is a composite number $n$ for which $b^n\equiv b(modn)$ for all $b\in \mathbb{N}$. …

20: 27.11 Asymptotic Formulas: Partial Sums

… ►

27.11.5 $$\sum _{n\le x}{\sigma }_{\alpha }\left(n\right)=\frac{\zeta \left(\alpha +1\right)}{\alpha +1}{x}^{\alpha +1}+O\left(x^{\beta }\right),$$ $\alpha >0$, $\alpha \ne 1$, $\beta =\max (1,\alpha )$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E5

Encodings:

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See also:

Annotations for §27.11 and Ch.27

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27.11.9 $$\sum _{\begin{array}{c}p\le x\\ p\equiv h(modk)\end{array}}\frac{1}{p}=\frac{1}{\varphi \left(k\right)}\ln \ln x+B+O\left(\frac{1}{\ln x}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\equiv$: modular equivalence, $\ln z$: principal branch of logarithm function, $k$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E9

Encodings:

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Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.11 $$\sum _{\begin{array}{c}p\le x\\ p\equiv h(modk)\end{array}}\frac{\ln p}{p}=\frac{1}{\varphi \left(k\right)}\ln x+O\left(1\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\equiv$: modular equivalence, $\ln z$: principal branch of logarithm function, $k$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Referenced by:

§27.11, §27.11

Permalink:

http://dlmf.nist.gov/27.11.E11

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

… ►Letting $x\to \mathrm{\infty }$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h(modk)$ if $h,k$ are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. … ►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 $x\to \mathrm{\infty }$.