Riemann identity

SearchAdvancedHelp

(0.001 seconds)

1—10 of 25 matching pages

1: 21.6 Products

… ►

§21.6(i) Riemann Identity

… ►Then …This is the Riemann identity. On using theta functions with characteristics, it becomes …

2: 21.7 Riemann Surfaces

… ►These special Riemann theta functions satisfy many special identities, two of which appear in the following subsections. … ►

§21.7(ii) Fay’s Trisecant Identity

… ► ►

§21.7(iii) Frobenius’ Identity

…

3: 25.10 Zeros

… ►

25.10.1 $$Z(t)\equiv \exp \left(\mathrm{i}\vartheta (t)\right)\zeta \left(\frac{1}{2}+\mathrm{i}t\right),$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\equiv$: equals by definition, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

definition

Source:

Titchmarsh (1986b, (4.17.2), p. 89)

Permalink:

http://dlmf.nist.gov/25.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

…

4: Bibliography S

… ►

Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.

ⓘ

External Links:

ISSN 0019-5588, MathReview (Jean-Marie De Koninck), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

…

5: 24.17 Mathematical Applications

… ►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …

6: 27.5 Inversion Formulas

… ►For example, the equation $\zeta \left(s\right)\cdot (1/\zeta \left(s\right))=1$ is equivalent to the identity …

7: 25.4 Reflection Formulas

§25.4 Reflection Formulas

… ►

25.4.1 $$\zeta \left(1-s\right)=2{(2\pi )}^{-s}\cos \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(s\right)\zeta \left(s\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (12), p. 259)

Referenced by:

§25.10(i), §5.17

Permalink:

http://dlmf.nist.gov/25.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

… ►

25.4.3 $$\xi \left(s\right)=\xi \left(1-s\right),$$

ⓘ

Symbols:

$\xi \left(s\right)$: Riemann’s $\xi$-function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, p. 260)

Permalink:

http://dlmf.nist.gov/25.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

►where $\xi \left(s\right)$ is Riemann’s $\xi$-function, defined by: ►

25.4.4 $$\xi \left(s\right)=\frac{1}{2}s(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s\right){\pi }^{-s/2}\zeta \left(s\right).$$

ⓘ

Defines:

$\xi \left(s\right)$: Riemann’s $\xi$-function

Symbols:

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

Keywords:

definition

Source:

Apostol (1976, p. 260)

Permalink:

http://dlmf.nist.gov/25.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

…

8: 27.4 Euler Products and Dirichlet Series

… ►Every multiplicative $f$ satisfies the identity … ►The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)): ►

27.4.3 $$\zeta \left(s\right)=\sum _{n=1}^{\mathrm{\infty }}{n}^{-s}=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$.

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

A&S Ref:

23.2.1 and 23.2.2

Permalink:

http://dlmf.nist.gov/27.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

►The Riemann zeta function is the prototype of series of the form …In (27.4.12) and (27.4.13) ${\zeta }^{\prime }\left(s\right)$ is the derivative of $\zeta \left(s\right)$.

9: 25.12 Polylogarithms

… ►

25.12.1 $${\mathrm{Li}}_2\left(z\right)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^2},$$ $|z|\le 1$.

ⓘ

Defines:

${\mathrm{Li}}_2\left(z\right)$: dilogarithm

Symbols:

$\equiv$: equals by definition, $n$: nonnegative integer, $x$: real variable and $z$: complex variable

Keywords:

definition

Source:

Maximon (2003, (3.1), p. 2808)

A&S Ref:

27.7.2 (with $z=1-x$)

Referenced by:

§25.12(i), §25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

… ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►

25.12.11 $${\mathrm{Li}}_s\left(z\right)\equiv \frac{z}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x-z}dx,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\int$: integral, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $x$: real variable, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Lewin (1981, (7.188), p. 236)

Referenced by:

(25.12.16), §25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

… ►Further properties include ►

25.12.12 $${\mathrm{Li}}_s\left(z\right)=\mathrm{\Gamma }\left(1-s\right){\left(\ln \frac{1}{z}\right)}^{s-1}+\sum _{n=0}^{\mathrm{\infty }}\zeta \left(s-n\right)\frac{{(\ln z)}^n}{n!},$$ $s\ne 1,2,3,\mathrm{\dots }$, ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

infinite series, series representation

Source:

Erdélyi et al. (1953a, (1.11.8), p. 29); take $v=1$ and use (25.14.3), (25.11.2)

Permalink:

http://dlmf.nist.gov/25.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

…

10: 25.13 Periodic Zeta Function

… ►

25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

ⓘ

Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

… ► $F(x,s)$ is periodic in $x$ with period 1, and equals $\zeta \left(s\right)$ when $x$ is an integer. …