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21: 31.18 Methods of Computation

… ►Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of $z$; see Laĭ (1994) and Lay et al. (1998). …

22: 25.10 Zeros

… ►

§25.10(ii) Riemann–Siegel Formula

… ►Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp \left(\mathrm{i}\vartheta (t)\right)$ to obtain the Riemann–Siegel formula: ►

25.10.3 $$Z(t)=2\sum _{n=1}^m\frac{\cos \left(\vartheta (t)-t\ln n\right)}{n^{1/2}}+R(t),$$ $m=\lfloor \sqrt{t/(2\pi )}\rfloor$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\lfloor x\rfloor$: floor of $x$ , $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $n$: nonnegative integer, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

Riemann–Siegel formula

Source:

Titchmarsh (1986b, (4.17.4), p. 89)

Referenced by:

§25.18(i), Erratum (V1.1.4) for Equation (25.10.3)

Permalink:

http://dlmf.nist.gov/25.10.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint $m=\lfloor \sqrt{t/(2\pi )}\rfloor$ was added.

Suggested 2021-08-23 by Gergő Nemes

See also:

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

… ►Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of $\zeta \left(s\right)$ in the critical strip are on the critical line (van de Lune et al. (1986)). …

23: 24.17 Mathematical Applications

… ►

Euler–Maclaurin Summation Formula

… ►

Boole Summation Formula

…

24: 36.4 Bifurcation Sets

… ►

§36.4(i) Formulas

… ► $K=2$, cusp bifurcation set: … ► $K=3$, swallowtail bifurcation set: … ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►Hyperbolic umbilic bifurcation set (codimension three): …

25: 35.2 Laplace Transform

… ►

Inversion Formula

…

26: 24.5 Recurrence Relations

… ►

§24.5(iii) Inversion Formulas

…

27: 1.8 Fourier Series

… ►

Parseval’s Formula

… ►

§1.8(iv) Poisson’s Summation Formula

… ►

1.8.16 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n+x)}^2\omega }=\sqrt{\frac{\pi }{\omega }}\left(1+2\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2{\pi }^2/\omega }\cos \left(2n\pi x\right)\right),$$ $\mathrm{\Re }\omega >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $n$: nonnegative integer

Referenced by:

§1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(iv), §1.8 and Ch.1

…

28: 16.12 Products

… ►The following formula is often referred to as Clausen’s formula …

29: 5.19 Mathematical Applications

…

30: 25.13 Periodic Zeta Function

… ►

25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

►

25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25