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1: 25.10 Zeros

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

2: Bibliography B

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Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

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External Links:

ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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Prolate angular spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 187–192.

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External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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Prolate radial spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 177–185.

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External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(iii), §7.17(iii).

Encodings:

BibTeX

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More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

Encodings:

BibTeX

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3: 30 Spheroidal Wave Functions

Chapter 30 Spheroidal Wave Functions

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4: Errata

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Version 1.1.6 (June 30, 2022)

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In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

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5: 24.2 Definitions and Generating Functions

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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$4$	$-\frac{1}{30}$	$5$
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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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$4$	$-1$	$30$	$-\mathrm{3.33333\hspace{0.33em}3333}$	$\mathrm{\times 10⁻²}$
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$8$	$-1$	$30$	$-\mathrm{3.33333\hspace{0.33em}3333}$	$\mathrm{\times 10⁻²}$
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$30$	$\mathrm{861\hspace{0.33em}58412\hspace{0.33em}76005}$	$14322$	$\mathrm{6.01580\hspace{0.33em}8739}$	$\mathrm{\times 10⁸}$
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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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$30$	$-\mathrm{44\hspace{0.33em}15438\hspace{0.33em}93249\hspace{0.33em}02310\hspace{0.33em}45536\hspace{0.33em}82821}$
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6: 26.9 Integer Partitions: Restricted Number and Part Size

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Table 26.9.1: Partitions $p_k\left(n\right)$.

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$n$	$k$
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9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42

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7: 27.2 Functions

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Table 27.2.1: Primes.

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$n$	$p_n$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
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Table 27.2.2: Functions related to division.

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$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
4	2	3	7	17	16	2	18	30	8	8	72	43	42	2	44
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
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11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
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8: 14.33 Tables

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Zhang and Jin (1996, Chapter 4) tabulates ${𝖯}_n\left(x\right)$ for $n=2(1)5,10$, $x=0(.1)1$, 7D; ${𝖯}_n\left(\cos \theta \right)$ for $n=1(1)4,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖰}_n\left(x\right)$ for $n=0(1)2,10$, $x=0(.1)0.9$, 8S; ${𝖰}_n\left(\cos \theta \right)$ for $n=0(1)3,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖯}_n^m\left(x\right)$ for $m=1(1)4$, $n-m=0(1)2$, $n=10$, $x=0,0.5$, 8S; ${𝖰}_n^m\left(x\right)$ for $m=1(1)4$, $n=0(1)2,10$, 8S; ${𝖯}_{\nu }^m\left(\cos \theta \right)$ for $m=0(1)3$, $\nu =0(.25)5$, $\theta =0({15}^{\circ }){90}^{\circ }$, 5D; $P_n\left(x\right)$ for $n=2(1)5,10$, $x=1(1)10$, 7S; $Q_n\left(x\right)$ for $n=0(1)2,10$, $x=2(1)10$, 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$-zeros of ${𝖯}_{\nu }^m\left(\cos \theta \right)$ and of its derivative for $m=0(1)4$, $\theta ={10}^{\circ },{30}^{\circ },{150}^{\circ }$.

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Žurina and Karmazina (1964, 1965) tabulate the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when .

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Žurina and Karmazina (1963) tabulates the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when .

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9: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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Table 26.4.1: Multinomials and partitions.

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$n$	$m$	$\lambda$	$M_1$	$M_2$	$M_3$
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$5$	$2$	$1^1,4^1$	$5$	$30$	$5$
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$5$	$3$	$1^1,2^2$	$30$	$15$	$15$
$5$	$4$	$1^3,2^1$	$60$	$10$	$10$
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10: 3.4 Differentiation

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$$B_0^5=-\frac{1}{12}(4+30t-15t^2-12t^3+5t^4),$$

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$$B_{-2}^6=\frac{1}{60}(9-9t-30t^2+20t^3+5t^4-3t^5),$$

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$$B_2^6=-\frac{1}{60}(9+9t-30t^2-20t^3+5t^4+3t^5),$$

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$$B_{-1}^7=-\frac{1}{240}(144-360t-48t^2+260t^3-45t^4-30t^5+7t^6),$$

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3.4.23 $$\frac{{\partial }^2{u}_{0,0}}{{\partial x}^2}=\frac{1}{12h^2}(-u_{2,0}+16u_{1,0}-30u_{0,0}+16u_{-1,0}-u_{-2,0})+O\left(h^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.24

Permalink:

http://dlmf.nist.gov/3.4.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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