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1: 25.10 Zeros

… ►Riemann developed a method for counting the total number

N(T)

of zeros of

\zeta\left(s\right)

in that portion of the critical strip with

0<t<T

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

2: Bibliography B

… ►

E. A. Bender (1974) Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.

ⓘ

External Links:: Document, MathReview (Bernard Harris), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

… ►

R. L. Bishop (1981) Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

ⓘ

External Links:: ISSN 0035-9149, Document, MathReview
Referenced by:: Sidebar 9.SB1.
Encodings:: BibTeX

►

Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

ⓘ

External Links:: ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt
Referenced by:: §3.11(v).
Encodings:: BibTeX

… ►

J. M. Borwein and P. B. Borwein (1991) A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §15.8(v).
Encodings:: BibTeX

… ►

H. M. Bui, B. Conrey, and M. P. Young (2011) More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

ⓘ

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

…

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the

icon) for links to defining formula. …Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …

4: Errata

… ►

Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, $\mathsf{P}_{n},\mathsf{Q}_{n},P_{n},Q_{n},\boldsymbol{Q}_{n}$ and the Laguerre polynomial, $L_{n}$ , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

… ►

Subsection 25.10(ii)

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).

… ►

Equation (3.3.34)

In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

Reported by Nico Temme on 2021-06-01

… ►

Table 22.4.3

Originally a minus sign was missing in the entries for $\operatorname{cd}u$ and $\operatorname{dc}u$ in the second column (headed $z+K+iK^{\prime}$ ). The correct entries are $-k^{-1}\operatorname{ns}z$ and $-k\operatorname{sn}z$ . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\operatorname{sn},\operatorname{cn},\operatorname{dn}$ , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}K^{\prime}$	$z+\mathrm{i}K^{\prime}$	$z+2K$	$z+2K+2\mathrm{i}K^{\prime}$	$z+2\mathrm{i}K^{\prime}$
$\vdots$
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
$\vdots$
$\operatorname{dc}u$	$-\operatorname{ns}z$	$-k\operatorname{sn}z$	$k\operatorname{cd}z$	$-\operatorname{dc}z$	$-\operatorname{dc}z$	$\operatorname{dc}z$
$\vdots$

Reported 2014-02-28 by Svante Janson.

… ►

References

Bibliographic citations were added in §§3.5(iv), 4.44, 8.22(ii), 22.4(i), and minor clarifications were made in §§19.12, 20.7(vii), 22.9(i). In addition, several minor improvements were made affecting only ancilliary documents and links in the online version.

…

5: How to Cite

… ►The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version! ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …

6: Frank W. J. Olver

… ►In 1945–1961 he was a founding member of the Mathematics Division and Head of the Numerical Methods Section at the National Physical Laboratory, Teddington, U. … ►Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … ►Most notably, he served as the Editor-in-Chief and Mathematics Editor of the online NIST Digital Library of Mathematical Functions and its 966-page print companion, the NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). … ►

1 Algebraic and Analytic Methods

…

7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space

V

is called an inner product space if an inner product

\left\langle u,v\right\rangle\in\mathbb{C}

is defined for all

u,v\in V

with the properties: (i)

\left\langle u,v\right\rangle

is complex linear in

u

; (ii)

\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}

; (iii)

\left\langle v,v\right\rangle\geq 0

; (iv) if

\left\langle v,v\right\rangle=0

then

v=0

. …Two elements

u

and

v

V

are orthogonal if

\left\langle u,v\right\rangle=0

. … ►Functions

f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)

for which

\left\langle f-g,f-g\right\rangle=0

are identified with each other. … ►,

\left\langle u_{\lambda},u_{\lambda^{\prime}}\right\rangle=0

, for

\lambda\neq\lambda^{\prime}

. … ►The adjoint

{T}^{*}

T

does satisfy

\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle

where

\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x

. …

8: DLMF Project News

error generating summary

9: 1.2 Elementary Algebra

… ►

1.2.41

\left\langle\mathbf{u},\mathbf{v}\right\rangle=\overline{\left\langle\mathbf{v% },\mathbf{u}\right\rangle},

ⓘ

Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E41
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.43

\left\langle\mathbf{v},\mathbf{v}\right\rangle=0,

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $\mathbf{v}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E43
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.44

\left\langle\mathbf{u},\mathbf{v}\right\rangle=0.

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : column vector
Permalink:: http://dlmf.nist.gov/1.2.E44
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii). … ►Numerical methods and issues for solution of (1.2.72) appear in §§3.2(iv) to 3.2(vii). …

10: Viewing DLMF Interactive 3D Graphics

… ►Below we provide some notes and links to online material which might be helpful in viewing our visualizations, but please see our Disclaimer. …