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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: 19.15 Advantages of Symmetry

… ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

3: 19.33 Triaxial Ellipsoids

… ►The surface area of an ellipsoid with semiaxes $a,b,c$, and volume $V=4\pi abc/3$ is given by … ►If a conducting ellipsoid with semiaxes $a,b,c$ bears an electric charge $Q$, then the equipotential surfaces in the exterior region are confocal ellipsoids: … ►Let a homogeneous magnetic ellipsoid with semiaxes $a,b,c$, volume $V=4\pi abc/3$, and susceptibility $\chi$ be placed in a previously uniform magnetic field $H$ parallel to the principal axis with semiaxis $c$. …

4: Bibliography B

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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.

ⓘ

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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5: Possible Errors in DLMF

… ►Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …

6: 19.37 Tables

… ►Here ${\sigma }^2=\frac{2}{3}({(\ln a)}^2+{(\ln b)}^2+{(\ln c)}^2)$, $\cos \left(3\gamma \right)=(4/{\sigma }^3)(\ln a)(\ln b)(\ln c)$, and $a,b,c$ are semiaxes of an ellipsoid with the same volume as the unit sphere. …

7: 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. …

8: Errata

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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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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

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Originally a minus sign was missing in the entries for $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime }$). The correct entries are $-k^{-1}\mathrm{ns}z$ and $-k\mathrm{sn}z$. 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 $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

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

Reported 2014-02-28 by Svante Janson.

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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.

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9: 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. …

10: Foreword

… ►The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium. …