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1: 25.10 Zeros

… ►Calculations relating to the zeros on the critical line make use of the real-valued function … ►

25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

pMML, png, TeX

See also:

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

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

2: Bibliography B

… ►

The Scattering of Electromagnetic Waves from Rough Surfaces. Pergamon Press, New York.

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Notes:

Reprinted in 1987 by Artech House Publishers, Norwood, Massachusetts.

External Links:

ISBN 0-89006-238-2, ZentralBlatt

Referenced by:

§10.73(i).

Encodings:

BibTeX

… ►

Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.

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External Links:

ISSN 1431-0643, FullText, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

… ►

Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces. J. Phys. A 8 (4), pp. 566–584.

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External Links:

Document, MathReview

Referenced by:

§36.14(iii).

Encodings:

BibTeX

… ►

Focusing and twinkling: Critical exponents from catastrophes in non-Gaussian random short waves. J. Phys. A 10 (12), pp. 2061–2081.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§36.6.

Encodings:

BibTeX

… ►

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

…

3: Barry I. Schneider

… ► in chemistry from Brooklyn College, his M. … in chemistry from Yale University and a Ph. … in theoretical chemistry from the University of Chicago. … ►Schneider was awarded a Poste Rouge by the CNRS in 1980, was elected a Fellow of the American Physical Society (APS) in 1983 and received the prestigious Humboldt prize from the German government in 1987. He was a visiting scientist at NIST from 1995 to 2013 and spent a sabbatical year at NIST in 2000-2001. …

4: Brian Antonishek

… ►Antonishek received the Bachelor of Science degree in computer science from the University of Pittsburgh, Johnstown, Pennsylvania, in 1990 and the Master of Science degree in computer science from the University of Pittsburgh, Pittsburgh, Pennsylvania, in 1996. …His research areas include human robot interaction, medical data anlysis, and 3D web interactions using VRML and X3D.

5: Marjorie A. McClain

… ►More recently she has become an expert in the use of LaTeX and BibTex, and she has applied those skills in the preparation of the Latex documents for the DLMF chapters and in the development of the DLMF bibliography. … ►McClain received a degree in mathematics from Bucknell University in 1978 and an M. …in mathematics from the University of Maryland, College Park, in 1984.

6: Errata

… ►

The paragraph immediately below (9.8.23) was updated to include new information from Nemes (2021) pertaining to (9.8.22) and (9.8.23).

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The paragraph immediately following (10.18.21) was updated to include new information from Nemes (2021) pertaining to (10.18.17) and (10.18.18).

… ►

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).

… ►

The statements “If $c$ and $d$ are real” and “If $c$ and $d$ are not both real” have been further clarified (suggested by Alan Barnes on 2021-03-26).

… ►

The title was changed from Physical to Physical Applications.

…

7: 5.21 Methods of Computation

… ►An effective way of computing $\mathrm{\Gamma }\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). Or we can use forward recurrence, with an initial value obtained e. …from (5.7.3). For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). … ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …

8: Morris Newman

… ► 2007) received his early training in mathematics from New York University and Columbia University, obtaining his Ph. …from the University of Pennsylvania in 1952. From 1952–1977 he served as a research mathematician at the National Bureau of Standards. …Department of Commerce Gold Medal in 1966 for his work on algorithms for solving integral linear systems exactly by using congruence techniques. … ►Newman was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …

9: Bruce R. Miller

… ►D in experimental physics from the University of Texas at Austin in 1983. … ►He is the developer of the tools used to process the DLMF into both book and web forms. In particular, he developed the LaTeXML system used for converting the LaTeX source documents into XML and MathML from which the DLMF website is constructed. …

10: 27.17 Other Applications

… ►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. …Apostol and Zuckerman (1951) uses congruences to construct magic squares. … ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.