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1: 25.10 Zeros

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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

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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

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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

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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

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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

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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: George E. Andrews

… ►He holds honorary degrees from the Universities of Parma, Florida, Waterloo, Illinois, and SASTRA University (India). Andrews served as President of the AMS from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012. …

4: Ranjan Roy

… ► 1948 in Secunderabad, India, d. …(1974) degrees in mathematics from the Indian Institute of Technology Kharagpur, Indian Institute of Technology Kanpur, and the State University of New York at Stony Brook, respectively. … ►He also authored another two advanced mathematics books: Sources in the development of mathematics (Roy, 2011), Elliptic and modular functions from Gauss to Dedekind to Hecke (Roy, 2017). …

5: Errata

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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).

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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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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).

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The title was changed from Physical to Physical Applications.

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6: Joyce E. Conlon

… ► 2007) served as a NIST staff member from 1985 until her retirement in 2005. … ►in Mathematics from the University of Maryland Baltimore County in 1979. From 1980–85 she worked as a computer programmer for the Defense Mapping Agency. …

7: Roderick S. C. Wong

… ►He was the Dean of the Faculty of Science and Engineering from 1998 to 2004. From 2004 to 2006, he was Dean of Research and Graduate Studies. From 2006 to 2009, he was Vice-President (Research)/Dean of Graduate Studies. From 2009 to early 2015, he served as Vice-President for Development and External Relations. … ►He was recipient of the Killam Research Fellowship from the Canada Council in 1982–1984, and of the Rh Award for Outstanding Contributions to Scholarship and Research from the University of Manitoba in 1984. …

8: 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. …

9: Marjorie A. McClain

… ►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.

10: Gergő Nemes

… ►in mathematics (with honours) from Loránd Eötvös University, Budapest, Hungary and a Ph. … in mathematics from Central European University in Budapest, Hungary. …