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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: 8.24 Physical Applications

… ►The function $I_x(a,b)$ appears in: Monte Carlo sampling in statistical mechanics (Kofke (2004)); analysis of packings of soft or granular objects (Prellberg and Owczarek (1995)); growth formulas in cosmology (Hamilton (2001)). …

6: 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. …

7: Diego Dominici

… ►He was elected as Program Director for the period 2011–2016 and served as OPSF-Talk moderator from 2010–2022 with Bonita Saunders, and co-editor for OPSF-Net from 2006–2015 with Martin Muldoon. ►In 2008 Dominici received a Research Fellowship from the Alexander von Humboldt Foundation and visited the Technische Universität Berlin in Germany. …

8: Charles W. Clark

… ►in mathematics and physics from Western Washington University, and a Ph. …in physics from the University of Chicago. … ►He has been a Visiting Fellow at the Australian National University, a Dr. Lee Fellow at Christ Church College of the University of Oxford, and Visiting Professor at the National University of Singapore. …

9: Leonard C. Maximon

… ►He had been an Assistant Professor in the Graduate Division of Applied Mathematics at Brown University, a Visiting Professor at the Norwegian Technical University in Trondheim, Norway, and a Physicist at the Center for Radiation Research at the National Bureau of Standards. … ►Maximon was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …

10: Tom M. Apostol

… ►He received his bachelor of science in chemical engineering in 1944 and a master’s degree in mathematics in 1946, both from the University of Washington, Seattle. … in mathematics from the University of California, Berkeley. … ►He was a visiting professor at the University of Patras in Greece in 1978, and was elected a Corresponding Member of the Academy of Athens in 2001 (where he delivered his inaugural lecture in Greek). … ►In 1982, Apostol received an award for teaching excellence from the Associated Students of the California Institute of Technology. …He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …