The Physics and Computational Exploration of Zeta and L-Functions Chris King Feb 2016 Genotype 1.1.5
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In Search of the Riemann Zeros
In Search of the Riemann Zeros Strings, Fractal Membranes and Noncommutative Spacetimes In Search of the Riemann Zeros Strings, Fractal Membranes and Noncommutative Spacetimes Michel L. Lapidus 2000 Mathematics Subject Classification. P rim a ry 1 1 A 4 1 , 1 1 G 20, 1 1 M 06 , 1 1 M 26 , 1 1 M 4 1 , 28 A 8 0, 3 7 N 20, 4 6 L 5 5 , 5 8 B 3 4 , 8 1 T 3 0. F o r a d d itio n a l in fo rm a tio n a n d u p d a tes o n th is b o o k , v isit www.ams.org/bookpages/mbk-51 Library of Congress Cataloging-in-Publication Data Lapidus, Michel L. (Michel Laurent), 1956 In search o f the R iem ann zero s : string s, fractal m em b ranes and no nco m m utativ e spacetim es / Michel L. Lapidus. p. cm . Includes b ib lio g raphical references. IS B N 97 8 -0 -8 2 18 -4 2 2 2 -5 (alk . paper) 1. R iem ann surfaces. 2 . F unctio ns, Z eta. 3 . S tring m o dels. 4 . N um b er theo ry. 5. F ractals. 6. S pace and tim e. 7 . G eo m etry. I. T itle. Q A 3 3 3 .L3 7 2 0 0 7 515.93 dc2 2 2 0 0 7 0 60 8 4 5 Cop ying and rep rinting. Indiv idual readers o f this pub licatio n, and no npro fi t lib raries acting fo r them , are perm itted to m ak e fair use o f the m aterial, such as to co py a chapter fo r use in teaching o r research. -
The Weil Conjectures for Curves
The Weil Conjectures for Curves Caleb Ji Summer 2021 1 Introduction We will explain Weil’s proof of his famous conjectures for curves. For the Riemann hypothe- sis, we will follow Grothendieck’s argument [1]. The main tools used in these proofs are basic results in algebraic geometry: Riemann-Roch, intersection theory on a surface, and the Hodge index theorem. For references: in Section 2 we used [3], while the rest can be found in Hartshorne [2] V.1 and Appendix C (some of it in the form of exercises). 1.1 Statements of the Weil conjectures We recall the statements. Let X be a smooth projective variety of dimension n over Fq. We define its zeta function by 1 ! X tr Z(X; t) := exp N ; r r r=1 where Nr is the number of closed points of X where considered over Fqr . Theorem 1.1 (Weil conjectures). Use the above notation. 1. (Rationality) Z(X; t) is a rational function of t. 2. (Functional equation) Let E be the Euler characteristic of X considered over C. Then 1 Z = ±qnE=2tEZ(t): qnt 3. (Riemann hypothesis) We can write P (t) ··· P (t) Z(t) = 1 2n−1 P0(t) ··· P2n(t) n where P0(t) = 1 − t; P2n(t) = 1 − q t and all the Pi(t) are integer polynomials that can be written as Y Pi(t) = (1 − αijt): j i=2 Finally, jαijj = q . 4. (Betti numbers) The degree of the polynomials Pi are the Betti numbers of X considered over C. 1 Caleb Ji The Weil Conjectures for Curves Summer 2021 d P1 r Note that dt log Z(X; t) = r=0 Nr+1t . -
Counting Points and Acquiring Flesh
Innovations in Incidence Geometry Volume 00 (XXXX), Pages 000–000 ISSN 1781-6475 Counting points and acquiring flesh Koen Thas Abstract This set of notes is based on a lecture I gave at “50 years of Finite Ge- ometry — A conference on the occasion of Jef Thas’s 70th birthday,” in November 2014. It consists essentially of three parts: in a first part, I in- troduce some ideas which are based in the combinatorial theory underlying F1, the field with one element. In a second part, I describe, in a nutshell, the fundamental scheme theory over F1 which was designed by Deitmar. The last part focuses on zeta functions of Deitmar schemes, and also presents more recent work done in this area. Keywords: Field with one element, Deitmar scheme, loose graph, zeta function, Weyl geometry MSC 2000: 11G25, 11D40, 14A15, 14G15 Contents 1 Introduction 2 2 Combinatorial theory 5 3 Deninger-Manin theory 8 4 Deitmar schemes 10 5 Acquiring flesh (1) 14 arXiv:1508.03997v1 [math.AG] 17 Aug 2015 6 Kurokawa theory 15 7 Graphs and zeta functions 18 2 Thas 8 Acquiring flesh (2) — The Weyl functor depicted 25 1 Introduction For a class of incidence geometries which are defined (for instance coordina- tized) over fields, it often makes sense to consider the “limit” of these geome- tries when the number of field elements tends to 1. As such, one ends up with a guise of a “field with one element, F1” through taking limits of geometries. A general reference for F1 is the recent monograph [21]. -
Arxiv:1604.01256V4 [Math.NT] 14 Aug 2017 As P Tends to Infinity, We Are Happy to Ignore Such Primes, Which Are Necessarily finite in Number
SATO-TATE DISTRIBUTIONS ANDREW V.SUTHERLAND ABSTRACT. In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures≤ presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry. 1. AN INTRODUCTION TO SATO-TATE DISTRIBUTIONS Before discussing the Sato-Tate conjecture and Sato-Tate distributions for abelian varieties, we first consider the more familiar setting of Artin motives, which can be viewed as varieties of dimension zero. 1.1. A first example. Let f Z[x] be a squarefree polynomial of degree d; for example, we may take 3 2 f (x) = x x + 1. Since f has integer coefficients, we can reduce them modulo any prime p to obtain − a polynomial fp with coefficients in the finite field Z=pZ Fp. For each prime p define ' Nf (p) := # x Fp : fp(x) = 0 , f 2 g which we note is an integer between 0 and d. We would like to understand how Nf (p) varies with p. 3 The table below shows the values of Nf (p) when f (x) = x x + 1 for primes p < 60: − p : 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Nf (p) 00111011200101013 There does not appear to be any obvious pattern (and we should know not to expect one, the Galois group lurking behind the scenes is nonabelian). -
Bolles E.B. Einstein Defiant.. Genius Versus Genius in the Quantum
Selected other titles by Edmund Blair Bolles The Ice Finders: How a Poet, a Professor, and a Politician Discovered the Ice Age A Second Way of Knowing: The Riddle of Human Perception Remembering and Forgetting: Inquiries into the Nature of Memory So Much to Say: How to Help Your Child Learn Galileo’s Commandment: An Anthology of Great Science Writing (editor) Edmund Blair Bolles Joseph Henry Press Washington, DC Joseph Henry Press • 500 Fifth Street, NW • Washington, DC 20001 The Joseph Henry Press, an imprint of the National Academies Press, was created with the goal of making books on science, technology, and health more widely available to professionals and the public. Joseph Henry was one of the founders of the National Academy of Sciences and a leader in early American science. Any opinions, findings, conclusions, or recommendations expressed in this volume are those of the author and do not necessarily reflect the views of the National Academy of Sciences or its affiliated institutions. Library of Congress Cataloging-in-Publication Data Bolles, Edmund Blair, 1942- Einstein defiant : genius versus genius in the quantum revolution / by Edmund Blair Bolles. p. cm. Includes bibliographical references. ISBN 0-309-08998-0 (hbk.) 1. Quantum theory—History—20th century. 2. Physics—Europe—History— 20th century. 3. Einstein, Albert, 1879-1955. 4. Bohr, Niels Henrik David, 1885-1962. I. Title. QC173.98.B65 2004 530.12′09—dc22 2003023735 Copyright 2004 by Edmund Blair Bolles. All rights reserved. Printed in the United States of America. To Kelso Walker and the rest of the crew, volunteers all. -
Chao-Dyn/9402001 7 Feb 94
chao-dyn/9402001 7 Feb 94 DESY ISSN Quantum Chaos January Einsteins Problem of The study of quantum chaos in complex systems constitutes a very fascinating and active branch of presentday physics chemistry and mathematics It is not wellknown however that this eld of research was initiated by a question rst p osed by Einstein during a talk delivered in Berlin on May concerning Quantum Chaos the relation b etween classical and quantum mechanics of strongly chaotic systems This seems historically almost imp ossible since quantum mechanics was not yet invented and the phenomenon of chaos was hardly acknowledged by physicists in While we are celebrating the seventyfth anniversary of our alma mater the Frank Steiner Hamburgische Universitat which was inaugurated on May it is interesting to have a lo ok up on the situation in physics in those days Most I I Institut f urTheoretische Physik UniversitatHamburg physicists will probably characterize that time as the age of the old quantum Lurup er Chaussee D Hamburg Germany theory which started with Planck in and was dominated then by Bohrs ingenious but paradoxical mo del of the atom and the BohrSommerfeld quanti zation rules for simple quantum systems Some will asso ciate those years with Einsteins greatest contribution the creation of general relativity culminating in the generally covariant form of the eld equations of gravitation which were found by Einstein in the year and indep endently by the mathematician Hilb ert at the same time In his talk in May Einstein studied the -
Arithmetic Zeta-Function
Arithmetic Zeta-Function Gaurish Korpal1 [email protected] Summer Internship Project Report 14th year Int. MSc. Student, National Institute of Science Education and Research, Jatni (Bhubaneswar, Odisha) Certificate Certified that the summer internship project report \Arithmetic Zeta-Function" is the bona fide work of \Gaurish Korpal", 4th year Int. MSc. student at National Institute of Science Ed- ucation and Research, Jatni (Bhubaneswar, Odisha), carried out under my supervision during June 4, 2018 to July 4, 2018. Place: Mumbai Date: July 4, 2018 Prof. C. S. Rajan Supervisor Professor, Tata Institute of Fundamental Research, Colaba, Mumbai 400005 Abstract We will give an outline of the motivation behind the Weil conjectures, and discuss their application for counting points on projective smooth curves over finite fields. Acknowledgements Foremost, I would like to express my sincere gratitude to my advisor Prof. C. S. Rajan for his motivation. I am also thankful to Sridhar Venkatesh1, Rahul Kanekar 2 and Monalisa Dutta3 for the enlightening discussions. Last but not the least, I would like to thank { Donald Knuth for TEX { Michael Spivak for AMS-TEX { Sebastian Rahtz for TEX Live { Leslie Lamport for LATEX { American Mathematical Society for AMS-LATEX { H`anTh^e´ Th`anhfor pdfTEX { Heiko Oberdiek for hyperref package { Steven B. Segletes for stackengine package { David Carlisle for graphicx package { Javier Bezos for enumitem package { Hideo Umeki for geometry package { Peter R. Wilson & Will Robertson for epigraph package { Jeremy Gibbons, Taco Hoekwater and Alan Jeffrey for stmaryrd package { Lars Madsen for mathtools package { Philipp Khl & Daniel Kirsch for Detexify (a tool for searching LATEX symbols) { TeX.StackExchange community for helping me out with LATEX related problems 1M.Sc. -
Front Matter
Cambridge University Press 978-0-521-88508-9 - New Directions in Linear Acoustics and Vibration: Quantum Chaos, Random Matrix Theory, and Complexity Edited by Matthew Wright and Richard Weaver Frontmatter More information NEW DIRECTIONS IN LINEAR ACOUSTICS AND VIBRATION The field of acoustics is of immense industrial and scientific importance. The subject is built on the foundations of linear acoustics, which is widely re- garded as so mature that it is fully encapsulated in the physics texts of the 1950s. This view was changed by developments in physics such as the study of quantum chaos. Developments in physics throughout the last four decades, often equally applicable to both quantum and linear acoustic problems but overwhelmingly more often expressed in the language of the former, have explored this. There is a significant new amount of theory that can be used to address problems in linear acoustics and vibration, but only a small amount of reported work does so. This book is an attempt to bridge the gap be- tween theoreticians and practitioners, as well as the gap between quantum and acoustic. Tutorial chapters provide introductions to each of the major aspects of the physical theory and are written using the appropriate termi- nology of the acoustical community. The book will act as a quick-start guide to the new methods while providing a wide-ranging introduction to the phys- ical concepts. Matthew Wright is a senior lecturer in Acoustics at the Institute of Sound and Vibration Research (ISVR). His B.Eng. was in engineering acoustics and vibration, and his Ph.D. -
Geometry of Chaos
Part I Geometry of chaos e start out with a recapitulation of the basic notions of dynamics. Our aim is narrow; we keep the exposition focused on prerequisites to the applications to W be developed in this text. We assume that the reader is familiar with dynamics on the level of the introductory texts mentioned in remark 1.1, and concentrate here on developing intuition about what a dynamical system can do. It will be a broad stroke description, since describing all possible behaviors of dynamical systems is beyond human ken. While for a novice there is no shortcut through this lengthy detour, a sophisticated traveler might bravely skip this well-trodden territory and embark upon the journey at chapter 18. The fate has handed you a law of nature. What are you to do with it? 1. Define your dynamical system (M; f ): the space M of its possible states, and the law f t of their evolution in time. 2. Pin it down locally–is there anything about it that is stationary? Try to determine its equilibria / fixed points (chapter2). 3. Cut across it, represent as a return map from a section to a section (chapter3). 4. Explore the neighborhood by linearizing the flow; check the linear stability of its equilibria / fixed points, their stability eigen-directions (chapters4 and5). 5. Does your system have a symmetry? If so, you must use it (chapters 10 to 12). Slice & dice it (chapter 13). 6. Go global: train by partitioning the state space of 1-dimensional maps. Label the regions by symbolic dynamics (chapter 14). -
The Sound of Fractal Strings and the Riemann Hypothesis
The Sound of Fractal Strings and the Riemann Hypothesis Michel L. Lapidus Abstract We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly dis- cuss closely related developments, including the theory of (fractal) complex dimen- sions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his col- laborators). Key words: Riemann zeta function, Riemann hypothesis (RH), quantization, quan- tized number theory,fractal strings, geometry and spectra, direct and inverse spectral problems for fractal strings, Minkowski dimension, Minkowski measurability, com- plex dimensions, Weyl–Berry conjecture, fractal drums, infinitesimal shift, spec- tral operator, invertbility, quantized Dirichlet series and Euler product, universality, phase transitions, symmetric and asymmetric criteria for RH. arXiv:1505.01548v1 [math-ph] 7 May 2015 Suggested short title for the paper: ”Fractal Strings and the Riemann Hypothesis.” University of California Department of Mathematics 900 University Avenue Riverside, CA 92521–0135, USA e-mail: [email protected] 1 Contents The Sound of Fractal Strings and the Riemann Hypothesis............. 1 Michel L. Lapidus 1 Riemann Zeros and Spectra of Fractal Strings: An Informal Introduction....................................... ....... 4 2 FractalStringsandMinkowskiDimension ............... ..... 7 3 Characterization of Minkowski Measurability and Nondegeneracy 9 4 TheWeyl–BerryConjectureforFractalDrums............ ..... 13 4.1 Weyl’s asymptotic formula with sharp error term for fractaldrums .................................... -
The Sound of Fractal Strings and the Riemann Hypothesis
The Sound of Fractal Strings and the Riemann Hypothesis Michel L. LAPIDUS Institut des Hautes Etudes´ Scientifiques 35, route de Chartres 91440 – Bures-sur-Yvette (France) Juillet 2015 IHES/M/15/11 The Sound of Fractal Strings and the Riemann Hypothesis Michel L. Lapidus Abstract We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly dis- cuss closely related developments, including the theory of (fractal) complex dimen- sions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his col- laborators). Key words: Riemann zeta function, Riemann hypothesis (RH), quantization, quan- tized number theory, fractal strings, geometry and spectra, direct and inverse spectral problems for fractal strings, Minkowski dimension, Minkowski measurability, com- plex dimensions, Weyl–Berry conjecture, fractal drums, infinitesimal shift, spec- tral operator, invertbility, quantized Dirichlet series and Euler product, universality, phase transitions, symmetric and asymmetric criteria for RH. Suggested short title for the paper: ”Fractal Strings and the Riemann Hypothesis.” University of California Department of Mathematics 900 University Avenue Riverside, CA 92521–0135, USA e-mail: [email protected] 1 Contents The Sound of Fractal Strings and the Riemann Hypothesis............. 1 Michel L. Lapidus 1 Riemann Zeros and Spectra of Fractal Strings: An Informal Introduction . -
Chaos: Classical and Quantum Volume II: Semiclassicalchaos Chapter 31
Chaos: Classical and Quantum Volume II: SemiclassicalChaos Chapter 31 Prologue Anyone who uses words “quantum” and “chaos” in the same sentence should be hung by his thumbs on a tree in the park behind the Niels Bohr Institute. —Joseph Ford (G. Vattay, G. Tanner and P. Cvitanovi´c) ou have read the first volume of this book. So far, so good – anyone can play a game of classical pinball, and a skilled neuroscientist can poke Y rat brains. We learned that information about chaotic dynamics can be obtained by calculating spectra of linear operators such as the evolution operator of sect. 17.2 or the associated partial differential equations such as the Liouville equation (16.37). The spectra of these operators can be expressed in terms of periodic orbits of the deterministic dynamics by means of trace formulas and cycle expansions. But what happens quantum mechanically, i.e., if we scatter waves rather than point-like pinballs? Can we turn the problem round and study linear PDE’s in terms of the underlying deterministic dynamics? And, is there a link between structures in the spectrum or the eigenfunctions of a PDE and the dynamical properties of the underlying classical flow? The answer is yes, but ... things Predrag Cvitanovic´ – Roberto Artuso – Per Dahlqvist – Ronnie Mainieri – are becoming somewhat more complicated when studying 2nd or higher order Gregor Tanner – Gabor´ Vattay – Niall Whelan – Andreas Wirzba linear PDE’s. We can find classical dynamics associated with a linear PDE, just take geometric optics as a familiar example. Propagation of light follows a second order wave equation but may in certain limits be well described in terms of geometric rays.