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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. P erm issio n is g ranted to q uo te b rief passag es fro m this pub licatio n in rev iew s, pro v ided the custo m ary ack no w ledg m ent o f the so urce is g iv en. R epub licatio n, sy stem atic co py ing , o r m ultiple repro ductio n o f any m aterial in this pub licatio n is perm itted o nly under license fro m the A m erican Mathem atical S o ciety. R eq uests fo r such perm issio n sho uld b e addressed to the A cq uisitio ns D epartm ent, A m erican Mathem atical S o ciety, 2 0 1 C harles S treet, P ro v idence, R ho de Island 0 2 90 4 -2 2 94 , U S A . R eq uests can also b e m ade b y e-m ail to [email protected]. c 2 0 0 8 b y the A m erican Mathem atical S o ciety. A ll rig hts reserv ed. T he A m erican Mathem atical S o ciety retains all rig hts ex cept tho se g ranted to the U nited S tates G o v ernm ent. P rinted in the U nited S tates o f A m erica. ∞ T he paper used in this b o o k is acid-free and falls w ithin the g uidelines estab lished to ensure perm anence and durab ility. V isit the A MS ho m e pag e at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 13 12 11 10 0 9 0 8 God made the integers, all else is the work of man. L eopold K ronecker, 1 8 8 6 (quoted in [Web], [Bell,p.477] and [Boy,p.617]) S tring theory carries the seeds of a basic change in ou r ideas abou t spacetime and in other fu ndamental notions of p hysics. E dward W itten, 19 9 6 [Wit15 ,p.2 4] Contents Preface xiii A ck n o w led g em en ts xv ii C red its xxiii O v erv iew xxv A b o u t th e C o v er xxix C h ap ter 1 . In tro d u ctio n 1 1 .1 . A rith m etic an d S p acetim e G eo m etry 1 1 .2 . R iem an n ian , Q u an tu m an d N o n co m m u tativ e G eo m etry 2 1 .3 . S trin g T h eo ry an d S p acetim e G eo m etry 3 1 .4 . T h e R iem an n H y p o th esis an d th e G eo m etry o f th e Prim es 6 1 .5 . M o tiv atio n s, O b jectiv es an d O rg an izatio n o f T h is B o o k 9 C h ap ter 2 . S trin g T h eo ry o n a C ircle an d T -D u ality : A n alo g y w ith th e R iem an n Z eta F u n ctio n 2 1 2 .1 . Q u an tu m M ech an ical Po in t-Particle o n a C ircle 2 4 2 .2 . S trin g T h eo ry o n a C ircle: T -D u ality an d th e E xisten ce o f a F u n d am en tal L en g th 2 6 2 .2 .1 . String Theory on a Circle 2 9 2 .2 .2 . Circle D u ality (T-D u ality for Circu lar Spacetim es) 3 4 2 .2 .3 . T-D u ality and the E xistence of a F u nd am ental L ength 4 3 2 .3 . N o n co m m u tativ e S trin g y S p acetim es an d T -D u ality 4 5 2 .3 .1 . Target Space D u ality and N oncom m u tative G eom etry 4 8 2 .3 .2 . N oncom m u tative Stringy Spacetim es : F ock Spaces, V ertex A lgebras and Chiral D irac O perators 5 5 2 .4 . A n alo g y w ith th e R iem an n Z eta F u n ctio n : F u n ctio n al E q u atio n an d T -D u ality 6 6 2 .4 .1 . K ey P roperties of the R iem ann Z eta F u nction: E u ler P rod u ct and F u nctional E qu ation 6 7 2 .4 .2 . The F u nctional E qu ation, T -D u ality and the R iem ann H ypothesis 7 5 2 .5 . N o tes 8 0 C h ap ter 3 . F ractal S trin g s an d F ractal M em b ran es 8 9 ix x CONTENTS 3.1. Fractal Strings: Geometric Zeta Functions, Complex Dimensions and Self-Similarity 91 3.1.1. The Spectrum of a Fractal String 102 3.2. Fractal (and Prime) Membranes: Spectral Partition Functions and Euler Products 103 3.2.1. Prime M embranes 104 3.2.2. Fractal M embranes and Euler Products 109 3.3. Fractal Membranes vs. Self-Similarity: Self-Similar Membranes 123 3.3.1. Partition Functions View ed as Dynamical Zeta Functions 138 3.4. Notes 145 Chapter 4. Noncommutative Models of Fractal Strings: Fractal Membranes and Beyond 155 4.1. Connes Spectral Triple for Fractal Strings 157 4.2. Fractal Membranes and the Second Quantization of Fractal Strings 160 4.2.1. An Alternative Construction of Fractal M embranes 165 4.3. Fractal Membranes and Noncommutative Stringy Spacetimes 170 4.4. Towards a Cohomological and Spectral Interpretation of (Dynamical) Complex Dimensions 174 4.4.1. Fractal M embranes and Q uantum Deformations: A Possible Connection w ith Harans Real and Finite Primes 183 4.5. Notes 183 Chapter 5. Towards an Arithmetic Site: Moduli Spaces of Fractal Strings and Membranes 197 5.1. The Set of Penrose Tilings: Quantum Space as a Quotient Space 200 5.2. The Moduli Space of Fractal Strings: A Natural Receptacle for Zeta Functions 205 5.3. The Moduli Space of Fractal Membranes: A Quantized Moduli Space of Fractal Strings 208 5.4. Arithmetic Site, Frobenius Flow and the Riemann Hypothesis 215 5.4.1. The M oduli Space of Fractal Strings and Deningerss Arithmetic Site 217 5.4.2. The M oduli Space of Fractal M embranes and (Noncommutative) M odular Flow vs.Arithmetic Site and Frobenius Flow 219 5.4.2a. Factors and Their Classifi cation 220 5.4.2b. M odular Theory of von Neumann Algebras 223 5.4.2c. Type III Factors: Discrete vs.Continuous Flow s 227 5.4.2d. M odular Flow s and the Riemann Hypothesis 231 5.4.2e. Tow ards an Extended M oduli Space and Flow 236 5.5. Flows of Zeros and Zeta Functions: A Dynamical Interpretation of the Riemann Hypothesis 241 5.5.1. Introduction 241 5.5.2. Expected Properties of the Flow s of Zeros and Zeta Functions 243 CONTENTS xi 5.5.3. Analogies with Other Geometric, Analytical or Physical Flows 254 5.5.3a. Singular Potentials, Feynman Integrals and Renormalization Flow 255 5.5.3b. KMS-Flow and Deformation of Po´ lya Hilbert Operators 261 5.5.3c.
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