Preface ...................................................... xiii
Acknowledgements ............................................. xvii
Credits ..................................................... xxiii
Overview ...................................................... xxv
About the Cover .............................................. xxix
Chapter 1. Introduction ......................................... 1
1.1. Arithmetic and Spacetime Geometry .......................... 1
1.2. Riemannian, Quantum and Noncommutative Geometry ............ 2
1.3. String Theory and Spacetime Geometry ....................... 3
1.4. The Riemann Hypothesis and the Geometry of the Primes ...... 6
1.5. Motivations, Objectives and Organization of This Book ...... 9
Chapter 2. String Theory on a Circle and T-Duality: Analogy
with the Riemann Zeta Function ...................... 21
2.1. Quantum Mechanical Point-Particle on a Circle ............. 24
2.2. String Theory on a Circle: T-Duality and the Existence
of a Fundamental Length ................................... 26
2.2.1. String Theory on a Circle .......................... 29
2.2.2. Circle Duality (T-Duality for Circular
Spacetimes) ........................................ 34
2.2.3. T-Duality and the Existence of a Fundamental
Length ............................................. 43
2.3. Noncommutative Stringy Spacetimes and T-Duality ........... 45
2.3.1. Target Space Duality and Noncommutative Geometry ... 48
2.3.2. Noncommutative Stringy Spacetimes: Fock Spaces,
Vertex Algebras and Chiral Dirac Operators ......... 55
2.4. Analogy with the Riemann Zeta Function: Functional
Equation and T-Duality .................................... 66
2.4.1. Key Properties of the Riemann Zeta Function:
Euler Product and Functional Equation .............. 67
2.4.2. The Functional Equation, T-Duality and
the Riemann Hypothesis ............................. 75
2.5. Notes ..................................................... 80
Chapter 3. Fractal Strings and Fractal Membranes ............... 89
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 Membranes ................................... 104
3.2.2. Fractal Membranes and Euler Products .............. 109
3.3. Fractal Membranes vs. Self-Similarity: Self-Similar
Membranes ................................................ 123
3.3.1. Partition Functions Viewed 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
Membranes ......................................... 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 Membranes and Quantum Deformations:
A Possible Connection with Haran's 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 Moduli Space of Fractal Strings and
Deningers's Arithmetic Site ...................... 217
5.4.2. The Moduli Space of Fractal Membranes and
(Noncommutative) Modular Flow vs. Arithmetic
Site and Frobenius Flow .......................... 219
5.4.2a. Factors and Their Classification ................. 220
5.4.2b. Modular Theory of von Neumann Algebras ........... 223
5.4.2c. Type III Factors: Discrete vs. Continuous
Flows ............................................ 227
5.4.2d. Modular Flows and the Riemann Hypothesis ......... 231
5.4.2e. Towards an Extended Moduli 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 Flows of Zeros and
Zeta Functions ................................... 243
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 Polya-Hilbert
Operators ........................................ 261
5.5.3c. Ricci Flow and Geometric Symmetrization
(vs. Modular Flow and Arithmetic
Symmetrization) .................................. 270
5.5.3d. Towards a Noncommutative, Arithmetic KP-Flow ..... 293
5.6. Notes .................................................... 294
Appendix A. Vertex Algebras ................................... 315
A.l. Definition of Vertex Algebras: Translation and
Scaling Operators ................................... 315
A.2. Basic Properties of Vertex Algebras ................. 318
A.3. Notes ............................................... 321
Appendix B. The Weil Conjectures and the Riemann Hypothesis ... 325
B.l. Varieties Over Finite Fields and Their Zeta
Functions ........................................... 325
B.2. Zeta Functions of Curves Over Finite Fields and
the Riemann Hypothesis .............................. 335
B.3. The Weil Conjectures for Varieties Over Finite
Fields .............................................. 338
B.4. Notes ............................................... 344
Appendix C. The Poisson Summation Formula, with Applications .. 347
C.l. General PSF for Dual Lattices: Scalar Identity and
Distributional Form ................................. 348
C.2. Application: Modularity of Theta Functions .......... 350
C.3. Key Special Case: Self-Dual PSF ..................... 352
C.4. Proof of the General Poisson Summation Formula ...... 356
C.5. Modular Forms and Their Hecke L-Series .............. 358
C.5.1. Modular Forms and Cusp Forms ................. 358
C.5.2. Hecke Operators and Hecke Forms .............. 361
C.5.3. Hecke L-Series of a Modular Form ............. 362
C.5.4. Modular Forms of Higher Level and Their
L-Functions .................................. 366
C.6. Notes ............................................... 371
Appendix D. Generalized Primes and Beurling Zeta Functions .... 373
D.l. Generalized Primes  and Integers  ................ 373
D.2. Beurling Zeta Functions ζp .......................... 374
D.3. Analogues of the Prime Number Theorem ............... 375
D.4. Analytic Continuation and a Generalized
Functional Equation for ζp .......................... 378
D.5. Partial Orderings on Generalized Integers ........... 385
D.6. Notes ............................................... 387
Appendix E. The Selberg Class of Zeta Functions ............... 389
E.l. Definition of the Selberg Class ..................... 390
E.2. The Selberg Conjectures ............................. 393
E.3. Selected Consequences ............................... 394
E.4. The Selberg Class, Artin ISeries and Automorphic
L-Functions: Langlands' Reciprocity Laws ............ 398
E.4.1. Selberg's Orthonormality Conjecture and
Artin L-Series: Artin's Holomorphy
Conjecture .................................. 399
E.4.2. Selberg's Orthonormality Conjecture and
Automorphic Representations: Langlands'
Reciprocity Conjecture ...................... 400
E.4.2a. Adeles кA and Linear Group GLn{kA) .......... 400
E.4.2b. Automorphic Representations and
Automorphic L-Series ........................ 401
E.5. Notes ............................................... 407
Appendix F. The Noncommutative Space of Penrose Tilings and
Quasicrystals ..................................... 411
F.l. Combinatorial Coding of Penrose Tilings, and
Consequences ........................................ 412
F.2. Groupoid C*-Algebra and the Noncommutative Space
of Penrose Tilings .................................. 416
F.2.1. Groupoids: Definition and Examples ........... 416
F.2.2. The Groupoid Convolution Algebra ............. 420
F.2.3. Generalization: Groupoids, Quasicrystals
and Noncommutative Spaces .................... 424
F.3. Quasicrystals: Dynamical Hull and the
Noncommutative Brillouin Zone ....................... 427
F.3.1. Mathematical Quasicrystals and Their
Generalizations .............................. 427
F.3.2. Translation Dynamical System: The Hull of
a Quasicrystal ............................... 437
F.3.3. Typical Properties of Atomic
Configurations ............................... 444
F.3.4. The Noncommutative Brillouin Zone and
Groupoid C* -Algebra of a Quasicrystal ....... 446
F.4. Notes ............................................... 449
Bibliography .................................................. 453
Conventions ................................................... 491
Index of Symbols .............................................. 493
Subject Index ................................................. 503
Author Index .................................................. 551
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