Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings

Contents......Page 6
Preface......Page 12
List of Figures......Page 16
List of Tables......Page 19
Overview......Page 20
Introduction......Page 23
1.1 The Geometry of a Fractal String......Page 31
1.1.1 The Multiplicity of the Lengths......Page 34
1.1.2 Example: The Cantor String......Page 35
1.2 The Geometric Zeta Function of a Fractal String......Page 38
1.2.1 The Screen and the Window......Page 40
1.2.2 The Cantor String (continued)......Page 44
1.3 The Frequencies of a Fractal String and the Spectral Zeta Function......Page 45
1.4 Higher-Dimensional Analogue: Fractal Sprays......Page 48
1.5 Notes......Page 51
2.1 Construction of a Self-Similar Fractal String......Page 54
2.1.1 Relation with Self-Similar Sets......Page 56
2.2 The Geometric Zeta Function of a Self-Similar String......Page 59
2.2.1 Self-Similar Strings with a Single Gap......Page 61
2.3.1 The Cantor String......Page 62
2.3.2 The Fibonacci String......Page 64
2.3.3 The Modified Cantor and Fibonacci Strings......Page 67
2.3.5 Two Nonlattice Examples: the Two-Three String and the Golden String......Page 68
2.4 The Lattice and Nonlattice Case......Page 72
2.5 The Structure of the Complex Dimensions......Page 75
2.6 The Asymptotic Density of the Poles in the Nonlattice Case......Page 82
2.7 Notes......Page 83
3. Complex Dimensions of Nonlattice Self-Similar Strings: Quasiperiodic Patterns and Diophantine Approximation......Page 84
3.1 Dirichlet Polynomial Equations......Page 85
3.1.1 The Generic Nonlattice Case......Page 86
3.2.1 Generic and Nongeneric Nonlattice Equations......Page 87
3.3 The Structure of the Complex Roots......Page 92
3.4 Approximating a Nonlattice Equation by Lattice Equations......Page 99
3.4.1 Diophantine Approximation......Page 102
3.4.2 The Quasiperiodic Pattern of the Complex Dimensions......Page 105
3.4.3 Application to Nonlattice Strings......Page 108
3.5.1 Continued Fractions......Page 111
3.5.2 Two Generators......Page 113
3.5.3 More than Two Generators......Page 119
3.6 Dimension-Free Regions......Page 122
3.7.1 The Density of the Real Parts......Page 129
3.8 A Note on the Computations......Page 133
4. Generalized Fractal Strings Viewed as Measures......Page 135
4.1 Generalized Fractal Strings......Page 136
4.1.1 Examples of Generalized Fractal Strings......Page 139
4.2 The Frequencies of a Generalized Fractal String......Page 141
4.2.1 Completion of the Harmonic String: Euler Product......Page 144
4.4 The Measure of a Self-Similar String......Page 146
4.4.1 Measures with a Self-Similarity Property......Page 148
4.5 Notes......Page 151
5.1 Introduction......Page 152
5.1.1 Outline of the Proof......Page 154
5.1.2 Examples......Page 155
5.2 Preliminaries: The Heaviside Function......Page 157
5.3 Pointwise Explicit Formulas......Page 161
5.3.1 The Order of the Sum over the Complex Dimensions......Page 172
5.4 Distributional Explicit Formulas......Page 173
5.4.1 Extension to More General Test Functions......Page 178
5.4.2 The Order of the Distributional Error Term......Page 182
5.5 Example: The Prime Number Theorem......Page 189
5.5.1 The Riemann–von Mangoldt Formula......Page 191
5.6 Notes......Page 192
6. The Geometry and the Spectrum of Fractal Strings......Page 194
6.1.1 The Geometric Local Terms......Page 195
6.1.2 The Spectral Local Terms......Page 197
6.1.4 The Distribution x[sup(ω)] log[sup(m)] x......Page 198
6.2.1 The Geometric Counting Function of a Fractal String......Page 199
6.2.2 The Spectral Counting Function of a Fractal String......Page 200
6.2.3 The Geometric and Spectral Partition Functions......Page 201
6.3.1 The Density of Geometric and Spectral States......Page 203
6.3.2 The Spectral Operator and its Euler Product......Page 205
6.4 Self-Similar Strings......Page 208
6.4.1 Lattice Strings......Page 209
6.4.2 Nonlattice Strings......Page 212
6.4.3 The Spectrum of a Self-Similar String......Page 214
6.5.1 The α-String......Page 217
6.5.2 The Spectrum of the Harmonic String......Page 220
6.6 Fractal Sprays......Page 221
6.6.1 The Sierpinski Drum......Page 222
6.6.2 The Spectrum of a Self-Similar Spray......Page 225
7. Periodic Orbits of Self-Similar Flows......Page 227
7.1 Suspended Flows......Page 228
7.1.1 The Zeta Function of a Dynamical System......Page 229
7.2 Periodic Orbits, Euler Product......Page 230
7.3 Self-Similar Flows......Page 233
7.3.1 Examples of Self-Similar Flows......Page 236
7.3.2 The Lattice and Nonlattice Case......Page 238
7.4 The Prime Orbit Theorem for Suspended Flows......Page 239
7.4.1 The Prime Orbit Theorem for Self-Similar Flows......Page 241
7.4.2 Lattice Flows......Page 242
7.4.3 Nonlattice Flows......Page 243
7.5.1 Two Generators......Page 244
7.5.2 More Than Two Generators......Page 245
7.6 Notes......Page 248
8. Tubular Neighborhoods and Minkowski Measurability......Page 250
8.1 Explicit Formulas for the Volume of Tubular Neighborhoods......Page 251
8.1.1 The Pointwise Tube Formula......Page 256
8.1.2 Example: The a-String......Page 259
8.2 Analogy with Riemannian Geometry......Page 260
8.3 Minkowski Measurability and Complex Dimensions......Page 261
8.4.1 Generalized Cantor Strings......Page 266
8.4.2 Lattice Self-Similar Strings......Page 269
8.4.3 The Average Minkowski Content......Page 274
8.4.4 Nonlattice Self-Similar Strings......Page 277
8.5 Notes......Page 281
9. The Riemann Hypothesis and Inverse Spectral Problems......Page 284
9.1 The Inverse Spectral Problem......Page 285
9.2 Complex Dimensions of Fractal Strings and the Riemann Hypothesis......Page 288
9.3 Fractal Sprays and the Generalized Riemann Hypothesis......Page 291
9.4 Notes......Page 293
10.1 The Geometry of a Generalized Cantor String......Page 295
10.2 The Spectrum of a Generalized Cantor String......Page 298
10.2.1 Integral Cantor Strings: a-adic Analysis of the Geometric and Spectral Oscillations......Page 300
10.3 The Truncated Cantor String......Page 303
10.3.1 The Spectrum of the Truncated Cantor String......Page 306
10.4 Notes......Page 307
11. The Critical Zeros of Zeta Functions......Page 308
11.1 The Riemann Zeta Function: No Critical Zeros in Arithmetic Progression......Page 309
11.1.1 Finite Arithmetic Progressions of Zeros......Page 312
11.2 Extension to Other Zeta Functions......Page 318
11.3 Density of Nonzeros on Vertical Lines......Page 320
11.3.1 Almost Arithmetic Progressions of Zeros......Page 321
11.4 Extension to L-Series......Page 322
11.4.1 Finite Arithmetic Progressions of Zeros of L-Series......Page 323
11.5 Zeta Functions of Curves Over Finite Fields......Page 331
12. Concluding Comments, Open Problems, and Perspectives......Page 340
12.1 Conjectures about Zeros of Dirichlet Series......Page 342
12.2 A New Definition of Fractality......Page 345
12.2.1 Fractal Geometers\' Intuition of Fractality......Page 346
12.2.2 Our Definition of Fractality......Page 349
12.2.3 Possible Connections with the Notion of Lacunarity......Page 353
12.3 Fractality and Self-Similarity......Page 355
12.3.1 Complex Dimensions and Tube Formula for the Koch Snowflake Curve......Page 357
12.3.2 Towards a Higher-Dimensional Theory of Complex Dimensions......Page 364
12.4.1 Random Fractal Strings and their Zeta Functions......Page 369
12.4.2 Fractal Membranes: Quantized Fractal Strings......Page 376
12.5.1 The Weyl–Berry Conjecture......Page 384
12.5.2 The Spectrum of a Self-Similar Drum......Page 386
12.5.3 Spectrum and Periodic Orbits......Page 390
12.7 The Complex Dimensions as Geometric Invariants......Page 393
12.7.1 Connection with Varieties over Finite Fields......Page 395
12.7.2 Complex Cohomology of Self-Similar Strings......Page 397
12.8 Notes......Page 399
A.1 The Dedekind Zeta Function......Page 402
A.2 Characters and Hecke L-series......Page 403
A.3 Completion of L-Series, Functional Equation......Page 404
A.4 Epstein Zeta Functions......Page 405
A.5 Two-Variable Zeta Functions......Page 406
A.5.1 The Zeta Function of Pellikaan......Page 407
A.5.2 The Zeta Function of Schoof and van der Geer......Page 409
A.6 Other Zeta Functions in Number Theory......Page 411
B.1 Weyl\'s Asymptotic Formula......Page 413
B.2 Heat Asymptotic Expansion......Page 415
B.3 The Spectral Zeta Function and its Poles......Page 416
B.4 Extensions......Page 418
B.4.1 Monotonic Second Term......Page 419
B.5 Notes......Page 420
C. An Application of Nevanlinna Theory......Page 421
C.1 The Nevanlinna Height......Page 422
C.2 Complex Zeros of Dirichlet Polynomials......Page 423
Bibliography......Page 427
Acknowledgements......Page 452
Conventions......Page 456
Index of Symbols......Page 457
Author Index......Page 461
B......Page 463
C......Page 464
E......Page 465
F......Page 466
L......Page 467
N......Page 468
R......Page 469
S......Page 470
W......Page 471
Z......Page 472