Fractal Geometry, Complex Dimensions and Zeta Functions (eBook)

Contents 6
Preface 12
List of Figures 16
List of Tables 19
Overview 20
Introduction 23
Complex Dimensions of Ordinary Fractal Strings 31
1.1 The Geometry of a Fractal String 31
1.2 The Geometric Zeta Function of a Fractal String 38
1.3 The Frequencies of a Fractal String and the Spectral Zeta Function 45
1.4 Higher-Dimensional Analogue: Fractal Sprays 48
1.5 Notes 51
Complex Dimensions of Self- Similar Fractal Strings 54
2.1 Construction of a Self-Similar Fractal String 54
2.2 The Geometric Zeta Function of a Self- Similar String 59
2.3 Examples of Complex Dimensions of Self- Similar Strings 62
2.4 The Lattice and Nonlattice Case 72
2.5 The Structure of the Complex Dimensions 75
2.6 The Asymptotic Density of the Poles in the Nonlattice Case 82
2.7 Notes 83
Complex Dimensions of Nonlattice Self- Similar Strings: Quasiperiodic Patterns and Diophantine Approximation 84
3.1 Dirichlet Polynomial Equations 85
3.2 Examples of Dirichlet Polynomial Equations 87
3.3 The Structure of the Complex Roots 92
3.4 Approximating a Nonlattice Equation by Lattice Equations 99
3.5 Complex Roots of a Nonlattice Dirichlet Polynomial 111
3.6 Dimension-Free Regions 122
3.7 The Dimensions of Fractality of a Nonlattice String 129
3.8 A Note on the Computations 133
Generalized Fractal Strings Viewed as Measures 135
4.1 Generalized Fractal Strings 136
4.2 The Frequencies of a Generalized Fractal String 141
4.3 Generalized Fractal Sprays 146
4.4 The Measure of a Self-Similar String 146
4.5 Notes 151
Explicit Formulas for Generalized Fractal Strings 152
5.1 Introduction 152
5.2 Preliminaries: The Heaviside Function 157
5.3 Pointwise Explicit Formulas 161
5.4 Distributional Explicit Formulas 173
5.5 Example: The Prime Number Theorem 189
5.6 Notes 192
The Geometry and the Spectrum of Fractal Strings 194
6.1 The Local Terms in the Explicit Formulas 195
6.2 Explicit Formulas for Lengths and Frequencies 199
6.3 The Direct Spectral Problem for Fractal Strings 203
6.4 Self-Similar Strings 208
6.5 Examples of Non-Self-Similar Strings 217
6.6 Fractal Sprays 221
Periodic Orbits of Self-Similar Flows 227
7.1 Suspended Flows 228
7.2 Periodic Orbits, Euler Product 230
7.3 Self-Similar Flows 233
7.4 The Prime Orbit Theorem for Suspended Flows 239
7.5 The Error Term in the Nonlattice Case 244
7.6 Notes 248
Tubular Neighborhoods and Minkowski Measurability 250
8.1 Explicit Formulas for the Volume of Tubular Neighborhoods 251
8.2 Analogy with Riemannian Geometry 260
8.3 Minkowski Measurability and Complex Dimensions 261
8.4 Tube Formulas for Self-Similar Strings 266
8.5 Notes 281
The Riemann Hypothesis and Inverse Spectral Problems 284
9.1 The Inverse Spectral Problem 285
9.2 Complex Dimensions of Fractal Strings and the Riemann Hypothesis 288
9.3 Fractal Sprays and the Generalized Riemann Hypothesis 291
9.4 Notes 293
Generalized Cantor Strings and their Oscillations 295
10.1 The Geometry of a Generalized Cantor String 295
10.2 The Spectrum of a Generalized Cantor String 298
10.3 The Truncated Cantor String 303
10.4 Notes 307
The Critical Zeros of Zeta Functions 308
11.1 The Riemann Zeta Function: No Critical Zeros in Arithmetic Progression 309
11.2 Extension to Other Zeta Functions 318
11.3 Density of Nonzeros on Vertical Lines 320
11.4 Extension to L-Series 322
11.5 Zeta Functions of Curves Over Finite Fields 331
Concluding Comments, Open Problems, and Perspectives 340
12.1 Conjectures about Zeros of Dirichlet Series 342
12.2 A New Definition of Fractality 345
12.3 Fractality and Self-Similarity 355
12.4 Random and Quantized Fractal Strings 369
12.5 The Spectrum of a Fractal Drum 384
12.6 The Complex Dimensions as the Spectrum of Shifts 393
12.7 The Complex Dimensions as Geometric Invariants 393
12.8 Notes 399
Zeta Functions in Number Theory 402
A.1 The Dedekind Zeta Function 402
A.2 Characters and Hecke L-series 403
A.3 Completion of L-Series, Functional Equation 404
A.4 Epstein Zeta Functions 405
A.5 Two-Variable Zeta Functions 406
A.6 Other Zeta Functions in Number Theory 411
Zeta Functions of Laplacians and Spectral Asymptotics 413
B.1 Weyl’s Asymptotic Formula 413
B.2 Heat Asymptotic Expansion 415
B.3 The Spectral Zeta Function and its Poles 416
B.4 Extensions 418
B.5 Notes 420
An Application of Nevanlinna Theory 421
C.1 The Nevanlinna Height 422
C.2 Complex Zeros of Dirichlet Polynomials 423
Bibliography 427
Acknowledgements 452
Conventions 456
Index of Symbols 457
Author Index 461
Subject Index 463