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

Contents

Preface

List of Figures

List of Tables

Overview

Introduction

Complex Dimensions of Ordinary Fractal Strings

1.1 The Geometry of a Fractal String

1.2 The Geometric Zeta Function of a Fractal String

1.3 The Frequencies of a Fractal String and the Spectral Zeta Function

1.4 Higher-Dimensional Analogue: Fractal Sprays

1.5 Notes

Complex Dimensions of Self- Similar Fractal Strings

2.1 Construction of a Self-Similar Fractal String

2.2 The Geometric Zeta Function of a Self- Similar String

2.3 Examples of Complex Dimensions of Self- Similar Strings

2.4 The Lattice and Nonlattice Case

2.5 The Structure of the Complex Dimensions

2.6 The Asymptotic Density of the Poles in the Nonlattice Case

2.7 Notes

Complex Dimensions of Nonlattice Self- Similar Strings: Quasiperiodic Patterns and Diophantine Approximation

3.1 Dirichlet Polynomial Equations

3.2 Examples of Dirichlet Polynomial Equations

3.3 The Structure of the Complex Roots

3.4 Approximating a Nonlattice Equation by Lattice Equations

3.5 Complex Roots of a Nonlattice Dirichlet Polynomial

3.6 Dimension-Free Regions

3.7 The Dimensions of Fractality of a Nonlattice String

3.8 A Note on the Computations

Generalized Fractal Strings Viewed as Measures

4.1 Generalized Fractal Strings

4.2 The Frequencies of a Generalized Fractal String

4.3 Generalized Fractal Sprays

4.4 The Measure of a Self-Similar String

4.5 Notes

Explicit Formulas for Generalized Fractal Strings

5.1 Introduction

5.2 Preliminaries: The Heaviside Function

5.3 Pointwise Explicit Formulas

5.4 Distributional Explicit Formulas

5.5 Example: The Prime Number Theorem

5.6 Notes

The Geometry and the Spectrum of Fractal Strings

6.1 The Local Terms in the Explicit Formulas

6.2 Explicit Formulas for Lengths and Frequencies

6.3 The Direct Spectral Problem for Fractal Strings

6.4 Self-Similar Strings

6.5 Examples of Non-Self-Similar Strings

6.6 Fractal Sprays

Periodic Orbits of Self-Similar Flows

7.1 Suspended Flows

7.2 Periodic Orbits, Euler Product

7.3 Self-Similar Flows

7.4 The Prime Orbit Theorem for Suspended Flows

7.5 The Error Term in the Nonlattice Case

7.6 Notes

Tubular Neighborhoods and Minkowski Measurability

8.1 Explicit Formulas for the Volume of Tubular Neighborhoods

8.2 Analogy with Riemannian Geometry

8.3 Minkowski Measurability and Complex Dimensions

8.4 Tube Formulas for Self-Similar Strings

8.5 Notes

The Riemann Hypothesis and Inverse Spectral Problems

9.1 The Inverse Spectral Problem

9.2 Complex Dimensions of Fractal Strings and the Riemann Hypothesis

9.3 Fractal Sprays and the Generalized Riemann Hypothesis

9.4 Notes

Generalized Cantor Strings and their Oscillations

10.1 The Geometry of a Generalized Cantor String

10.2 The Spectrum of a Generalized Cantor String

10.3 The Truncated Cantor String

10.4 Notes

The Critical Zeros of Zeta Functions

11.1 The Riemann Zeta Function: No Critical Zeros in Arithmetic Progression

11.2 Extension to Other Zeta Functions

11.3 Density of Nonzeros on Vertical Lines

11.4 Extension to L-Series

11.5 Zeta Functions of Curves Over Finite Fields

Concluding Comments, Open Problems, and Perspectives

12.1 Conjectures about Zeros of Dirichlet Series

12.2 A New Definition of Fractality

12.3 Fractality and Self-Similarity

12.4 Random and Quantized Fractal Strings

12.5 The Spectrum of a Fractal Drum

12.6 The Complex Dimensions as the Spectrum of Shifts

12.7 The Complex Dimensions as Geometric Invariants

12.8 Notes

Zeta Functions in Number Theory

A.1 The Dedekind Zeta Function

A.2 Characters and Hecke L-series

A.3 Completion of L-Series, Functional Equation

A.4 Epstein Zeta Functions

A.5 Two-Variable Zeta Functions

A.6 Other Zeta Functions in Number Theory

Zeta Functions of Laplacians and Spectral Asymptotics

B.1 Weyl’s Asymptotic Formula

B.2 Heat Asymptotic Expansion

B.3 The Spectral Zeta Function and its Poles

B.4 Extensions

B.5 Notes

An Application of Nevanlinna Theory

C.1 The Nevanlinna Height

C.2 Complex Zeros of Dirichlet Polynomials

Bibliography

Acknowledgements

Conventions

Index of Symbols

Author Index

Subject Index