Fractal Geometry and Number TheoryComplex Dimensions of Fractal Strings and Zeros of Zeta Functions

Michel L. Lapidus, University of California, Riverside, CA

Machiel van Frankenhuysen, University of California, Riverside, CA0-8176-4098-3 * 2000 * $54.95 * Hardcover * 280 pages * 26 Illustrations

In this book, the authors take the viewpoint that number theory and fractal geometry can be fruitfully combined. They study, in particular, the vibrations of fractal strings (one-dimensional drums with fractal boundary) and zeros of zeta-functions.

In earlier publications on fractal and spectral geometry, the Riemann Hypothesis was studied and this hinted at the notion of

complex dimensionas a means to describe certain geometric properties of a fractal, such as its fractal (Minkowski) dimension or the oscillations in the volume of its tubular neighborhoods. This notion of complex dimension is now precisely defined in this book.A central problem in contemporary mathematics-often expressed as

"Can one hear the shape of a drum?"-- consists in describing the relationship between the shape (geometry) of a drum and its sound (its spectrum). In the case of fractal strings, the complex dimensions provide a unified description of the oscillations in the geometry and the spectrum. This description is provided by anexplicit formula-- an analytical tool, originally developed for the proof of the Prime Number Theorem, which is extended here to apply to the zeta-functions associated with fractals.The context of vibrating fractal strings enables the authors to put the Riemann Hypothesis in a geometric setting. This famous conjecture states that the zeros r in the critical strip 0 {\leq} Re

{\rho}{\leq} 1 of the Riemann zeta-function all lie on the critical line Re{\rho}= ½. Here, this conjecture becomes an inverse spectral problem, and its interpretation in the language of fractal strings, which have complex dimensions with real part between 0 and 1, is "One canhearif a fractal string is Minkowski measurable provided that its fractal dimension is not ½".In the more restricted context of fractal Cantor strings, the complex dimensions of which form an infinite vertical arithmetic progression, the inverse spectral problem gets an affirmative answer. The number-theoretical interpretation of this insight is that the Riemann zeta-function does not have an infinite vertical arithmetic progression of zeros. This result is generalized to apply to many other zeta-functions.

This highly original, self-contained monograph will appeal to geometers, fractalists, mathematical physicists, and number theorists, as well as to graduate students in these fields.

Contents

Overview

Introduction

1. Complex Dimensions of Ordinary Fractal Strings

1.1 The Geometry of a Fractal String

1.1.1The Multiplicity of the Lengths

1.1.2 Example: The Cantor String

1.2 The Geometric Zeta Function of a Fractal String

1.2.1 The Screen and the Window

1.2.2 The Cantor String (Continued)

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

1.4 Higher-Dimensional Analogue: Fractal Sprays

2. Complex Dimensions of Self-Similar Fractal Strings

2.1 The Geometric Zeta Function of a Self-Similar String

2.1.1 Dynamical Interpretation, Euler Product

2.2 Examples of Complex Dimensions of Self-Similar Strings

2.2.1 The Cantor String

2.2.2 The Fibonacci String

2.2.3 A String with Multiple Poles

2.2.4 Two Nonlattice Examples

2.3 The Lattice and Nonlattice Case

2.3.1 Generic Nonlattice Strings

2.4 The Structure of the Complex Dimensions

2.5 The Density of the Poles in the Nonlattice Case

2.5.1 Nevanlinna Theory

2.5.2 Complex Zeros of Dirichlet Polynomials

2.6 Approximating a Fractal String and Its Complex Dimensions

2.6.1 Approximating a Nonlattice String by Lattice Strings

3. Generalized Fractal Strings Viewed as Measures

3.1 Generalized Fractal Strings

3.1.1 Examples of Generalized Fractal Strings

3.2 The Frequencies of a Generalized Fractal String

3.3 Generalized Fractal Sprays

3.4 The Measure of a Self-Similar String

3.4.1 Measures with a Self-Similarity Property

4. Explicit Formulas for Generalized Fractal Strings

4.1 Introduction

4.1.1 Outline of the Proof

4.1.2 Examples

4.2 Preliminaries: The Heaviside Function

4.3 The Pointwise Explicit Formulas

4.3.1 The Order of the Sum over the Complex Dimensions

4.4 The Distributional Explicit Formulas

4.4.1 Alternative Proof of Theorem 4.12

4.4.2 Extension to More General Test Functions

4.4.3 The Order of the Distributional Error Term

4.5 Example: The Prime Number Theorem

4.5.1 The Riemann-von Mangoldt Formula

5. The Geometry and the Spectrum of Fractal Strings

5.1 The Local Terms in the Explicit Formulas

5.1.1 The Geometric Local Terms

5.1.2 The Spectral Local Terms

5.1.3 The Weyl Term

5.1.4 The Distributionxlog^{w}^{m}x

5.2 Explicit Formulas for Lengths and Frequencies

5.2.1 The Geometric Counting Function of a Fractal String

5.2.2 The Spectral Counting Function of a Fractal String

5.2.3 The Geometric and Spectral Partition Functions

5.3 The Direct Spectral Problem for Fractal Strings

5.3.1 The Density of Geometric and Spectral States

5.3.2 The Spectral Operator

5.4 Self-Similar Strings

5.4.1 Lattice Strings

5.4.2 Nonlattice Strings

5.4.3 The Spectrum of a Self-Similar String

5.4.4 The Prime Number Theorem for Suspended Flows

5.5 Examples of Non-Self-Similar Strings

5.5.1 The

a-String

5.5.2 The Spectrum of the Harmonic String

5.6 Fractal Sprays

5.6.1 The Sierpinski Drum

5.6.2 The Spectrum of a Self-Similar Spray

6. Tubular Neighborhoods and Minkowski Measurability

6.1 Explicit Formula for the Volume of a Tubular Neighborhood

6.1.1 Analogy with Riemannian Geometry

6.2 Minkowski Measurability and Complex Dimensions

6.3 Examples

6.3.1 Self-Similar Strings

6.3.2 Thea-String

7. The Riemann Hypothesis, Inverse Spectral Problems and Oscillatory Phenomena

7.1 The Inverse Spectral Problem

7.2 Complex Dimensions of Fractal Strings and the Riemann Hypothesis

7.3 Fractal Sprays and the Generalized Riemann Hypothesis

8. Generalized Cantor Strings and their Oscillations

8.1 The Geometry of a Generalized Cantor String

8.2 The Spectrum of a Generalized Cantor String

8.2.1 Integral Cantor Strings:

a-adic Analysis of the Geometric and Spectral Oscillations

8.2.2 Nonintegral Cantor Strings: Analysis of the Jumps in the Spectral Counting Function

9. The Critical Zeros of Zeta Functions

9.1 The Riemann Zeta Function: No Critical Zeros in an Arithmetic Progression

9.2 Extension to Other Zeta Functions

9.2.1 Density of Nonzeros on Vertical Lines

9.2.2 Almost Arithmetic Progressions of Zeros

9.3 Extension toL-Series

9.4 Zeta Functions of Curves Over Finite Fields

10. Concluding Comments

10.1 Conjectures about Zeros of Dirichlet Series

10.2 A New Definition of Fractality

10.2.1 Comparison with Other Definitions of Fractality

10.2.2 Possible Connections with the Notion of Lacunarity

10.3 Fractality and Self-Similarity

10.4 The Spectrum of a Fractal Drum

10.4.1 The Weyl-Berry Conjecture

10.4.2 The Spectrum of a Self-Similar Drum

10.4.3 Spectrum and Periodic Orbits

10.5 The Complex Dimensions as Geometric Invariants

Appendices

A. Zeta Functions in Number Theory

A.1 The Dedekind Zeta Function

A.2 Characters and HeckeL-Series

A.3 Completion ofL-Series, Functional Equation

A.4 Epstein Zeta Functions

A.5 Other Zeta Functions in Number Theory

B. 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.4.1 Monotonic Second Term

References

Conventions

Symbol Index

Index

List of Figures

Acknowledgements