Fractal Geometry and Number Theory
Complex Dimensions of Fractal Strings and Zeros of Zeta Functions
Michel L. Lapidus, University of California, Riverside, CA
Machiel van Frankenhuysen, University of California, Riverside, CA
0-8176-4098-3 * 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 dimension as 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 an explicit 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} = . 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 can hear if a fractal string is Minkowski measurable provided that its fractal dimension is not ".

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

2. Complex Dimensions of Self-Similar Fractal Strings

3. Generalized Fractal Strings Viewed as Measures

4. Explicit Formulas for Generalized Fractal Strings

5. The Geometry and the Spectrum of Fractal Strings

6. Tubular Neighborhoods and Minkowski Measurability

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

8. Generalized Cantor Strings and their Oscillations

9. The Critical Zeros of Zeta Functions

10. Concluding Comments

Appendices

A. Zeta Functions in Number Theory

B. Zeta Functions of Laplacians and Spectral Asymptotics

References

Conventions

Symbol Index

Index

List of Figures

Acknowledgements