Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

277 Pages·2000·11.76 MB·English

by  Michel L. Lapidus

To my Riemann zeros:
Odile, Julie and Michae"l, my muse and offspring
Michel L" Lapidus
To my father and mother,
Ad van Frankenhuysen and Mieke Arts
:Machiel van Frankenhuysen
Michel L. Lapidus
Machiel van Frankenhuysen
Fractal Geometry
and Number Theory
Complex Dimensions of Fractal Strings
and Zeros of Zeta Functions
with 26 illustrations
Birkhauser
Boston • Basel • Berlin
Michel L. Lapidus Machiel van Frankenhuysen
University of California University of California
Department of Mathematics Department of Mathematics
Sproul Hall Sproul Hall
Riverside, CA 92521-0135 Riverside, CA 92521-0135
USA USA
[e-mail] [e-mail]
Library of Congress Cataloging-in-Publication Data
Lapidus, Michel L. (Michel Laurent), 1956-
Fractal geometry and number theory: complex dimensions of fractal strings and zeros
of zeta functions / Michel L. Lapidus, Machiel van Frankenhuysen.
p. cm.
Includes bibliographical references and indexes.
(acid-free paper)
1. Fractals. 2. Number theory. 3. Functions, Zeta. I. van Frankenhuysen, Machiel,
1967- II. Title.
OA614.86.L361999
514'.742-dc21 99-051583
CIP
AMS Subject Classifications: Primary-llM26, llM41, 28A75, 28A80, 35P20, 58G25
Secondary-llM06, llN05, 28A12, 30D35, 58F19, 58F20, 81020
ro
Printed on acid-free paper ®
©2000 Birkhauser Boston Birkhauser ll@
Softcover reprint of the hardcover I st edition 2000
All rights reserved. This work may not be translated or copied in whole or in part without the written
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The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by the
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ISBN-13: 978-1-4612-5316-7 e-ISBN-I3: 978-1-4612-5314-3
001: 10.1007/978-1-4612-5314-3
Typeset by the authors in It\TEX.
Cover design by Jeff Cosloy, Newton, MA.
The front cover shows the complex dimensions of the golden string. See page 35, Figure 2.6.
Printed and bound by Hamilton Printing, Rensselaer, NY.
9 8 765 432 1
Contents
Overview ix
Introduction 1
1 Complex Dimensions of Ordinary Fractal Strings 7
1.1 The Geometry of a Fractal String . . . . 7
1.1.1 The Multiplicity of the Lengths. . . . . . 10
1.1.2 Example: The Cantor String ....... 11
1.2 The Geometric Zeta Function of a Fractal String 13
1.2.1 The Screen and the Window ....... 14
1.2.2 The Cantor String (Continued) . . . . . . 16
1.3 The Frequencies of a Fractal String and the Spectral Zeta
Function . . . . . . . . . . . . . . . . . . . . . 17
1.4 Higher-Dimensional Analogue: Fractal Sprays . . . . . 19
2 Complex Dimensions of Self-Similar Fractal Strings 23
2.1 The Geometric Zeta Function of a Self-Similar String. 23
2.1.1 Dynamical Interpretation, Euler Produc1: ... 26
2.2 Examples of Complex Dimensions of Self-Similar Strings. 28
2.2.1 The Cantor String . . . . . . 28
2.2.2 The Fibonacci String. . . . . 28
2.2.3 A String with Multiple Poles 30
2.2.4 Two Nonlattice Examples 31
2.3 The Lattice and Nonlattice Case . . 34
vi Contents
2.3.1 Generic Nonlattice Strings. . . . . . . . . 36
2.4 The Structure of the Complex Dimensions . . . . 37
2.5 The Density of the Poles in the Nonlattice Case. 42
2.5.1 Nevanlinna Theory . . . . . . . . . . . . . 42
2.5.2 Complex Zeros of Dirichlet Polynomials . 43
2.6 Approximating a Fractal String and Its Complex Dimensions 47
2.6.1 Approximating a Nonlattice String by Lattice Strings 49
3 Generalized Fractal Strings Viewed as Measures 55
3.1 Generalized Fractal Strings . . . . . . . . . . . . 55
3.1.1 Examples of Generalized Fractal Strings . 58
3.2 The Frequencies of a Generalized Fractal String. 60
3.3 Generalized Fractal Sprays ............ 64
3.4 The Measure of a Self-Similar String . . . . . . . 65
3.4.1 Measures with a Self-Similarity Property 67
4 Explicit Formulas for Generalized Fractal Strings 71
4.1 Introduction......... 71
4.1.1 Outline of the Proof . . . . . . 73
4.1.2 Examples............ 74
4.2 Preliminaries: The Heaviside Function 76
4.3 The Pointwise Explicit Formulas . . . 79
4.3.1 The Order of the Sum over the Complex Dimensions 89
4.4 The Distributional Explicit Formulas . . . . . . . . 90
4.4.1 Alternate Proof of Theorem 4.12 . . . . . . . 95
4.4.2 Extension to More General Test Functions. . 95
4.4.3 The Order of the Distributional Error Term . 99
4.5 Example: The Prime Number Theorem ... 106
4.5.1 The Riemann-von Mangoldt Formula . . . . 108
5 The Geometry and the Spectrum of Fractal Strings 111
5.1 The Local Terms in the Explicit Formulas 112
5.1.1 The Geometric Local Terms. 112
5.1.2 The Spectral Local Terms . 113
5.1.3 The Weyl Term. . . . . . . . 114
5.1.4 The Distribution XW logm x . 114
5.2 Explicit Formulas for Lengths and Frequencies 115
5.2.1 The Geometric Counting Function of a Fractal String 115
5.2.2 The Spectral Counting Function of a Fractal String 116
5.2.3 The Geometric and Spectral Partition Functions 118
5.3 The Direct Spectral Problem for Fractal Strings. . . . 119
5.3.1 The Density of Geometric and Spectral States 119
5.3.2 The Spectral Operator. 121
5.4 Self-Similar Strings. . 121
5.4.1 Lattice Strings . . . . . 122
Contents vii
5.4.2 Nonlattice Strings ............... . .. 126
5.4.3 The Spectrum of a Self-Similar String . . . . . .. 127
5.4.4 The Prime Number Theorem for Suspended Flows 130
5.5 Examples of Non-Self-Similar Strings . . . . . 132
5.5.1 The a-String . . . . . . . . . . . . . . 133
5.5.2 The Spectrum of the Harmonic String 136
5.6 Fractal Sprays. . . . . . . . . . . . . . . . . . 136
5.6.1 The Sierpinski Drum. . . . . . . . . . 138
5.6.2 The Spectrum of a Self-Similar Spray 141
6 Tubular Neighborhoods and Minkowski Measurability 143
6.1 Explicit Formula for the Volume of a Tubular Neighborhood 144
6.1.1 Analogy with Riemannian Geometry. . . . 147
6.2 Minkowski Measurability and Complex Dimensions 148
6.3 Examples . . . . . . . . . . 153
6.3.1 Self-Similar Strings. 154
6.3.2 The a-String . . . . 160
1 The Riemann Hypothesis, Inverse Spectral Problems
and Oscillatory Phenomena 163
7.1 The Inverse Spectral Problem . . . . . . . . . . . . . . . 163
7.2 Complex Dimensions of Fractal Strings and the Riemann
Hypothesis .......................... 167
7.3 Fractal Sprays and the Generalized Riemann Hypothesis. 170
8 Generalized Cantor Strings and their Oscillations 113
8.1 The Geometry of a Generalized Cantor String . . . . 173
8.2 The Spectrum of a Generalized Cantor String . . . . 176
8.2.1 Integral Cantor Strings: a-adic Analysis of the
Geometric and Spectral Oscillations . . . . . . 176
8.2.2 Nonintegral Cantor Strings: Analysis of the Jumps
in the Spectral Counting Function . . . . . . . . . . 179
9 The Critical Zeros of Zeta Functions 181
9.1 The Riemann Zeta Function: No Critical Zeros
in an Arithmetic Progression ... . . . . . . 182
9.2 Extension to Other Zeta Functions ...... . 184
9.2.1 Density of Nonzeros on Vertical Lines . 186
9.2.2 Almost Arithmetic Progressions of Zeros. 187
9.3 Extension to L-Series ............ . 188
9.4 Zeta Functions of Curves Over Finite Fields 189
10 Concluding Comments 191
10.1 Conjectures about Zeros of Dirichlet Series 198
10.2 A New Definition of Fractality ...... . 201
viii Contents
10.2.1 Comparison with Other Definitions of Fractality 205
10.2.2 Possible Connections with the Notion of Lacunarity 206
10.3 Fractality and Self-Similarity . . . . 208
10.4 The Spectrum of a Fractal Drum . . . . . . . 212
10.4.1 The Weyl-Berry Conjecture. . . . . . 212
10.4.2 The Spectrum of a Self-Similar Drum 214
10.4.3 Spectrum and Periodic Orbits. . . . . 217
10.5 The Complex Dimensions as Geometric Invariants 219
Appendices
A Zeta Functions in Number Theory 221
A.1 The Dedekind Zeta Function ......... 221
A.2 Characters and Hecke L-series. . . . . . . . . 222
A.3 Completion of L-Series, Functional Equation 223
A.4 Epstein Zeta Functions. . . . . . . . . . . 224
A.5 Other Zeta Functions in Number Theory. . . 225
B Zeta Functions of Laplacians and Spectral Asymptotics 221
B.l Weyl's Asymptotic Formula. . . . . . . . 227
B.2 Heat Asymptotic Expansion. . . . . . . . 229
B.3 The Spectral Zeta Function and Its Poles 231
B.4 Extensions. . . . . . . . . . . . . 232
B.4.1 Monotonic Second Term. . . . . . 233
References 235
Conventions 253
Symbol Index 254
Index 251
List of Figures 265
Acknowledgements 261
Overview
In this book, we develop a theory of complex dimensions of fractal strings
(Le., one-dimensional drums with fractal boundary). These complex dimen
sions are defined as the poles of the corresponding (geometric or spectral)
zeta function. They describe the oscillations in the geometry or the fre
quency spectrum of a fractal string by means of an explicit formula.
A long-term objective of this work is to merge aspects of fractal, spectral,
and arithmetic geometries. From this perspective, the theory presented in
this book enables us to put the theory of Dirichlet series (and of other zeta
functions) in the geometric setting of fractal strings. It also allows us to
view certain fractal geometries as arithmetic objects by applying number
theoretic methods to the study of the geometry and the spectrum of fractal
strings.
In Chapter 1, we first give an introduction to fractal strings and their
spectrum, and we precisely define the notion of complex dimension. We
then make an extensive study of the complex dimensions of self-similar
fractal strings. This study provides a large class of examples to which our
theory can be applied fruitfully. In particular, we show in the latter part
of Chapter 2 that self-similar strings always have infinitely many complex
dimensions with positive real part, and that their complex dimensions are
almost periodically distributed. This is established by proving that the
lattice strings-the complex dimensions of which are shown to be periodi
cally distributed along finitely many vertical lines-are dense (in a suitable
sense) in the set of all self-similar strings.
In Chapter 3, we extend the notion of fractal string to include (possibly
virtual) geometries that are needed later on in our work. Then, in Chap-