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

A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo- metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di- mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref- erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap- pendix B. ) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex- tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.

Table of Contents:
1 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.- 2 Complex Dimensions of Self-Similar Fractal Strings.- 2.1 The Geometric Zeta Function of a Self-Similar String.- 2.2 Examples of Complex Dimensions of Self-Similar Strings.- 2.3 The Lattice and Nonlattice Case.- 2.4 The Structure of the Complex Dimensions.- 2.5 The Density of the Poles in the Nonlattice Case.- 2.6 Approximating a Fractal String and Its Complex Dimensions.- 3 Generalized Fractal Strings Viewed as Measures.- 3.1 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.- 4 Explicit Formulas for Generalized Fractal Strings.- 4.1 Introduction.- 4.2 Preliminaries: The Heaviside Function.- 4.3 The Pointwise Explicit Formulas.- 4.4 The Distributional Explicit Formulas.- 4.5 Example: The Prime Number Theorem.- 5 The Geometry and the Spectrum of Fractal Strings.- 5.1 The Local Terms in the Explicit Formulas.- 5.2 Explicit Formulas for Lengths and Frequencies.- 5.3 The Direct Spectral Problem for Fractal Strings.- 5.4 Self-Similar Strings.- 5.5 Examples of Non-Self-Similar Strings.- 5.6 Fractal Sprays.- 6 Tubular Neighborhoods and Minkowski Measurability.- 6.1 Explicit Formula for the Volume of a Tubular Neighborhood.- 6.2 Minkowski Measurability and Complex Dimensions.- 6.3 Examples.- 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.- 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.- 9.3 Extension to L-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.- 10.3 Fractality and Self-Similarity.- 10.4 The Spectrum of a Fractal Drum.- 10.5 The Complex Dimensions as Geometric Invariants.- Appendices.- A Zeta Functions in Number Theory.- A.l The Dedekind Zeta Function.- A.3 Completion of L-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.l 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.- List of Figures.- Acknowledgements.

Review :
"This highly original self-contained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style." -Mathematical Reviews (Review of First Edition) "It is the reviewer's opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is self-contained, intelligent and well paced." -Bulletin of the London Mathematical Society (Review of First Edition) "The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics." -Simulation News Europe (Review of First Edition)