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Birkhauser Fractal Geometry and Number Theory Complex Dimensions of Fractal Strings and Zeros of Zeta Functions by Michel L. Lapidus, Machiel van Frankenhuysen
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._x000D_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._x000D_ Table of contents : - _x000D_
1 Complex Dimensions of Ordinary Fractal Strings.- 1.1 The Geometry of a Fractal String.- 1.1.1 The 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 Alternate 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 Distribution x?logmx.- 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 The a-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 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.- 10.2.1 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.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._x000D_