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

Bibliographic Details
Main Authors: Lapidus, Michel L., van Frankenhuysen, Machiel (Author)
Format: eBook
Language:English
Published: Boston, MA Birkhäuser 2000, 2000
Edition:1st ed. 2000
Subjects:
Number Theory
Geometry, Differential
Algebraic Geometry
Differential Geometry
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
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
  • 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
  • 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

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