عنوان

Fractal Geometry and Number Theory

9781461253143

9781461253167

b402552

Fractal Geometry and Number Theory

[Book]

Complex Dimensions of Fractal Strings and Zeros of Zeta Functions /

by Michel L. Lapidus, Machiel Frankenhuysen.

Boston, MA :

Birkhäuser Boston,

2000.

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.

0

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.

9781461253167

Springer eBooks

Geometry, Algebraic.

Global differential geometry.

Mathematics.

Number theory.

Lapidus, Michel L.

Frankenhuysen, Machiel.

SpringerLink (Online service)

20190301082100.0

[Book]

Y