Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions / Edition 1

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Can one hear the shape of a drum? The relationship between the shape (geometry) of a drum and its sound (its spectrum is an interesting and difficult problem to describe. In this monograph, an extensive study is made of this problem for one-dimensional drums with fractal strings. The work centers around the author's original research on a newly developed theory of "complex dimensions of fractal strings." All of the material is self-contained and provide a unique geometric characterization of the Riemann hypothesis.
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Editorial Reviews

From the Publisher

"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)

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Product Details

  • ISBN-13: 9780817640989
  • Publisher: Birkhauser Verlag
  • Publication date: 12/10/1999
  • Edition description: 1999
  • Edition number: 1
  • Pages: 284
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.75 (d)

Table of Contents

Introduction 1
1 Complex Dimensions of Ordinary Fractal Strings 7
2 Complex Dimensions of Self-Similar Fractal Strings 23
3 Generalized Fractal Strings Viewed as Measures 55
4 Explicit Formulas for Generalized Fractal Strings 71
5 The Geometry and the Spectrum of Fractal Strings 111
6 Tubular Neighbourhoods and Minkowski Measurability 143
7 The Riemann Hypothesis, Inverse Spectral Problems and Oscillatory Phenomena 163
8 Generalized Cantor Strings and their Oscillations 173
9 The Critical Zeros of Zeta Functions 181
10 Concluding Comments 197
A Zeta Functions in Number Theory 221
B Zeta Functions of Laplacians and Spectral Asymptotics 227
References 235
Conventions 253
Symbol Index 254
Index 257
List of Figures 265
Acknowledgements 267
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