## Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal StringsNumber theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Key Features of this Second Edition: The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula The method of Diophantine approximation is used to study self-similar strings and flows Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions Throughout, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Fractal Geometry, Complex Dimensions and Zeta Functions, Second Edition will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics. Review of the First Edition: "In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actual and has many applications." -- Nicolae-Adrian Secelean for Zentralblatt MATH |

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### Contents

1 | |

1 Complex Dimensions of Ordinary Fractal Strings | 9 |

2 Complex Dimensions of SelfSimilar Fractal Strings | 33 |

3 Complex Dimensions of Nonlattice SelfSimilar Strings Quasiperiodic Patterns and Diophantine Approximation | 65 |

4 Generalized Fractal Strings Viewed as Measures | 118 |

5 Explicit Formulas for Generalized Fractal Strings | 137 |

6 The Geometry and the Spectrum of Fractal Strings | 179 |

7 Periodic Orbits of SelfSimilar Flows | 213 |

12 Fractality and Complex Dimensions | 333 |

13 New Results and Perspectives | 373 |

Appendix A Zeta Functions in Number Theory | 485 |

Appendix B Zeta Functions of Laplacians and Spectral Asymptotics | 497 |

Appendix C An Application of Nevanlinna Theory | 505 |

Acknowledgements | 511 |

Bibliography | 515 |

548 | |

8 Fractal Tube Formulas | 236 |

9 Riemann Hypothesis and Inverse Spectral Problems | 271 |

10 Generalized Cantor Strings and their Oscillations | 283 |

11 Critical Zeros of Zeta Functions | 296 |

553 | |

563 | |

Conventions | 567 |