Table of contents

  1. Front Matter
    Pages i-xxvi
  2. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 9-32
  3. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 33-63
  4. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 119-135
  5. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 137-178
  6. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 179-212
  7. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 213-235
  8. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 237-270
  9. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 271-281
  10. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 283-295
  11. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 297-332
  12. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 333-371
  13. Michel L. Lapidus, Machiel van Frankenhuijsen
    Pages 373-483
  14. Back Matter
    Pages 485-567

About this book

Introduction

Number 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. This second edition of Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, complex analysis, distribution theory, and mathematical physics. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level.

Key Features include:

·         The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings

·         Complex dimensions of a fractal string are studied in detail, and 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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

Key Features include:

·         The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings

·         Complex dimensions of a fractal string are studied in detail, and 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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

·         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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

 

Key Features include:

·         The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings

·         Complex dimensions of a fractal string are studied in detail, and 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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

·         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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

 

·         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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

 

 

Key Features include:

·         The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings

·         Complex dimensions of a fractal string are studied in detail, and 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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

·         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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

 

·         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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

 

 

 

·         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

The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Review of the First Edition:

" The book is self contained, the material organized in chapters preceded 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 actually has many applications."

—Nicolae-Adrian Secelean, Zentralblatt

Keywords

Riemann hypothesis cantor strings complex dimensions fractality inverse spectral problems minkowski measurability nonlattice self-similar strings self-similar flows tubular neighborhoods

Authors and affiliations

  • Michel L. Lapidus
    • 1
  • Machiel van Frankenhuijsen
    • 2
  1. 1.Dept. MathematicsUniversity of California, RiversideRiversideUSA
  2. 2.Department of MathematicsUtah Valley State CollegeOremUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-2176-4
  • Copyright Information Springer Science+Business Media New York 2013
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-2175-7
  • Online ISBN 978-1-4614-2176-4
  • Series Print ISSN 1439-7382
  • About this book