skip to content
Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions Preview this item
ClosePreview this item

Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions

Author: Michel L Lapidus; Machiel Van Frankenhuysen
Publisher: Boston : Birkhäuser, ©2000.
Edition/Format:   Print book : EnglishView all editions and formats

Number theory and fractal geometry are combined in this study of the vibrations of fractal strings. The book centres around a notion of complex dimension extended here to apply to the zeta functions  Read more...


(not yet rated) 0 with reviews - Be the first.

More like this


Find a copy online

Links to this item

Find a copy in the library

&AllPage.SpinnerRetrieving; Finding libraries that hold this item...


Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Michel L Lapidus; Machiel Van Frankenhuysen
ISBN: 0817640983 9780817640989 3764340983 9783764340988
OCLC Number: 52404467
Description: x, 268 pages : illustrations ; 24 cm
Contents: 1 Complex Dimensions of Ordinary Fractal Strings.- 1.1 The Geometry of a Fractal String.- 1.1.1 The Multiplicity of the Lengths.- 1.1.2 Example: The Cantor String.- 1.2 The Geometric Zeta Function of a Fractal String.- 1.2.1 The Screen and the Window.- 1.2.2 The Cantor String (Continued).- 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.- 2.1.1 Dynamical Interpretation, Euler Product.- 2.2 Examples of Complex Dimensions of Self-Similar Strings.- 2.2.1 The Cantor String.- 2.2.2 The Fibonacci String.- 2.2.3 A String with Multiple Poles.- 2.2.4 Two Nonlattice Examples.- 2.3 The Lattice and Nonlattice Case.- 2.3.1 Generic Nonlattice Strings.- 2.4 The Structure of the Complex Dimensions.- 2.5 The Density of the Poles in the Nonlattice Case.- 2.5.1 Nevanlinna Theory.- 2.5.2 Complex Zeros of Dirichlet Polynomials.- 2.6 Approximating a Fractal String and Its Complex Dimensions.- 2.6.1 Approximating a Nonlattice String by Lattice Strings.- 3 Generalized Fractal Strings Viewed as Measures.- 3.1 Generalized Fractal Strings.- 3.1.1 Examples of 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.- 3.4.1 Measures with a Self-Similarity Property.- 4 Explicit Formulas for Generalized Fractal Strings.- 4.1 Introduction.- 4.1.1 Outline of the Proof.- 4.1.2 Examples.- 4.2 Preliminaries: The Heaviside Function.- 4.3 The Pointwise Explicit Formulas.- 4.3.1 The Order of the Sum over the Complex Dimensions.- 4.4 The Distributional Explicit Formulas.- 4.4.1 Alternate Proof of Theorem 4.12.- 4.4.2 Extension to More General Test Functions.- 4.4.3 The Order of the Distributional Error Term.- 4.5 Example: The Prime Number Theorem.- 4.5.1 The Riemann-von Mangoldt Formula.- 5 The Geometry and the Spectrum of Fractal Strings.- 5.1 The Local Terms in the Explicit Formulas.- 5.1.1 The Geometric Local Terms.- 5.1.2 The Spectral Local Terms.- 5.1.3 The Weyl Term.- 5.1.4 The Distribution x?logmx.- 5.2 Explicit Formulas for Lengths and Frequencies.- 5.2.1 The Geometric Counting Function of a Fractal String.- 5.2.2 The Spectral Counting Function of a Fractal String.- 5.2.3 The Geometric and Spectral Partition Functions.- 5.3 The Direct Spectral Problem for Fractal Strings.- 5.3.1 The Density of Geometric and Spectral States.- 5.3.2 The Spectral Operator.- 5.4 Self-Similar Strings.- 5.4.1 Lattice Strings.- 5.4.2 Nonlattice Strings.- 5.4.3 The Spectrum of a Self-Similar String.- 5.4.4 The Prime Number Theorem for Suspended Flows.- 5.5 Examples of Non-Self-Similar Strings.- 5.5.1 The a-String.- 5.5.2 The Spectrum of the Harmonic String.- 5.6 Fractal Sprays.- 5.6.1 The Sierpinski Drum.- 5.6.2 The Spectrum of a Self-Similar Spray.- 6 Tubular Neighborhoods and Minkowski Measurability.- 6.1 Explicit Formula for the Volume of a Tubular Neighborhood.- 6.1.1 Analogy with Riemannian Geometry.- 6.2 Minkowski Measurability and Complex Dimensions.- 6.3 Examples.- 6.3.1 Self-Similar Strings.- 6.3.2 The a-String.- 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.- 8.2.1 Integral Cantor Strings: a-adic Analysis of the Geometric and Spectral Oscillations.- 8.2.2 Nonintegral Cantor Strings: Analysis of the Jumps in the Spectral Counting Function.- 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.- 9.2.1 Density of Nonzeros on Vertical Lines.- 9.2.2 Almost Arithmetic Progressions of Zeros.- 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.- 10.2.1 Comparison with Other Definitions of Fractality...- 10.2.2 Possible Connections with the Notion of Lacunarity.- 10.3 Fractality and Self-Similarity.- 10.4 The Spectrum of a Fractal Drum.- 10.4.1 The Weyl-Berry Conjecture.- 10.4.2 The Spectrum of a Self-Similar Drum.- 10.4.3 Spectrum and Periodic Orbits.- 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.
Responsibility: Michel L. Lapidus, Machiel van Frankenhuysen.


Editorial reviews

Publisher Synopsis

"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 Read more...

User-contributed reviews
Retrieving GoodReads reviews...
Retrieving DOGObooks reviews...


Be the first.
Confirm this request

You may have already requested this item. Please select Ok if you would like to proceed with this request anyway.

Linked Data

Primary Entity

<> # Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions
    a schema:CreativeWork, schema:Book ;
   library:oclcnum "52404467" ;
   library:placeOfPublication <> ;
   library:placeOfPublication <> ; # Boston
   schema:about <> ; # Number theory
   schema:about <> ; # Zetafunktion
   schema:about <> ; # Functions, Zeta
   schema:about <> ; # Fraktalgeometrie
   schema:about <> ; # Fraktal
   schema:about <> ; # Fractals
   schema:about <> ; # Nullstelle
   schema:about <> ; # Zahlentheorie
   schema:about <> ;
   schema:bookFormat bgn:PrintBook ;
   schema:contributor <> ; # Machiel Van Frankenhuysen
   schema:copyrightYear "2000" ;
   schema:creator <> ; # Michel Laurent Lapidus
   schema:datePublished "2000" ;
   schema:exampleOfWork <> ;
   schema:inLanguage "en" ;
   schema:name "Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions"@en ;
   schema:productID "52404467" ;
   schema:publication <> ;
   schema:publisher <> ; # Birkhäuser
   schema:url <> ;
   schema:workExample <> ;
   schema:workExample <> ;
   umbel:isLike <> ;
   wdrs:describedby <> ;

Related Entities

<> # Boston
    a schema:Place ;
   schema:name "Boston" ;

<> # Number theory
    a schema:Intangible ;
   schema:name "Number theory"@en ;

<> # Fractals
    a schema:Intangible ;
   schema:name "Fractals"@en ;

<> # Functions, Zeta
    a schema:Intangible ;
   schema:name "Functions, Zeta"@en ;

<> # Michel Laurent Lapidus
    a schema:Person ;
   schema:birthDate "1956" ;
   schema:familyName "Lapidus" ;
   schema:givenName "Michel Laurent" ;
   schema:givenName "Michel L." ;
   schema:name "Michel Laurent Lapidus" ;

<> # Machiel Van Frankenhuysen
    a schema:Person ;
   schema:birthDate "1967" ;
   schema:familyName "Van Frankenhuysen" ;
   schema:givenName "Machiel" ;
   schema:name "Machiel Van Frankenhuysen" ;

    a schema:ProductModel ;
   schema:isbn "0817640983" ;
   schema:isbn "9780817640989" ;

    a schema:ProductModel ;
   schema:isbn "3764340983" ;
   schema:isbn "9783764340988" ;

    a genont:InformationResource, genont:ContentTypeGenericResource ;
   schema:about <> ; # Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions
   schema:dateModified "2018-03-11" ;
   void:inDataset <> ;

Content-negotiable representations

Close Window

Please sign in to WorldCat 

Don't have an account? You can easily create a free account.