Fractal Geometry, Complex Dimensions and Zeta Functions
Geometry and Spectra of Fractal Strings
Authors: Lapidus, Michel, van Frankenhuijsen, Machiel
Free Preview The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings
 Number theory, spectral geometry, and fractal geometry are interlinked in this indepth study of the vibrations of fractal strings, that is, onedimensional drums with fractal boundary
 Numerous theorems, examples, remarks and illustrations enrich the text
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 About this book

Number theory, spectral geometry, and fractal geometry are interlinked in this indepth study of the vibrations of fractal strings, that is, onedimensional 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 selfsimilar flows, and a geometric tube formula
The method of Diophantine approximation is used to study selfsimilar strings and flows
Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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.
 Reviews

“This interesting volume gives a thorough introduction to an active field of research and will be very valuable to graduate students and researchers alike.” (C. Baxa, Monatshefte für Mathematik, Vol. 180, 2016)
“In this research monograph the authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a selfcontained manner the results … are completely proved. I appreciate that the book is useful for mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying the fractals and dimension theory.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, April, 2013)
“The authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a selfcontained manner, the results (including some fundamental ones) are completely proved. … the book will be useful to mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying fractals and dimension theory.” (NicolaeAdrian Secelean, Zentralblatt MATH, Vol. 1261, 2013)
"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."  NicolaeAdrian Secelean for Zentralblatt MATH
"This highly original selfcontained 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 previous book by authors)
"It is the reviewera (TM)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 selfcontained, intelligent and well paced."  Bulletin of the London Mathematical Society (Review of previous book by authors)
"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 previous book by authors)
 Table of contents (13 chapters)


Complex Dimensions of Ordinary Fractal Strings
Pages 932

Complex Dimensions of SelfSimilar Fractal Strings
Pages 3363

Complex Dimensions of Nonlattice SelfSimilar Strings: Quasiperiodic Patterns and Diophantine Approximation
Pages 65117

Generalized Fractal Strings Viewed as Measures
Pages 119135

Explicit Formulas for Generalized Fractal Strings
Pages 137178

Table of contents (13 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Fractal Geometry, Complex Dimensions and Zeta Functions
 Book Subtitle
 Geometry and Spectra of Fractal Strings
 Authors

 Michel Lapidus
 Machiel van Frankenhuijsen
 Series Title
 Springer Monographs in Mathematics
 Copyright
 2013
 Publisher
 SpringerVerlag New York
 Copyright Holder
 Springer Science+Business Media New York
 eBook ISBN
 9781461421764
 DOI
 10.1007/9781461421764
 Hardcover ISBN
 9781461421757
 Softcover ISBN
 9781489988386
 Series ISSN
 14397382
 Edition Number
 2
 Number of Pages
 XXVI, 570
 Topics