Fractal Geometry, Complex Dimensions and Zeta Functions Geometry and Spectra of Fractal Strings
Key Features 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 cor...
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Format:  eBook 
Language:  English 
Published: 
New York, NY
Springer New York
2006, 2006

Edition:  1st ed. 2006 
Series:  Springer Monographs in Mathematics

Subjects:  
Online Access:  
Collection:  Springer eBooks 2005  Collection details see MPG.ReNa 
Summary:  Key Features 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. 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. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts, and discusses several open problems and extensions. 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 will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics. From Reviews of Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, by Michel Lapidus and Machiel van Frankenhuysen, Birkhäuser Boston Inc., 2000. "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 "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 selfcontained, intelligent and well paced." –Bulletin of the London Mathematical Society 

Physical Description:  XXIV, 460 p. 54 illus online resource 
ISBN:  9780387352084 