Fractal Geometry, Complex Dimensions and Zeta Functions

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.

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 dynamic zeta functions associated with a fractal

- Examples of such formulas include Prime Orbit Theorem with error term for self-similar flows, and a 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. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts.

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.

• Diophantine approximation
• Number theory
• Prime
• Riemann hypothesis
• cantor strings
• complex dimensions
• inverse spectral problems
• minkowski measurability
• nonlattice self-similar strings
• self-similar flows
• tubular neighborhoods
• partial differential equations

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

"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 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 self-contained, 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)

• Department of Mathematics, University of California, Riverside, USA

  Michel L. Lapidus

• Department of Mathematics, Utah Valley State College, Orem, USA

  Machiel Frankenhuijsen

Policies and ethics