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Fractal Geometry, Complex Dimensions and Zeta Functions

Geometry and Spectra of Fractal Strings

By: Michel L. Lapidus, Machiel van Frankenhuijsen

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eText | 8 August 2007

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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.