Summary

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.

Choice Review

Riemann's so-called explicit formula holds place among the crown jewels of number theory. This equation, on the one hand, involves prime numbers in an explicit way and, on the other, zeros of the Riemann zeta function. With additional information about zeta zeros, it yields the prime number theorem with best-known error estimates. Lapidus (Univ. of California, Riverside) and van Frankenhuijsen (Utah Valley State College) reinterpret the explicit formula in a geometric context, where it admits vast generalization. The title's fractal strings mean bounded open subsets of the real line with a fractal boundary. The usual computation of dimension for self-similar fractals requires finding roots of Dirichlet series, a class including the Riemann zeta function, so the zeta zeros receive interpretation as complex fractal dimensions. This material develops in directions too numerous to summarize, but students of number theory will certainly find interest in a novel proof of the prime number theory, reinterpretation of the celebrated Riemann hypothesis, and information about possible arithmetic progressions among the zeta zeros. This interdisciplinary work connecting analytic number theory to fractal geometry and spectral theory will attract readers interested in any of these subjects. ^BSumming Up: Recommended. Upper-division undergraduates through professionals. D. V. Feldman University of New Hampshire