Fractal geometry, complex dimensions and zeta functions : geometry and spectra of fractal strings /

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