In Search of the Riemann Zeros: Strings, Fractal Membranes, and Noncommutative Spacetimes is an ambitious book, as the title indicates. The author describes it as part “research program to pursue” and part “continuing dialogue.” Largely, the reader is offered a series of “heuristic arguments based on mathematical or physical analogies.” One of the proposed analogies is between the functional equation for the Riemann Zeta function and the so-called “target space duality” of string theory, that is, the physical equivalence of string theories for different target geometries. A second proposed analogy is between the Riemann hypothesis and the existence of a minimum length in string theory.
The author also introduces the concept of a fractal string, and its multiplicative or quantized version, a fractal membrane, which is associated to the vibrations of an infinite dimensional (adelic) torus, and leads to an analogy with the Euler product expansion of the Riemann Zeta function.
The exposition begins with a substantial survey of selected topics from string theory and number theory needed in preparation for the analogies mentioned above. Many lengthy quotations from expository and technical sources alike are interspersed in these sections. A development of the notion of fractal strings and fractal membranes follows, and the book culminates by laying out the aforementioned analogies.
Because no results are claimed there are no proofs to be presented, but all terms are rigorously defined (requiring a highly mathematically sophisticated reader). Those desiring additional details and development on any topic will be satisfied by the thorough references provided throughout. Additionally, over 100 pages of further details are included in six illuminating appendices.
The analogies proposed by Lapidus are intriguing. Although full comprehension of the dialogue contained within In Search of the Riemann Zeros may be demanding, it has the potential to be both enjoyable and inspiring for the interested and motivated reader.
Lee Stemkoski is Assistant Professor of Mathematics at Adelphi University.