According to G. Harder, “zeta functions know everything. One just has to know how to ask them.” We have suspected for a long time that this is certainly true in Number Theory. Something similar could be said for a large portion of Algebraic Geometry, starting with the fact that its modern incarnation is due, in part, to the need to formulate and solve some conjectures of Weil about zeta functions of algebraic varieties. But in recent decades we have seen in other areas of mathematics, such as dynamical systems and Riemannian geometry, that certain zeta functions play an increasingly important role, e.g., spectral zeta functions associated to the eigenvalues of Laplace-Beltrami operators. Harder’s assertion no longer seems far-fetched.

That having been said, there is nonetheless more than a small element of surprise that there are applications of zeta functions in Physics, even taking for granted that Physicists are always surprising us with the depth of the mathematical insights needed to read the book of nature.

The book under review is a collection of four introductory lectures and four research lectures. The survey lectures range from the basic properties of Riemann’s zeta function (analytic continuation, functional equation, special values, Euler product expansion, L-functions, modular forms and L-functions attached to modular forms) to some initial applications: Planck’s radiation density integral formula, zeta regularization (a procedure for assigning to some infinite quantities a finite value, usually given by a special value of a zeta function). One example of this method appears in Bosonic string theory where the amusing “equation”

1 + 2 + 3 + 4 + 5 + … = –1/12

begs for a meaningful interpretation. Noticing that this “equation” looks like the “value” at s = –1 of Riemann’s zeta function, given for Re(s)>1 by the series

ζ(s) = 1 + 1/2^{s }+ 1/3^{s }+ 1/4^{s }+ … + 1/n^{s }+ …

it is natural to interpret ζ(–1) as the value at s = –1 of the analytic continuation of ζ, and then one recalls Euler’s computation of the special values of ζ at the negative integers. The first article gives several more examples of zeta regularization, and includes other applications such as a computation of Kaluza-Klein modes for the gravitational potential in d+4 dimensions.

The second paper focuses on spectral zeta functions by considering various differential operators with motivations varying from Riemannian geometry, the Casimir effect, and Bose-Einstein condensation of gases in statistical mechanics. The third paper gives an introduction to zeta functions coming from random matrix theory and quantum chaos. The fourth paper shifts attention to certain chiral algebras, their representations and characters, and their relations to modular forms and elliptic functions. Examples of these algebras coming from Physics are the Heisenberg algebra and the Virasoro algebra. This chapter is a well-written introduction to some mathematical topics of interest to physicists working on two-dimensional conformal field theory and its relation to string theory.

The second part of the book has four papers reporting some applications of special functions, e.g., elliptic or theta functions, to cosmology, two-dimensional gravitation models, or formulas for functional determinants.

The essays collected in this book give a panoramic introduction to an important topic of current interest, and they should appeal to a broad audience, either researchers approaching this area or interested graduate students.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is mfzc@oso.izt.uam.mx.