The article under review is a survey article and is a rich source of information about zeta function regularization and its applications to infinite dimensional geometry. To give an example about the topics discussed in the paper under review, consider an infinite-dimensional Laplacian
on some Hilbert space, say where is a compact manifold. Then observe that for the radial coordinate variable ,
diverges. The function is such a useful function in geometry (for example, in polar coordinates) so we would like this sum to converge. To overcome this divergence as well as divergences of other infinite-dimensional geometric quantities, the author uses a [systematic] zeta-regularized calculus. The neat idea is as follows. Take an elliptic (pseudo) differential operator on and let denote the eigenvalues of , and consider
For a function we define the regularized Laplacian of by the formula
provided, of course, that the right-hand side exists. Now consider our function again. In this case,
where is the zeta function of defined by . Therefore,
where represents a generalized dimension of the Hilbert space.
By a systematic use of zeta function regularization, the paper defines infinite-dimensional regularized volume forms, integrals, fractional derivatives, logarithmic derivatives, exterior derivatives, and many other quantities. An important application is that the author gives a mathematical justification of the appearance of the Ray-Singer determinant in the evaluation of a Gaussian path integral.