| Summary: | Motivated by questions in theoretical physics, there has been much interest in the problem of defining and calculating the determinants of differential operators, such as the Laplacian on manifolds. The most common method that makes sense out of the product of a discrete sequence, such as the spectrum of an operator, is the zeta function regularization, in which one defines the zeta regularized product by analytically continuing the zeta function to the origin. When the analytic continuation exists and how to implement the process are major questions in zeta regularization. === The purpose of this dissertation is to develop a systematic approach for operating with zeta regularized products which will make use of some basic properties to calculate zeta regularized products without finding the actual continuation. === First we establish a number of properties similar to that of ordinary products, second we use these properties to compute various zeta regularized products, in particular, the determinants of Laplacian on p-dimensional flat tori. Without involving the analytic continuation, our computation turns out to be much simpler and easier to understand. === Some analytic properties are also discussed, namely the Weierstrass factorization and the Laplace-Mellin transform. Relationships between gamma functions and zeta regularized products are established. It turns out that the gamma function as well as Barnes' multiple gamma functions can be represented as special zeta regularized products. Also some asymptotic expansions of zeta regularized products are obtained. As a result, the classic Stirling formula is generalized to a double Stirling formula. === Source: Dissertation Abstracts International, Volume: 54-12, Section: B, page: 6242. === Major Professor: John R. Quine. === Thesis (Ph.D.)--The Florida State University, 1993.
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