## ABSTRACT

This conclusion summarizes the concepts discussed in the book. The book provides and applies techniques for the analysis of the most important spectral functions frequently appearing in mathematics and physics. Examples treated are the heat kernel, determinants and partition functions of statistical ensemble theory. The central object for dealing with all these entities is the zeta function associated with a suitable elliptic differential operator. Within a specific class of examples, the book shows how to find by analytical as well as numerical means all properties of the zeta functions needed. In addition, approximation schemes useful in finite temperature theory have been developed. Given the broad range of applications provided, including mathematical problems such as the heat equation asymptotics and phenomenological ones such as Bose-Einstein condensation, it is clear that spectral functions indeed play a crucial role in many different fields.