1. Introduction

It is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the Casimir effect and Bose–Einstein condensation. A brief introduction into these areas is given in the respective sections; for recent monographs on these topics see [8; 22; 33; 34; 57; 67; 68; 71; 72]. We will consider exclusively spectral zeta functions, that is, zeta functions arising from the eigenvalue spectrum of suitable differential operators. Applications like those in number theory [3; 4; 23; 79] will not be considered in this contribution.

There is a set of technical tools that are at the very heart of understanding analytical properties of essentially every spectral zeta function. Those tools are introduced in Section 2 using the well-studied examples of the Hurwitz [54], Epstein [38; 39] and Barnes zeta function [5; 6]. In Section 3 it is explained how these different examples can all be thought of as being generated by the same mechanism, namely they all result from eigenvalues of suitable (partial) differential operators. It is this relation with partial differential operators that provides the motivation for analyzing the zeta functions considered in these lectures. Motivations come for example from the questions “Can one hear the shape of a drum?”, “What does the Casimir effect know about a boundary?”, and “What does a Bose gas know about its container?” The first two questions are considered in detail in Section 4. The last question is examined in Section 5, where we will see how zeta functions can be used to analyze the phenomenon of Bose–Einstein condensation. Section 6 will point towards recent developments for the analysis of spectral zeta functions and their applications.

2. Some basic zeta functions

In this section we will construct analytical continuations of basic zeta functions. From these we will determine the meromorphic structure, residues at singular points and special function values.