1. Introduction

Functional determinants of second-order differential operators are of great importance in many different fields. In physics, functional determinants provide the one-loop approximation to quantum field theories in the path integral formulation [21; 48]. In mathematics they describe the analytical torsion of a manifold [47].

Although there are various ways to evaluate functional determinants, the zeta function scheme seems to be the most elegant technique to use [9; 16; 17; 31]. This is the method introduced by Ray and Singer to define analytical torsion [47]. In physics its origin goes back to ambiguities in dimensional regularization when applied to quantum field theory in curved spacetime [11; 29].

For many second-order ordinary differential operators surprisingly simple answers can be given. The determinants for these situations have been related to boundary values of solutions of the operators, see, e.g., [8; 10; 12; 22; 23; 26; 36; 39; 40]. Recently, these results have been rederived with a simple and accessible method which uses contour integration techniques [33; 34; 35]. The main advantage of this approach is that it can be easily applied to general kinds of boundary conditions [35] and also to cases where the operator has zero modes [34; 35]; see also [37; 38; 42]. Equally important, for some higher dimensional situations the task of finding functional determinants remains feasible. Once again closed answers can be found but compared to one dimension technicalities are significantly more involved [13; 14]. It is the aim of this article to choose specific higher dimensional examples where technical problems remain somewhat confined. The intention is to illustrate that also for higher dimensional situations closed answers can be obtained which are easily evaluated numerically.

Some exciting, bold new cooperative explorations of various interconnections between traditional domains of “pure” mathematics and exotic new developments in theoretical physics have continued to emerge in recent years. The beautiful interlacing of theory and application, and cross-discipline interaction has led, as is usual, to notable, fruitful, and bonus outcomes.

These interconnections range from topology, algebraic geometry, modular forms, Eisenstein series, zeta functions, vertex operators, and knot theory to gauge theory, strings and branes, quantum fields, cosmology, general relativity, and Bose–Einstein condensation. They are broad enough in scope to present the average reader with not only a measure of enchantment but with some mild bewilderment as well. A new journal, Communications in Number Theory and Physics, has recently been launched to follow and facilitate interactions and dynamics between these two disciplines, for example. Various books that are now available, in addition to an array of conference and workshop activity, accent this fortunate merger of mathematics and physical theory and assist greatly in bridging the divide, although in some cases the themes are pitched at a level more suitable for advanced readers and researchers.

In an attempt to further bridge the divide, at least in some modest way for students and non-experts, and to provide a window into this adventurous arena of intertwining ideas, a graduate workshop entitled “A Window into Zeta and Modular Physics” was presented at MSRI during the period from 16 to 27 June 2008. The workshop consisted of daily expository lectures, speakers’ seminar lectures (where the material was more technical and represented their research, but which at the same time did connect to and enlarge on the daily lectures), and four special student lectures. Given the excellent preparation and presentation of the lectures, it was proposed that it would be of further benefit to the students (especially given their enthusiastic response), and to the mathematical physics community in general, if the lectures were eventually molded in some form as a book. Thus the present volume, with the workshop title and similar style, evolved. Besides the editors, there were three workshop speakers: Geoffrey Mason, Audrey Terras, and Michael Tuite.

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.