  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. We
will consider exclusively spectral zeta functions, that is zeta functions
arising from the eigenvalue spectrum of suitable differential operators. 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 using the well-studied examples of the Hurwitz, Epstein and
Barnes zeta function. It is explained how these different examples of zeta
functions 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?" and "What does the Casimir effect know about a boundary?".
Finally "What does a Bose gas know about its container?"
