Zeta functions have-function regularization is a powerful method in spectral theory, with many profound applications in physics, as explained inincluding the Casimir effect, gravity and string theory, high-temperature phase transition, topological symmetry breaking, and non-commutative spacetime. See Ten Physical Applications of Spectral Zeta Functions, or these lecture notes:
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. 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. Motivations come 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?"