Zeta-function regularization is a powerful method in perturbation theory. This text is meant as a guide for the student of this subject. It gives explanations, in particular on the mathematical difficulties and tricky points, and several applications are given to show how the procedure works in practice (for example, Casimir effect, gravity and string theory, high-temperature phase trasition, topological symmetry breaking). The formulas can be used for accurate numerical calculations.
From the contents: Introduction and Outlook.- Mathematical Formulas Involving the Different Zeta Functions.- A Treatment of the Non-Polynomial Contributions: Application to Calculate Partition Functions of Strings and Membranes.- Analytical and Numerical Study of Inhomogeneous Epstein and Epstein-Hurwitz Zeta Functions.- Physical Application: Casimir Effect.- Four Physical Applications of the Inhomogeneous Generalized Epstein-Hurwitz Zeta Functions.- Miscellaneous Applications Combining Zeta with Other Regularization Procedures.- Applications to Gravity, Strings and P-Branes.- Last Application: Topological Symmetry Breaking in Self-Interacting Theories.