Zeta Function on a Generalised Cone

The analytic properties of the ζ-function for a Laplace operator on a generalised cone \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}^2 \times \mathcal{M}^{\text{N}}$$ \end{document} are studied in some detail using Cheeger's approach and explicit expressions are given. In the compact case, the ζ-function of the Laplace operator turns out to be singular at the origin. As a result, strictly speaking, the ζ-function regularisation does not ‘regularise’ and a further subtraction is required for the related one-loop effective potential.