5 - Determinants of the Laplacians

Abstract

The problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention. This chapter is focused on the determinants of the Laplacians. The chapter emphasizes on the evaluation of the functional determinant for the n-dimensional sphere Sn with the standard metric, which is potentially useful to compute. In computations of the determinants of the Laplacians on manifolds of constant curvature, an important role is played by the closed-form evaluations of the series involving the zeta function and by the theory of the multiple Gamma functions. One uses factorization while computing through multiple Gamma functions and the other way is to make use of lemma on zeta regularized products.