Zeta regularized determinant of the Laplacian for classes of spherical space forms

We derive the spectral zeta function in terms of certain Dirichlet series for a variety of spherical space forms MGMG. Extending results in [C. Nash, D. O’Connor, Determinants of Laplacians on lens spaces, J. Math. Phys. 36 (3) (1995) 1462–1505] the zeta-regularized determinant of the Laplacian on MGMG is obtained explicitly from these formulas. In particular, our method applies to manifolds of dimension higher than 3 and it includes the case where GG arises from the dihedral group of order 2m2m. As a crucial ingredient in our analysis we determine the dimension of eigenspaces of the Laplacian in form of some combinatorial quantities for various infinite classes of manifolds from the explicit form of the generating function in [A. Ikeda, On the spectrum of a Riemannian manifold of positive constant curvature, Osaka J. Math. 17 (1980) 75–93].