“…The zeta-regularized determinant of Laplacian on a compact Riemannian manifold was introduced in [11] and since then was studied and used in an immense number of papers in string theory and geometric analysis, for our future purposes we mention here the memoir [5], where the determinant of Laplacian is studied as a functional on the space of smooth Riemannian metrics on a compact two-dimensional manifold, and the papers [6,13], where the reader may find explicit calculation of the determinant of Laplacian for three-dimensional flat tori and for the sphere S 3 (respectively). The main result of the present paper is a comparison formula relating det(Δ α,P − λ) to det(Δ − λ), for λ ∈ C \ (Spectrum(Δ) ∪ Spectrum(Δ α,P )) (see Theorem 1 in Section 4 and Theorem 2 in Section 5).…”