Dirichlet Scalar Determinants On Two-Dimensional Constant Curvature Disks

We compute exactly the scalar determinants (+M^2) on the two-dimensional round disks of constant curvature R=0, 2, for any finite boundary length and mass M, with Dirichlet boundary conditions, using the -function prescription. When M^2= q (q+1), q N, a simple expression involving only elementary functions and the Euler function is found. Applications to two-dimensional Liouville and Jackiw-Teitelboim quantum gravity are presented in a separate paper.