  We solve the Riemann-Hilbert problem on the sphere topology for three
singularities of finite strength and a fourth one infinitesimal, by determining
perturbatively the Poincare' accessory parameters. In this way we compute the
semiclassical four point vertex function with three finite charges and a fourth
infinitesimal. Some of the results are extended to the case of n finite charges
and m infinitesimal. With the same technique we compute the exact Green
function on the sphere with three finite singularities. Turning to the full
quantum problem we address the calculation of the quantum determinant on the
background of three finite charges and the further perturbative corrections.
The zeta function technique provides a theory which is not invariant under
local conformal transformations. Instead by employing a regularization
suggested in the case of the pseudosphere by Zamolodchikov and Zamolodchikov we
obtain the correct quantum conformal dimensions from the one loop calculation
and we show explicitly that the two loop corrections do not change such
dimensions. We expect such a result to hold to all order perturbation theory.
