Regularized determinant of the Laplacian on forms over odd dimensional projective spaces

We establish formulae for the regularized determinant of the twisted Laplacianson forms over odd dimensional real projective spaces. This work corresponds toa generalization of the previous formula for this type of space and we prove theequivalence in the common cases, what leads to interesting, if simple, identitiesinvolving special values of Bernoulli polynomials and the Riemann zeta function.As application, we calculate the Analytic Torsion of these spaces in relation to allunitary representations of their fundamental group.