the general definition of the polarization is [itex]\int_{\infty}^t j dt [/itex] (up to maybe some minus) whose longitudinal part (the expression you gave is only for the longitudinal dielectric function) can be expressed in terms of rho (and q and omega). In linear response rho will be induced by the coupling of the charges to the electromagnetic field V in the hamiltonian proportional to rho V.
So the polarization will become proportional to a chargecharge correlation function and all the q´s and omegas flying around can be shown to give the Coulomb factor v.
