In this presentation, we discuss the effective diffusion coefficient associated with the generalized Langevin equation (GLE) in a periodic potential. We begin by presenting estimates on the rate of convergence to equilibrium for a simple Markovian approxmimation of the GLE. We then present asymptotic results in the small correlation time regime, as well as in the overdamped and underdamped limits. Finally, we apply a recently developed numerical method in order to calculate the effective diffusion coefficient for a wide range of friction coefficients, confirming our asymptotic results and corroborating the findings of earlier studies on the subject.