  We prove that the orbit of a non-periodic point at prime values of the
horocycle flow in the modular surface is dense in a set of positive measure.
For some special orbits we also prove that they are dense in the whole space
(assuming the Ramanujan/Selberg conjectures for GL(2)/Q). In the process, we
derive an effective version of Dani's Theorem for the (discrete) horocycle
flow.
