We have analysed the nature of persistent currents in open coupled mesoscopic rings. Our system is comprised of two ideal loops connected to an electron reservoir. We have obtained analytical expressions for the persistent current densities in two rings in the presence of a magnetic field. We show that the known even-odd parity effects in isolated single loops have to be generalised for the case of coupled rings. We also show that when the two rings have unequal circumferences, it is possible to observe opposite currents (diamagnetic or paramagnetic) in the two rings for a given Fermi level.

Submitted to PRB. 9 figures availabel on request