Motivated by the recent results from Daya Bay, Reno and Double Chooz Collaborations, we study the consequences of small departures from exact $mu-tau$ symmetry in the neutrino sector, to accommodate a non-vanishing value of the element $V_{e3}$ from the leptonic mixing matrix. Within the see-saw framework, we identify simple patterns of Dirac mass matrices that lead to approximate $mu-tau$ symmetric neutrino mass matrices, which are consistent with the neutrino oscillation data and lead to non-vanishing mixing angle $V_{e3}$ as well as precise predictions for the CP violating phases. We also show that there is a transparent link between neutrino mixing angles and see-saw parameters, which we further explore within the context of leptogenesis as well as double beta decay phenomenology.
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