Uniform approximation in the spherical distance by functions meromorphic on Riemann surfaces

Given a function f:E→C‾ from a closed subset of a Riemann surface R   to the Riemann sphere C‾, we seek to approximate f in the spherical distance by functions meromorphic on R. As a consequence we generalize a recent extension of Mergelyan's theorem, due to Fragoulopoulou, Nestoridis and Papadoperakis [12]. The problem of approximating by meromorphic functions pole-free on E is equivalent to that of approximating by meromorphic functions zero-free on E, which in turn is related to Voronin's spectacular universality theorem for the Riemann zeta-function.