Mergelyan’s approximation theorem with nonvanishing polynomials and universality of zeta-functions

We prove a variant of the Mergelyan approximation theorem that allows us to approximate functions that are analytic and nonvanishing in the interior of a compact set KK with connected complement, and whose interior is a Jordan domain, with nonvanishing polynomials. This result was proved earlier by the author in the case of a compact set KK without interior points, and independently by Gauthier for this case and the case of strictly starlike compact sets. We apply this result on the Voronin universality theorem for compact sets KK, where the usual condition that the function is nonvanishing on the boundary can be removed. We conjecture that this version of Mergelyan’s theorem might be true for a general set KK with connected complement and show that this conjecture is equivalent to a corresponding conjecture on Voronin Universality.