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

Abstract

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 K 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 K 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 K of this type, 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 K with connected complement and show that this conjecture is equivalent to a corresponding conjecture on Voronin Universality.

v1: 13 pages. v2+v3: 14 pages. Fixed error in the statement of the Lemma + minor changes. v4: 13 pages, Removed Lemma since it does not give any non trivial sets + minor changes. v5+v6: 13 pages, Clarified the last half of the proof of Theorem 6 + minor changes

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