Mergelyan's approximation theorem with nonvanishing polynomials and
universality of zeta-functions
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by
Johan Andersson
2011
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.
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