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

Johan Andersson

doi:10.1016/j.jat.2012.12.005

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arXiv · 1010.0850

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

Johan Andersson

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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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Johan Andersson. 2010-10-05. Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions. https://doi.org/10.1016/j.jat.2012.12.005

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