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

TL;DR

This paper extends Mergelyan's approximation theorem to nonvanishing polynomials for certain compact sets and applies it to enhance the universality results of zeta-functions, removing boundary nonvanishing conditions.

Contribution

It introduces a new variant of Mergelyan's theorem for nonvanishing functions on specific sets, enabling broader applications in universality theorems for zeta-functions.

Findings

Proves a new approximation theorem for nonvanishing functions on Jordan domains.

Removes boundary nonvanishing conditions in Voronin's universality theorem.

Establishes a conjecture linking Mergelyan's theorem and zeta-function universality.

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.