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

TL;DR

This paper extends approximation and universality theorems for zeta-functions to more general compact sets, removing the nonvanishing condition and connecting to the Riemann hypothesis.

Contribution

It introduces a variant of Lavrentiev's approximation theorem for nonvanishing polynomials on complex sets and applies it to universality results for zeta-functions.

Findings

Proves a new approximation theorem for continuous functions on complex sets.

Establishes a universality theorem for zeta-functions on sets without interior points.

Links the results to a criterion equivalent to the Riemann hypothesis.

Abstract

We prove a variant of the Lavrentiev's approximation theorem that allows us to approximate a continuous function on a compact set K in C without interior points and with connected complement, with polynomial functions that are nonvanishing on K. We use this result to obtain a version of the Voronin universality theorem for compact sets K, without interior points and with connected complement where it is sufficient that the function is continuous on K and the condition that it is nonvanishing can be removed. This implies a special case of a criterion of Bagchi, which in the general case has been proven to be equivalent to the Riemann hypothesis.