On the Mishou Theorem with an Algebraic Parameter

Abstract

The Riemann zeta-function and the Hurwitz zeta-function with transcendental or rational parameter are universal in the sense of Voronin: their shifts approximate broad classes of analytic functions. The universality of the Hurwitz zeta-function with an algebraic irrational parameter is an open problem since 1979. Mishou proved the joint universality of the Riemann zeta-function and the Hurwitz zeta-function with transcendental parameter. Mishou’s theorem with an algebraic irrational parameter is also an open problem. Here we obtain first results in this direction. We prove that there exists a nonempty closed subset of a two-dimensional set of analytic functions such that every pair in it is approximated by the shifts mentioned.