Self-approximation of Hurwitz zeta-functions with rational parameter

In this paper, we show the self-approximation property for Hurwitz zeta-functions with rational parameters. Namely, we prove that ζ(s?+?iατ, a/b) approximates uniformly ζ(s?+?iβτ, a/b) for infinitely many real τ , where α, β are arbitrary real numbers linearly independent over $ \mathbb{Q} $ , and s is in a compact set lying in the open right half of the critical strip.