Hurwitz zeta universality conjecture for algebraic irrational parameters

Let $\alpha$ be a parameter with $0<\alpha<1/2$ or $1/2<\alpha<1$, and let Theorem 1 denote the universality statement for the Hurwitz zeta-function in the preceding theorem. Hurwitz zeta universality conjecture. Theorem 1 is true for all $0<\alpha<1/2$ and $1/2<\alpha<1$. Proving this would extend universality beyond the currently known transcendental and rational parameters to algebraic irrational parameters; this is described as one of the main open problems in the theory of universality of $L$-functions and zeta-functions.