Universality of the Hurwitz zeta-function on the half plane of absolute convergence

Abstract

Let $K$ be a compact set with connected complement on the half-plane Re$(s)>0$, and let $f$ be a continuous function on $K$ which is analytic in its interior. We prove that for any parameter $0<\alpha<1, \alpha \neq \frac 1 2$ then $f(s)$ may be uniformly approximated arbitrarily closely by $\zeta(1+iT+i\delta s,\alpha)$ on $K$ for some $T,\delta>0$, where $\zeta(s,\alpha)$ denote the Hurwitz zeta-function. This is the first known universality result that is also known to hold for the Hurwitz zeta-function with an algebraic irrational parameter.