On the value-distribution of Hurwitz zeta-functions with algebraic parameter (Q2672286)

Let \(s=\sigma+it\) denote a complex variable. The Hurwitz zeta-function with a real parameter \(\alpha \in (0, 1]\) is for \(\sigma > 1\) defined by the Dirichlet series expansion \[\zeta(s;\alpha)=\sum_{n=0}^\infty(n+\alpha)^{-s},\] and by analytic continuation elsewhere except for a simple pole at \(s = 1.\) This function generalizes the Riemann zeta function \(\zeta(s)=\zeta(s;1)\) and satisfies the analogue of Voronin's universality theorem whenever \(\alpha\) is rational or transcendental. However, it is not clear whether the Hurwitz zeta-function with an algebraic irrational parameter \(\alpha\) is universal. In this direction, the authors study the behavior of \(\zeta(s;\alpha)\) on the left of \(1+i\mathbb{R}\), where the parameter \(\alpha\) is an algebraic irrational number and prove quantitative results on the denseness of the Hurwitz zeta-function.