On the distribution of zeros of the Hurwitz zeta-function

Assuming the Riemann hypothesis, we prove asymptotics for the sum of values of the Hurwitz zeta-function ζ(s, α) taken at the nontrivial zeros of the Riemann zeta-function ζ(s) = ζ(s, 1) when the parameter α either tends to 1/2 and 1, respectively, or is fixed; the case α = 1/2 is of special interest since ζ(s, 1/2) = (2s − 1)ζ(s). If α is fixed, we improve an older result of Fujii. Besides, we present several computer plots which reflect the dependence of zeros of ζ(s, α) on the parameter α. Inspired by these plots, we call a zero of ζ(s, α) stable if its trajectory starts and ends on the critical line as α varies from 1 to 1/2, and we conjecture an asymptotic formula for these zeros. 1. Motivation Let, as usual, s = σ+ it denote a complex variable and define e(z) = exp(2πiz). For σ > 1, the Hurwitz zeta-function is given by ζ(s, α) = ∞ ∑