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 $\zeta(s, \alpha)$ taken at the nontrivial zeros of the Riemann zeta-function $\zeta(s)=\zeta(s,1)$ when the parameter $\alpha$ either tends to $1/2$ and $1$, respectively, or is fixed; the case $\alpha=1/2$ is of special interest since $\zeta(s,1/2)=(2^s-1)\zeta(s)$. If $\alpha$ is fixed, we improve an older result of Fujii. Besides, we present several computer plots which reflect the dependence of zeros of $\zeta(s, \alpha)$ on the parameter $\alpha$. Inspired by these plots, we call a zero of $\zeta(s,\alpha)$ stable if its trajectory starts and ends on the critical line as $\alpha$ varies from $1$ to $1/2$, and we conjecture an asymptotic formula for these zeros.