Infinitely many sinks and sources among critical-line zeros for the holomorphic zeta flow

Determine whether there are infinitely many nontrivial zeros ρ on the critical line Re(ρ) = 1/2 such that Re ζ'(ρ) < 0 (sinks) and infinitely many such that Re ζ'(ρ) > 0 (sources) for the holomorphic flow s'(t) = ζ(s(t)).

Background

In the holomorphic Riemann flow s'=ζ(s), the zeros of ζ are equilibrium points whose stability is determined by the sign of Re ζ'(ρ): negative sign corresponds to a sink (stable), positive to a source (unstable).

Building on prior work by Broughan and Barnett, the paper recalls a conjecture concerning the distribution of these dynamical types among nontrivial zeros on the critical line; confirming or refuting this would link fine dynamical properties of the flow to zero statistics on the critical line.