  The main aim of this paper is twofold. First we generalize, in a novel way,
most of the known non-vanishing results for the derivatives of the Riemann zeta
function by establishing the existence of an infinite sequence of regions in
the right half-plane where these derivatives cannot have any zeros; and then,
in the rare regions of the complex plane that do contain zeros of the k-th
derivative of the zeta function, we describe a unexpected phenomenon, which
implies great regularities in their zero distributions. In particular, we prove
sharp estimates for the number of zeros in each of these new critical strips,
and we explain how they converge, in a very precise, periodic fashion, to their
central, critical lines, as k increases. This not only shows that the zeros are
not randomly scattered to the right of the line Re(s)=1, but that, in many
respects, their two-dimensional distribution eventually becomes much simpler
and more predictable than the one-dimensional behavior of the zeros of the zeta
function on the line Re(s)=1/2.
