  We review generalized zeta functions built over the Riemann zeros (in short:
"superzeta" functions). They are symmetric functions of the zeros that display
a wealth of explicit properties, fully matching the much more elementary
Hurwitz zeta function. As a concrete application, a superzeta function enters
an integral representation for the Keiper--Li coefficients, whose large-order
behavior thereby becomes computable by the method of steepest descents; then
the dominant saddle-point entirely depends on the Riemann Hypothesis being true
or not, and the outcome is a sharp exponential-asymptotic criterion for the
Riemann Hypothesis that only refers to the large-order Keiper--Li coefficients.
As a new result, that criterion, then Li's criterion, are transposed to a novel
sequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1
for Keiper--Li).
