  We study the effect of the fast rotation and vertical viscosity on the
lifespan of solutions to the three-dimensional primitive equations (also known
as the hydrostatic Navier-Stokes equations) with impermeable and stress-free
boundary conditions. Firstly, for a short time interval, independent of the
rate of rotation $|\Omega|$, we establish the local well-posedness of solutions
with initial data that is analytic in the horizontal variables and only $L^2$
in the vertical variable. Moreover, it is shown that the solutions immediately
become analytic in all the variables with increasing-in-time (at least
linearly) radius of analyticity in the vertical variable for as long as the
solutions exist. On the other hand, the radius of analyticity in the horizontal
variables might decrease with time, but as long as it remains positive the
solution exists. Secondly, with fast rotation, i.e., large $|\Omega|$, we show
that the existence time of the solution can be prolonged, with "well-prepared"
initial data. Finally, in the case of two spatial dimensions with $\Omega=0$,
we establish the global well-posedness provided that the initial data is small
enough. The smallness condition on the initial data depends on the vertical
viscosity and the initial radius of analyticity in the horizontal variables.
