  An asymptote is derived from Turing's local reestimation formula for
population frequencies, and a local reestimation formula is derived from Zipf's
law for the asymptotic behavior of population frequencies. The two are shown to
be qualitatively different asymptotically, but nevertheless to be instances of
a common class of reestimation-formula-asymptote pairs, in which they
constitute the upper and lower bounds of the convergence region of the
cumulative of the frequency function, as rank tends to infinity. The results
demonstrate that Turing's formula is qualitatively different from the various
extensions to Zipf's law, and suggest that it smooths the frequency estimates
towards a geometric distribution.
