  A landmark theorem in the metric theory of continued fractions begins this
way: Select a non-negative real function $f$ defined on the positive integers
and a real number $x$, and form the partial sums $s_n$ of $f$ evaluated at the
partial quotients $a_1,..., a_n$ in the continued fraction expansion for $x$.
Does the sequence $\{s_n/n\}$ have a limit as $n\rar\infty$? In 1935 A. Y.
Khinchin proved that the answer is yes for almost every $x$, provided that the
function $f$ does not grow too quickly. In this paper we are going to explore a
natural reformulation of this problem in which the function $f$ is defined on
the rationals and the partial sums in question are over the intermediate
convergents to $x$ with denominators less than a prescribed amount. By using
some of Khinchin's ideas together with more modern results we are able to
provide a quantitative asymptotic theorem analogous to the classical one
mentioned above.
