  We study the limiting distributions of Birkhoff sums of a large class of cost
functions (observables) evaluated along orbits, under the Gauss map, of
rational numbers in $(0,1]$ ordered by denominators. We show convergence to a
stable law in a general setting, by proving an estimate with power-saving error
term for the associated characteristic function. This extends results of Baladi
and Vall\'ee on Gaussian behaviour for costs of moderate growth.
  We apply our result to obtain the limiting distribution of values of several
key examples of quantum modular forms. We show that central values of the
Esterman function ($L$ function of the divisor function twisted by an additive
character) tend to have a Gaussian distribution, with a large variance. We give
a dynamical, "trace formula free" proof that central modular symbols associated
with a holomorphic cusp form for $SL(2,{\bf Z})$ have a Gaussian distribution.
We also recover a result of Vardi on the convergence of Dedekind sums to a
Cauchy law, using dynamical methods.
