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We study the limiting distributions of Birkhoff sums of a large class of costfunctions (observables) evaluated along orbits, under the Gauss map, ofrational numbers in $(0,1]$ ordered by denominators. We show convergence to astable law in a general setting, by proving an estimate with power-saving errorterm for the associated characteristic function. This extends results of Baladiand Vall\'ee on Gaussian behaviour for costs of moderate growth.  We apply our result to obtain the limiting distribution of values of severalkey examples of quantum modular forms. We show that central values of theEsterman function ($L$ function of the divisor function twisted by an additivecharacter) tend to have a Gaussian distribution, with a large variance. We givea dynamical, "trace formula free" proof that central modular symbols associatedwith 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 aCauchy law, using dynamical methods.

Limit laws for rational continued fractions and value distribution of quantum modular forms