Paolo Minelli, KTH Royal Institute of Technology
In this talk we discuss the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval (0,\alpha), \(alpha<1/2\), establishing that they behave differently on (0,\alpha) than they do on (1-\alpha,1). These results are tightly linked with the distribution of lengths of certain continued fraction expansions as well as the distribution of the involved partial quotients. As an application, we prove a conjecture of Ito on the distribution of values of Dedekind sums. This is joint work with A. Sourmelidis and M. Technau.