Paolo Minelli: Bias in a variant of the Euclidean algorithm and Dedekind sums

Abstract

We investigate the number of steps taken by a variant of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval (0,1/2), establishing that they behave differently on (0,1/2) than they do on (1/2,1). 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.