A note on Farey fractions with odd denominators

In this paper we examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of the values taken on by Δ=qa′−aq′ where a/q<a′/q′ are consecutive in . After proving an asymptotic result for these frequencies, we generalize the result to the subset of elements of formed by restriction to a subinterval α,β]⊆0,1].