A scaling property of Farey fractions

Abstract

The Farey sequence of order n consists of all reduced fractions a/b between 0 and 1 with positive denominator b less than or equal to n. The sums of the inverse denominators 1/b of the Farey fractions in prescribed intervals with rational bounds have a simple main term, but the deviations are determined by an interesting sequence of polygonal functions $$f_n$$ . For $$n \rightarrow \infty $$ we also obtain a certain limit function, which describes an asymptotic scaling property of functions $$f_n$$ in the vicinity of any fixed fraction a/b and which is independent of a/b. The result can be obtained by using only elementary methods. We also study this limit function and especially its decay behaviour by using the Mellin transform and analytical properties of the Riemann zeta function.