A Note on Farey Fractions With Odd Denominators

TL;DR

This paper investigates the distribution of Farey fractions with odd denominators, analyzing the frequency of numerator differences and extending results to subintervals using Kloosterman sum estimates.

Contribution

It provides an asymptotic analysis of Farey fractions with odd denominators and generalizes the results to subintervals with new estimates.

Findings

Asymptotic formulas for numerator differences in Farey fractions with odd denominators

Extension of results to subintervals of [0,1] using Kloosterman sums

Enhanced understanding of the distribution of odd-denominator Farey fractions

Abstract

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 values of numerators of differences of consecutive elements in this set. After proving an asymptotic result for these frequencies, we use estimates coming from incomplete Kloosterman sums to generalize our result to subintervals of [0,1].