On the distribution of index of Farey Sequences

TL;DR

This paper investigates the distribution and moments of the index of Farey fractions, providing new asymptotic formulas and extending previous results to square-free denominators and higher level correlations.

Contribution

It introduces new asymptotic formulas for moments of Farey index distributions twisted by Dirichlet characters, including novel results for square-free denominators and higher level correlations.

Findings

Asymptotic formulas for moments with Dirichlet twists

New results for square-free denominators

Analysis of higher level correlations

Abstract

In this article, we study the distribution of index of Farey fractions which was first introduced and studied by Hall and Shiu. We provide asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with $\mathcal{B}$-free denominators. Additionally, we reconsider the squarefree case earlier done in [ALVZ08], and obtain new results for moments of indices with square-free denominators. We also study higher level correlations of the index function generalizing earlier known results on two level correlations.