Distribution of Farey fractions with $k$-free denominators

TL;DR

This paper studies the distribution of Farey fractions with k-free denominators in residue classes, proving equidistribution, analogues of classical results, and analyzing local statistics including correlation functions.

Contribution

It introduces new results on the distribution and local statistics of Farey fractions with k-free denominators, including explicit correlation functions and equivalences related to the GRH.

Findings

Sequences are equidistributed modulo one.

Derived analogues of classical theorems for these sequences.

Established explicit formulas for correlation measures, including the pair correlation function.

Abstract

We investigate the distributional properties of the sequence of Farey fractions with $k$-free denominators in residue classes, defined as \[\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\ k\text{-free}\ \&\ q\equiv b\pmod{m} \right\}.\] We show that $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ is equidistributed modulo one, and prove analogues of the classical results of Franel, Landau, and Niederreiter for $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$, particularly, deriving an equivalent form of the generalized Riemann hypothesis (GRH) for Dirichlet $L$-functions in terms of the distribution of $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$. Beyond examining the global distribution, we also study the local statistics of these sequences. We establish formulas for all levels ($k\ge 2$) of correlation measure. Specifically, we show the existence of the limiting pair ($k=2$) correlation function and provide an explicit expression for it. Our results are based upon the estimation of weighted Weyl sums and weighted lattice point counting in restricted domains.