Distribution of Farey fractions with $k$-free denominators

Abstract: We study the global and local distribution 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) in terms of the distribution of $\left(\mathscr{F}{Q,k}^{(m)}\right){Q\ge 1}$. Additionally, we study the local distribution of these sequences. We establish formulas for all levels of correlation measure. Specifically, we show the existence of the limiting pair 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.