A conjecture of R. R. Hall on Farey points

Let I = (α, β] ⊆ (0, 1] be an interval, let γ₁, < γ₂ < ⋯ < γ_NI(Q) denote the Farey fractions of order Q from I and set S_r(Q) = ∑_j=1^NI(Q) (γ_j+1 - γ_j)(γ_j+r+1 - γ_j+r) for r ≧ 0. We prove that, for all r ≧ 2, there exists a constant I_r such that the following asymptotic formula holds: S_r(Q) = S_r(Q, (0,1)] = 6I_r/π² Q² + O_r (log ^1/(r+3)Q/Q^2+1/(r+3) This implies, for all h ≧ 3, the existence of a constant D(h) such that ∑_j=1^N(0,1)(Q) (γ_j+h-γ_j)² = 12(2h-1) logQ/π² Q² + D(h)/Q² + O_h(log^1/(h+2) Q/Q^2+1/(h+2)) providing a positive answer to a conjecture of R. R. Hall. Furthermore, we prove that, for any r ≧ 2 and any such interval I, the sums S_r(Q, I) and S_r(Q) are related by S_r(Q, I) = |I|S_r(Q) + O_ε(Q^{-2-1/(2r+4)+ε}). For r ∈ {0, 1} we prove, for any I with rational end points, the existence of a constant c_I such that S₀(Q,I) = |I|S₀(Q) + 2_CI/Q² + O_ε(Q^-2-1/10+ε) and S₁(Q,I) = |I|S_I(Q) + C_I/Q² + O_ε(Q^-2-1/12+ε).