On the uniformity of distribution of Farey fractions

Abstract

Let F _x be the set of nonnegative rationals m\n with 0 < n < x and (m, n) = 1. For some fixed interval I ⊂ (0;+∞), I = (λ₁;λ₂) let F (u|x, I) = #(F_x ∩ (λ₁;λ₁ + u(λ₂ - λ₁))/#(F_x ∩ I). The paper deals with the estimation of discrepancy |F (u|x, I) - u|, 0 < u < 1.