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We prove that the set of Farey fractions of order T, that is, the set {α/β ∈ Q: gcd(α,β) = 1, 1 p 1/2+e for any fixed e > 0. We apply this to obtain upper bounds for the Lang-Trotter conjectures on Frobenius traces and Frobenius fields "on average" over a one-parametric family of elliptic curves.

Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average