Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average
Abstract
We prove that the set of Farey fractions of order T, that is, the set \α/β ∈ : (α, β) = 1, 1 α, β T\, is uniformly distributed in residue classes modulo a prime p provided T p1/2 + for any fixed >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.
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