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 \{\alpha/\beta \in \Q : \gcd(\alpha, \beta) = 1, 1 \le \alpha, \beta \le T\}, is uniformly distributed in residue classes modulo a prime $p$ provided T \ge p^{1/2 +\eps} for any fixed \eps>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