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      Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average

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          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.

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          Author and article information

          Journal
          25 May 2007
          Article
          0705.3861
          08ac1b7a-14f6-4ba3-a1b3-280a55103776
          History
          Custom metadata
          11B57, 11G07, 14H52
          math.NT

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