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

A. C. Cojocaru; I. E. Shparlinski

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

Number Theory 2007-05-29 v1

Authors: A. C. Cojocaru , I. E. Shparlinski

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.

Keywords

farey fractions residue classes lang--trotter conjecture elliptic curves frobenius traces uniform distribution number theory

Cite

@article{arxiv.0705.3861,
  title  = {Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average},
  author = {A. C. Cojocaru and I. E. Shparlinski},
  journal= {arXiv preprint arXiv:0705.3861},
  year   = {2007}
}

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

Abstract

Keywords

Cite

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