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

Author(s): Alina Carmen Cojocaru; Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 136 (2008), 1977-1986.
MSC (2000): Primary 11B57, 11G07, 14H52
Posted: February 15, 2008
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Abstract: We prove that the set of Farey fractions of order $ T$, that is, the set $ \{\alpha/\beta \in \mathbb{Q} : \operatorname{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 +\varepsilon}$ for any fixed $ \varepsilon>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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Additional Information:

Alina Carmen Cojocaru
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607; and Institute of Mathematics of the Romanian Academy, Calea Grivitei 21, 010702, Bucharest, Romania
Email: cojocaru@math.uic.edu

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email: igor@ics.mq.edu.au

DOI: 10.1090/S0002-9939-08-09324-6
PII: S 0002-9939(08)09324-6
Received by editor(s): May 14, 2007
Posted: February 15, 2008
Communicated by: Wen-Ching Winnie Li
Copyright of article: Copyright 2008, American Mathematical Society

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