Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average
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- by Alina Carmen Cojocaru and Igor E. Shparlinski PDF
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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.References
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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
- MR Author ID: 703080
- Email: cojocaru@math.uic.edu
- Igor E. Shparlinski
- Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
- MR Author ID: 192194
- Email: igor@ics.mq.edu.au
- Received by editor(s): May 14, 2007
- Published electronically: February 15, 2008
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 1977-1986
- MSC (2000): Primary 11B57, 11G07, 14H52
- DOI: https://doi.org/10.1090/S0002-9939-08-09324-6
- MathSciNet review: 2383504