Distribution of farey fractions in residue classes and lang-trotter conjectures on average

Alina Carmen Cojocaru; Igor E. Shparlinski

doi:10.1090/S0002-9939-08-09324-6

Distribution of farey fractions in residue classes and lang-trotter conjectures on average

Alina Carmen Cojocaru, Igor E. Shparlinski

School of Computing

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10 Citations (Scopus)

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Abstract

We prove that the set of Farey fractions of order T,that is, the set {α/β ∈ ℚ :gcd(α, β) = 1, 1 ≤ α,β ≤ T}, is uniformly distributed in residue classes modulo a prime p provided T ≥; p^1/2+∈ for any fixed ∈ > 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.

Original language	English
Pages (from-to)	1977-1986
Number of pages	10
Journal	Proceedings of the American Mathematical Society
Volume	136
Issue number	6
DOIs	https://doi.org/10.1090/S0002-9939-08-09324-6
Publication status	Published - Jun 2008

Bibliographical note

Copyright 2008 American Mathematical Society. First published in Proceedings of the American Mathematical Society, Vol 136, Iss 6, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9939-08-09324-6

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abstract = "We prove that the set of Farey fractions of order T,that is, the set \{α/β ∈ ℚ :gcd(α, β) = 1, 1 ≤ α,β ≤ T\}, is uniformly distributed in residue classes modulo a prime p provided T ≥; p1/2+∈ for any fixed ∈ > 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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note = "Copyright 2008 American Mathematical Society. First published in Proceedings of the American Mathematical Society, Vol 136, Iss 6, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9939-08-09324-6",

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AU - Cojocaru, Alina Carmen

AU - Shparlinski, Igor E.

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N2 - We prove that the set of Farey fractions of order T,that is, the set {α/β ∈ ℚ :gcd(α, β) = 1, 1 ≤ α,β ≤ T}, is uniformly distributed in residue classes modulo a prime p provided T ≥; p1/2+∈ for any fixed ∈ > 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.

AB - We prove that the set of Farey fractions of order T,that is, the set {α/β ∈ ℚ :gcd(α, β) = 1, 1 ≤ α,β ≤ T}, is uniformly distributed in residue classes modulo a prime p provided T ≥; p1/2+∈ for any fixed ∈ > 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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ER -

Distribution of farey fractions in residue classes and lang-trotter conjectures on average

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