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

Bull. Polish Acad. Sci. Math.

2009

of Farey fractions of order N. There is an extensive literature where various distributional properties of Farey fractions are investigated. In particular, some of these properties are directly related to the Riemann Hypothesis. (See, for example, the survey [4] as well as the more recent works [1, 2, 3, 7, 8] and references therein.) Here we consider an apparently new problem of the distribution of F(N) in residue classes modulo an integer m ≥ 2. In particular, we show that for any interval I = [k, k + h − 1] ⊆ [0, m − 1] and any integer N ≥ m ε , where ε > 0 is fixed, the number R m (N, I) of Farey fractions s/r ∈ F(N) with gcd(r, m) = 1 and such that s/r ≡ z (mod m) for some z ∈ I is close to its expected value, provided that m is large enough. Naturally, our main tool is bounds for exponential sums. For any integer m ≥ 2 and any real z we put e m (z) = exp(2πiz/m)

mentioning

confidence: 95%

Exponential Sums with Farey Fractions

Bull. Polish Acad. Sci. Math.

2009

“…In addition, for the family of curves (1.2), Cojocaru and Hall [12] have given an upper bound on the frequency of the event a p (E(t)) = a for a fixed integer a, when the parameter t runs through the set F (T ). This bound has been improved by Cojocaru and Shparlinski [13] and then further improved by Sha and Shparlinski [23].…”

Section: 2mentioning

confidence: 97%

The Sato–Tate distribution in families of elliptic curves with a rational parameter of bounded height

Sha

Indagationes Mathematicae

2017

“…We now recall that the common feature of the approaches of both [6] and [17] is that they need two independently varying parameters u and v. This has been a part of the motivation for Cojocaru and Hall [11] and Cojocaru and Shparlinski [12] to consider the family of curves (2). However, even this family cannot be considered as a truly single parametric family of curves, because the simple exclusion-inclusion principle reduces a problem with the parameter t ∈ F (T ) to a series of problems with t = u/v, where u and v run independently through some intervals of consecutive integers.…”

Section: 2mentioning

confidence: 99%

Lang–Trotter and Sato–Tate distributions in single and double parametric families of elliptic curves

Sha

2015

Acta Arith.

Abstract. We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterized families of elliptic curves when the parameter runs through a set of rational numbers of bounded height. Some of the families we consider are much thinner than the ones previously studied.