Average distributions and products of special values of L-series

Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average

“…Several more interesting questions on elliptic curves have been studied "on average" for similar families of curves in [1,3,5,7,8,18,20,21].…”

Average Frobenius distribution of elliptic curves

“…We would also like to point out that in Theorem 1, and in the main theorems of [1] and [3], the minimal ranges of A and B can be reduced a little by estimating some exponential sums. This is of independent interest but not our focus in this paper.…”

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

“…Fouvry and R. M. Murty [13], and C. David and F. Pappalardi [8], have considered the average of Π E (a, x) for the family of curves Y 2 = X 3 + uX + v where the integers u and v satisfy the inequalities |u| ≤ U , |v| ≤ V ; they have shown that if UV ≥ x 3/2+ε and min{U, V } ≥ x 1/2+ε for some fixed positive ε > 0, then, "on average", the Lang-Trotter conjecture holds for such curves. This result has been extended in various directions [1,2,3,4,9,14,15,16]. A. C. Cojocaru and C. Hall [7] have recently considered the one parametric family of curves of the form (1) and established an improved upper bound on the average value of Π E (a, x) over curves of such families when the parameter t runs through the elements of F(T ) with T of the same order of magnitude as x.…”

“…We use some results of [20] to show that the elements of this set are uniformly distributed in residue classes modulo a prime p. More precisely, we prove: (1) .…”