The asymptotic number of integral cubic polynomials with bounded heights and discriminants

Let P denote a cubic integral polynomial, and let D(P) and H(P) denote the discriminant and height of P, respectively. Let N(Q,X) be the number of cubic integral polynomials P such that H(P) ≤ Q and |D(P)| ≤ X. We obtain an asymptotic formula of N(Q,X) for Q ^14/5 ? X ? Q ⁴ and Q → +∞. Using this result, for 0 ≤ η ≤ 9/10, we find the asymptotic value of $$ sumlimits_{{begin{array}{*{20}{c}} {H(P)leq Q} {1leq left| {D(P)} right|ll {Q^{{4-eta }}}} end{array}}} {{{{left| {D(P)} right|}}^{{-{1 left/ {2} right.}}}}}, $$ where the sum is taken over irreducible integral polynomials and Q → +∞. This improves upon a result of Davenport, who dealt with the case η = 0.