Short cubic exponential sums over primes

Abstract

For y ≥ x^4/5L^8B+151 (where L = log(xq) and B is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form S₃(α; x, y) = ∑_x−y<n≤x Λ(n)e(αn³), where α = a/q + θ/q², (a, q) = 1, L^32(B+20) < q ≤ y⁵x⁻²L^−32(B+20), |θ| ≤ 1, Λ is the von Mangoldt function, and e(t) = e^2πit.