Short cubic exponential sums over primes

Abstract

For y ≥ x 4/5 L 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 S 3(α; x, y) = ∑ x−y<n≤x Λ(n)e(αn 3), where α = a/q + θ/q 2, (a, q) = 1, L 32(B+20) < q ≤ y 5 x −2 L −32(B+20), |θ| ≤ 1, Λ is the von Mangoldt function, and e(t) = e 2πit.