Exponential sums over primes in short intervals

Let Lambda(n) be the von Mangoldt function, x real and 2 <= y <= x. This paper improves the estimate on the exponential sum over primes in short intervals S-k(x, y; alpha) = Sigma Lambda(n)e(n(k)alpha) x<n <= x+y when k >= 4 for all alpha is an element of [0,1]. And then combined with the Hardy-Littlewood method, this enables us to give some short interval variants of Hua's theorems in additive prime number theory. (C) 2014 Elsevier Inc. All rights reserved.