Exponential sums over primes in short intervals

Abstract

Let Λ(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 intervalsSk(x,y;α)=∑x<n≤x+yΛ(n)e(nkα) when k≥4 for all α∈[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.