On the exceptional set in Goldbach’s problem in short intervals

Abstract

Let 1/5 < θ ≤ 1. We prove that there exists a positive constant δ such that the number of even integers in the interval [X, X + Xθ] which are not a sum of two primes is 《 Xθ−δ. The proof uses the circle method, a sieve method, exponential sum estimates and zero-density estimates for L-functions.