The Exceptional Set for the Number of Primes in Short Intervals, Original Research Article

We investigate the exceptional set Eδ(X, h) associated with the asymptotic formula for the number of primes in short intervals; see Section 1 for the definition. We first obtain two results about the basic structure of this set, proving the inertia and decrease properties; see Theorem 1. Then we turn to estimates for the size of Eδ(X, h), showing that non-trivial bounds for Eδ(X, h) can be obtained when h(x)=xθ and 1/6<θ<7/12; see Theorem 2.