TWO CONDITIONAL RESULTS ABOUT PRIMES IN SHORT INTERVALS

In 1937, Ingham proved that ψ(x + xθ) - ψ(x) ~ xθ for x → ∞, under the assumption of the Lindelof hypothesis for θ > 1/2. In this paper we examine how the above asymptotic formula holds by assuming in turn two different heuristic hypotheses. It must be stressed that both the hypotheses are implied by the Lindelof hypothesis.