The Difference between Consecutive Primes, Ii

With enough effort, the value of x0 could be determined effectively. The paper has much in common with [1]; in particular we use the sieve method of Harman [4, 5]. We no longer use zero density estimates, however, but rather mean value results on Dirichlet polynomials similar to those that give rise to such estimates. Compare, for example, work of Iwaniec and Pintz [9] and Baker, Harman and Pintz [2]. Much of the improvement over [1] arises from the use of Watt's theorem [11] on a particular kind of mean value. More accurate estimates for six-dimensional integrals are also used to good effect. There is in addition a device which uses a two-dimensional sieve to get an asymptotic formula for a `one-dimensionally sieved' set; see Lemmas 16, 17. Unfortunately, these lemmas, which would be of great signi®cance for v ˆ 0:53, are not very numerically signi®cant when v drops to 0.525; the same applies to the `roÃle reversals' discussed below. Let us introduce enough notation to permit an outline of the proof. When E is a ®nite sequence of positive integers, counted with multiplicity, we write jEj for the number of terms of E, and Ed ˆ fm: dm 2 Eg: Let P…z† ˆ Y