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- R. C. Baker and G. Harman, "The difference between consecutive primes," Proc. Lond. Math. Soc., series 3, 72 (1996) 261--280. MR 96k:11111
The main result of the paper is that for all large x, the interval A=[x-x0.535,x] contains prime numbers. The most recent published result meeting rigorous standards is due to Iwaniec and Pintz (0.547... in place of 0.535). The idea is to begin with asymptotic formulas for sums over products such as pqm in A where p and q run over primes in suitably restricted intervals and m over some set of integers. One then builds on these formulae using the sieve method of Harman (`On the distribution of α p modulo one' J. London Math. Soc. 27 (1983), 9--18), to obtain asymptotic formula for sums of the type
∑m ∑n am bn S(Amn, z), the number z being a positive power of x depending on the size of m and n. From this point, the use of Buchstab's identity enables one to reach a lower bound for the number of primes in A of c times the expected value. Certain integrals in two and four dimensions must be bounded above, using a computer calculation, in order to ensure a positive value of c.