A simplified proof of primes in almost all short intervals

We give a simplified proof of Jia's 1996 result which states that the interval n, n+n^1{20+} contains prime numbers for almost all n. Furthermore, we show that for almost all n, the shorter interval n, n+n^1{21+} still contains prime numbers. A delicate sieve decomposition and more accurate estimates for integrals are used to good effect.