On the Riemann hypothesis and the difference between primes

We prove some results concerning the distribution of primes assuming the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval (x - 4/π x log x, x] for all x ≥ 2; this improves a result of Ramaré and Saouter. We then show that the constant 4/π may be reduced to (1 + ε) provided that x is taken to be sufficiently large. From this, we get an immediate estimate for a well-known theorem of Cramér, in that we show the number of primes in the interval (x, x+c x log x] is greater than x for c = 3 + ε and all sufficiently large values of x.