On the Riemann hypothesis and the difference between primes

Abstract

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 [Formula: see text] 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 [Formula: see text] is greater than [Formula: see text] for c = 3 + ϵ and all sufficiently large values of x.