On the Riemann Hypothesis and the Difference Between Primes

We prove some results concerning the distribution of primes on the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval $(x-\frac{4}{\pi }\sqrt{x}\log x,x]$ for all $x\ge 2$; this improves a result of Ramar'{e} and Saouter. We then show that the constant $4\mathrm{/}\pi$ 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 Cram'{e}r, in that we show the number of primes in the interval $(x,x+c\sqrt{x}\log x]$ is greater than $\sqrt{x}$ for $c=3+ϵ$ and all sufficiently large $x$.