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 $x \geq 2$; this improves a result of Ramar\'{e} and Saouter. We then show that the constant $4/\pi$ may be reduced to $(1+\epsilon)$ 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 $c=3+\epsilon$ and all sufficiently large $x$.