On the Riemann Hypothesis and the Difference Between Primes

Abstract

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 $x$.