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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 $(x-\frac{4}π \sqrt{x} \log x,x]$ for all $x \geq 2$; 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 $(x, x+c \sqrt{x} \log x]$ is greater than $\sqrt{x}$ for $c=3+ε$ and all sufficiently large $x$.