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 Ramaré 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 wellknown 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 $c=3+\epsilon$ and all sufficiently large $x$.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.6417
 Bibcode:
 2014arXiv1402.6417D
 Keywords:

 Mathematics  Number Theory
 EPrint:
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