TITLE: A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis
QUESTION [5 upvotes]: Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ that works? In Cramer's original paper (On the order of magnitude of the difference of consecutive prime numbers, Acta Arithmetica 2 26-46) Cramér makes $C=5\lambda$, but I was unable to give an upper bound value to $\lambda$ from a first reading of the paper. Perhaps one of you knows?

REPLY [2 votes]: I wanted to give some additional remarks to Will Sawin's answer and the associated comments.
Carneiro, Milinovich, and Soundararajan certainly have the best result in literature (as far as I'm aware). They showed that $C=21/25$ for sufficiently large $n$, but only $C=22/25$ for $p_n>3$.
It's still an open and interesting question (that I've been thinking about) to determine the lowest value of $C$ without assuming anything beyond the Riemann Hypothesis. Certainly, if one assumes Montgomery's pair correlation (or something stronger), then it's possible to take $C$ arbitrarily small as $n\to\infty$. See this paper by Goldston and Heath-Brown for instance.