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](https://eudml.org/doc/205441), 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?



A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis
[On the Riemann hypothesis and the difference between primes](https://doi.org/10.1142/S1793042115500426) by Adrian W. Dudek states the result (Theorem 3, at least in [the arXiv version](https://arxiv.org/abs/1402.6417)) that any $C>1$ works (for $n$ sufficiently large).


