# the exponent of convergence of $\frac{p_{n+1}}{p_n}$ to $1$

Let $$p_k$$ be the $$k$$-th prime.

Then $$\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$$ -- this is well known.

I was looking for more specific information: What is the exponent of convergence of $$\frac{p_{n+1}}{p_n}$$ to $$1$$, and found no answer.

I mean, what is $$\lim_{n\to\infty}n^a(\frac{p_{n+1}}{p_n}-1)$$ in terms of $$a$$ (of course $$a>0$$).

• Finding the $a$ such that your limit converges to $0$ is exactly the prime gap problem. The RH implies $a < 1/2$ works – reuns Apr 5 at 19:16

$$p_n^{\big(\frac{p_{n+1}}{p_n}\big)^n} \le n^{p_n}$$
A corollary of this conjecture is the weaker Firoozbakht conjecture $$p_{n+1}^{1/n+1} < p_n^{1/n}$$ which has been verified for $$n \le 10^{17}$$. Both these conjectures are however believed to be false in light of the Cramer-Granville heuristics.