Is $(\ln p)^\alpha$ a limit for gaps proceeding $p$ for some $\alpha$?

Question

By computational experiments I found that there are no primes $p_n$ with $7<p_n<100,000,000$ such that $p_{n+1}-p_n>(\ln p_n)^\alpha$, if $\alpha\ge 1.8932$.

Are there primes $p_n>7$ such that $p_{n+1}-p_n>(\ln p_n)^2$?

Answer 1

I didn't try to write a code for check if a "small" example exists. However, I just want to recall the Cramer's conjecture which states that $$ \limsup_{n\to \infty}\,\frac{p_{n+1}-p_n}{(\log p_n)^2}=1. $$ You can find therein related references about large gaps on consecutive primes. In particular, take a look at these recent Ann. of Math. articles here and here.

Answer 2

Not currently known.

See Nicely's discussion here: http://trnicely.net/#MaxMerit and elsewhere on his pages and publications.

The largest value of the CSG ratio $\frac{G}{\log(p1)^2}$ is 0.9206386 for G=1132 P1= 1693182318746371.

That comes out to an alpha for your formula of ~ 1.98.