An inequality on prime gaps

Question

Is there a good inequality for prime gaps. Like $p_{k}-p_{k-1}\leq f(k)$ ? In other words is there a known upper bound for $p_{k}-p_{k-1}$?

Answer 1

Bertrand's postulate gives that $p_k-p_{k-1}\le p_{k-1}.$ A result of Baker, Harman, & Pintz can be used to improve this to $p_k-p_{k-1}\ll p_{k-1}^{0.525}.$

It is conjectured that $p_k-p_{k-1}\ll \log^2 p_{k-1},$ perhaps with a constant as small as $2e^{-\gamma}\approx1.1229\ldots.$

Answer 2

You can find some results in the following link: https://primegaps.files.wordpress.com/2016/09/gap-between-consecutive-primes-by-using-a-new-approach.pdf