An asymptotic upper bound on prime gaps

Abstract

The Cram\'er-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \cCramer \, \log^2 p_n$ for some constant $\cCramer \geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\sum_{n=1}^N g_n < \sum_{n=1}^N \log^2 p_n$. In the remainder of the paper we investigate which properties of the fluctuations $\fluc (x) = \pi (x) - \Li(x)$ would imply the Cram\'er-Granville conjecture is true and present two such properties, one of which assumes the Riemann Hypothesis; however we are unable to prove these properties are indeed satisfied. We argue that the conjecture is related to the enormity of the Skewes number.