An asymptotic upper bound on prime gaps

TL;DR

This paper explores bounds on prime gaps, proving a weaker summed inequality using Selberg's formula, and investigates properties of prime counting fluctuations that could imply the Cramér-Granville conjecture, linking it to the Skewes number.

Contribution

It proves a summed version of the Cramér-Granville conjecture using Selberg's formula and analyzes fluctuation properties that might imply the conjecture, including assumptions related to the Riemann Hypothesis.

Findings

Proved the inequality: sum of prime gaps < sum of log^2 p_n.

Identified properties of prime counting fluctuations that could imply the conjecture.

Linked the conjecture's validity to the magnitude of the Skewes number.

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.