An asymptotic upper bound on prime gaps
arXiv:1506.03359
Abstract
The Cramér-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: . In the remainder of the paper we investigate which properties of the fluctuations $\fluc (x) = π(x) - \Li(x)$ would imply the Cramér-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.
The proof of the last theorem of the last version is incorrect, because the fluctuations were treated too smoothly. We could only replace it with a weaker result