# Cramér's Model - “The Prime Numbers and Their Distribution” - Part 4

Following a previous question (here you'll find an introduction):

A paper by Maier which refutes Cramer's Model suggests we should replace the heuristic "$\Bbb P(n\in\mathcal P)=1/\log n$" with $$\Bbb P(n\in\mathcal P|P^-(n)\gt z)=\frac{1}{\log n}\prod_{p\le z\atop p\in\mathcal P}(1-1/p)^{-1}\quad (z\approx \log n)$$ where $P^{-}(n)$ denotes the least prime number that divides n. The book states that the new heuristic leads us to expects that the strong Cramér's conjecture $$\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{(\log p_n)^2}=1$$ (which is derived in this paper by Cramer) is false and should be replaced by $$\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{(\log p_n)^2}=2\mathrm e^{-\gamma}$$ where $p_n$ denotes the $n^{th}$ prime number, and $\gamma$ denotes Euler's constant. For proving this last implication, I should mention Mertens' formula: $$\prod_{p\le z\atop p\in\mathcal P}(1-1/p)^{-1}=\mathrm e^\gamma\log z+O(1)\quad(z\ge 1)$$

I couldn't prove this.

• the revised Cramer conjecture is by Granville, should not be hard to find. As far as anyone has been able to compute, the Cramer conjecture holds but with the lim sup seeming a bit below 1. Yes available online reference 4 at en.wikipedia.org/wiki/… – Will Jagy Aug 1 '14 at 20:04
• I put the data in math.stackexchange.com/questions/459185/… where my formatting was not that good, but you can copy and paste the prime gap data section into a text file of your own. – Will Jagy Aug 1 '14 at 20:10
• Actually Granville won't conjecture that the limit is $2e^{-\gamma}$, merely that it's at least this big. As far as I know he's not on the record conjecturing that it's even finite. – Charles Aug 2 '14 at 5:54