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.
