Prime gap around $p_n$

Question

As ${\rm li}(x)\sim \pi(x) $ and Cramer's conjecture predicts that the maximal prime gap around $ p_n $ is $ O(\log^{2}p_n) $, does a strong heuristics suggest that this prime gap is approximately $ \int_{1}^{p_n}\left(\frac{dx}{dy}\right)\frac{dy}{y} $ where $ y=\pi(x) $? Indeed the derivative of $ \log^{2}x $ is $ 2\frac{\log x}{x} $ which is approximately $ \frac{2}{\pi(x)} $.

Answer 1

Probably the strongest "conceptual" heuristics of this kind can be found in Marek Wolf's articles:

• M. Wolf, Some heuristics on the gaps between consecutive primes, arXiv:1102.0481 (2011). http://arxiv.org/abs/1102.0481

• M. Wolf, Nearest neighbor spacing distribution of prime numbers and quantum chaos, Phys. Rev. E 89, 022922 (2014). http://arxiv.org/abs/1212.3841

Indeed, Wolf expresses the most likely value of $G(x)$, the maximal prime gap up to $x$, in terms of the prime counting function $\pi(x)$, somewhat like you suggest in your question: $$ G(x) \sim {x\over\pi(x)}(2\log\pi(x) - \log x + c), $$ which, for all practical purposes, is equivalent to $$ G(x) \sim \log^2 x − 2 \log x \log \log x + O(\log x). $$ Wolf's argument is more complicated than that of your question. Nevertheless, your conjecture and Wolf's formula give asymptotically the same (quite realistic) prediction for the size of maximal prime gaps.