On ratios of consecutive prime gaps (Q6546710)

Let \(p_n\) be the \(n\)-th prime number, \(d_n=p_{n+1}-p_n\) be the \(n\)-th prime gap, \(\pi(x)\) be the prime counting function, and for any fixed \(c \geq 0\) the function \(\pi_c(x)\) defined by\N\[\N\pi_c(x)=\#\{p_n\leq x : d_{n+1}/d_n\geq c\}.\N\]\NAn elementary heuristic argument suggests that \(\pi_c(x)\sim\pi(x)/(c+1)\) as \(x\to\infty\). In the paper under review, the author provides a more delicate heuristic argument, based on a quantitative form of the Hardy-Littlewood prime \(k\)-tuple conjecture, supporting the following asymptotic for any given \(c\geq 0\) and any \(\varepsilon>0\),\N\[\N\pi_c(x)=\frac{1}{c+1}\,\pi(x)+O_{c,\varepsilon}\left(x(\log x)^{-3/2+\varepsilon}\right).\N\]