Cramer's conjecture and Jacobsthal function

Question

This question is a follow-up to my comment to the answer to this question. Writing $g_{n}:=p_{n+1}-p_{n}$, and as all numbers between $p_{n}$ and $p_{n+1}$ are composite, one has $j(p_{n})=O(\log^{2}p_{n})$ from the result of Iwaniec. As one has the obvious inequality $\max_{a<x}(f(a)g(a))\leq \max_{a<x}f(a)\times\max_{b<x}g(b)$, with $f$ and $g$ both taking positive values on the integers, can one hope to get a weak form of Cramer's conjecture?

Answer 1

I refer to the version of the question that suggests $j(p_n)=O((\log p_n)^2)$. Actually, Iwaniec proved results of a qualitative character using the linear sieve (about $r^2$ many coprimes to $n$ appearing in an interval of length $O(\pi^{-1}(n)r^2\log r)$, where $n$ has $r$ distinct prime factors and $\pi^{-1}(n)$ is $O(\log r)$). As noted in a comment $j(p_n)$ is 2, while if we take $n$ to be $P_r$, the product of the first $r$ primes, Iwaniec's result gives $j(P_r)$ is $O((r\log r)^2)$, which is not far from $O(p_r^2)$.

While Jacobsthal's function $j(n)$ is useful in providing upper bounds to lengths of gaps between numbers coprime to $n$ and also lower bounds on prime gaps, Cramer's conjecture deals with gaps between primes which may contain totatives to a given number. Such a gap may look like sequences of such totative-free intervals laced with totatives which are composite numbers, making a prime gap potentially much larger than a totative-free interval. The best upper bound for the prime gap $g_n$ is the oft quoted Baker-Harman-Pintz bound of $O(p_n^{0.525})$ which is far from the lower bound found in http://arXiv.org/abs/1412.5029 which is strictly smaller than Cramer's $O((\log p_n)^2)$.

It is not clear to me that the inequality involving $f$ and $g$ is of any use here, since the theory at present says nothing about the totatives that are the endpoints of the intervals predicted by the Jacobsthal function.

Gerhard "Guesses A 'No' Answer Here" Paseman, 2016.07.31.