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The following quote comes from Wikipedia http://en.wikipedia.org/wiki/Prime_gap

"Usually the ratio of $g_n / \log p_n$ is called the ''merit'' of the gap $g_n$;. In 1931, E. Westzynthius proved that prime gaps grow more than logarithmically. That is,

$$\limsup_{n\to\infty}\frac{g_n}{\log p_n}=\infty.$$"

Cramér's conjecture and Firoozbakht's conjecture states

$$\limsup_{n\to\infty}\frac{g_n}{\log^2 p_n} < \infty.$$

Has the folloing been proved?:

$$\limsup_{n\to\infty}\frac{g_n}{\log^3 p_n} < \infty.$$

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    $\begingroup$ No, not even remotely; the only bounds known are polynomial in $p_n$. The current best result is still the Baker–Harman–Pintz bound $g_n\le O(p_n^{0{.}525})$ mentioned on the same Wikipedia page. $\endgroup$ – Emil Jeřábek May 8 '15 at 12:17
  • $\begingroup$ @EmilJeřábek: Post as an answer? $\endgroup$ – Charles May 8 '15 at 14:18
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This has not been proved, and results known so far do not get anywhere close: we only have upper bounds of the form $p_n^\theta$ for constant $\theta$. The best currently known unconditional bound is $$g_n=O(p_n^{0{.}525}),$$ due to Baker, Harman, and Pintz [1]. Even assuming the Riemann hypothesis, we only know $g_n=O(\sqrt{p_n}\log p_n)$. For a related problem, it is not known (without assuming Cramér’s conjecture or alike) whether the smallest prime larger than $x$ can be computed in deterministic polynomial time, when given $x$ in binary.

[1] Roger C. Baker, Glyn Harman, and János Pintz, The Difference Between Consecutive Primes, II, Proceedings of the London Mathematical Society 83 (2001), no. 3, pp. 532–562. http://dx.doi.org/10.1112/plms/83.3.532

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