In http://arxiv.org/pdf/1405.0682.pdf, the author gives a conditional proof of the twin prime conjecture under both a generalized version of the ElliottHalberstam conjecture and a hypothesis on the main terms of the GPY sieve. My question is: can a similar approach provide better upper bounds for the quantities $H_{k},k>1$ where $H_{k}:=\lim\inf_{n\to\infty} p_{n+k}p_{n}$ than those currently obtained thanks to the Polymath8b project?
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$\begingroup$ I think this is not a mathematical question but a psychological one. More precisely, there is only one way to know if a certain result can be improved: someone does it or proves that the improved result is false. $\endgroup$ – GH from MO May 11 '14 at 16:37

$\begingroup$ I've been to a series of talks by Maynard just recently (going beyond the "regular" presentation he gives). In the closing remarks, he spoke about Generalized EH conjecture and others, and he claimed (I hope I'm quoting him right) that the maximum that one can hope for in those sieve approaches is to get $p_{n+1)p_{n}\leq 3$, but not $2$. This somehow contradicts the claimed results in the paper, although the results in the paper you cited seem to use different sieve. Anyways, I would be extremely careful with the paper you've linked until it will be refereed. $\endgroup$ – Asaf May 11 '14 at 19:47

$\begingroup$ I'm sorry but I don't get it. As the gap between two consecutive primes is necessarily even, if one gets $H_{1}\leq 3$, then $H_{1}\leq 2$. $\endgroup$ – Sylvain JULIEN May 11 '14 at 20:35

1$\begingroup$ I think Asaf meant $H_1\leq 6$ instead of $H_1\leq 2$. This is now achieved by PolyMath8b under the generalized ElliottHalberstam conjecture (finding two primes in the translates of any admissible triple). $\endgroup$ – GH from MO May 11 '14 at 21:28

$\begingroup$ Yes I had a mistake there, I guess GH is much more informed than I am about the polymath project going on. Anyways, the bottom line is that according to Maynard, you won't be able to get twinprimes by his approach, no matter what kind of generalized feasible conjectures you'll make. Maybe the extra conjecture made in the paper you've linked to is equivalent to twinprime or so. $\endgroup$ – Asaf May 12 '14 at 6:30