In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound for the number of primes $p$ below $x$ such that $p'-p=g$, with $p'$ the smallest prime strictly greater than $p$?
Thanks in advance.
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asked Jul 23, 2014 at 14:06
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2$\begingroup$ "brothers-in-law primes"? $\endgroup$– Pietro MajerCommented Jul 23, 2014 at 20:47
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2 Answers
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Zhang's proof can be refined to show that the number of those primes is $\gg x/\log^k x$, where $k$ is the size of the tuple whose translates contain the relevant pairs, i.e. $k=3{,}500{,}000$ in the original proof and $k=50$ in the current record by PolyMath8b. For more details see the Main Theorem in Pintz's article here.
answered Jul 23, 2014 at 16:25
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$\begingroup$ So could we deduce from the twin prime conjecture that the number of twin primes below $x$ is $\gg x/log^{2} x$? $\endgroup$ Commented Jul 23, 2014 at 17:22
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7$\begingroup$ He didn't say it followed from any conjecture: he said it followed from Zhang's proof. When someone proves the twin prime conjecture, then it will be time to see what the proof implies about the number of twin primes below $x$. $\endgroup$ Commented Jul 24, 2014 at 0:43
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A paper related to the prime gaps, more accurately - maximal prime gaps, with a complete set of bounds including Supremum and Infimum with an estimation error at the last known maximal gap 80 being approx. 0.03. Are in "Maximal prime gaps bounds" by J. Feliksiak
https://www.scienceopen.com/document?vid=97deccbd-db11-4a0e-a55c-ec97fa313ad9
