Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
1  
"brothers-in-law primes"? –  Pietro Majer Jul 23 at 20:47

1 Answer 1

up vote 8 down vote accepted

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.

share|improve this answer
    
So could we deduce from the twin prime conjecture that the number of twin primes below $x$ is $\gg x/log^{2} x$? –  Sylvain JULIEN Jul 23 at 17:22
3  
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 $x$. –  Gerry Myerson Jul 24 at 0:43

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.