Comments on: The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoWed, 20 Jul 2016 03:33:39 +0000
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By: John
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-470793
Wed, 20 Jul 2016 03:33:39 +0000http://terrytao.wordpress.com/?p=6728#comment-470793Oh I guess it’s a simple consequence of the Prime Number Theorem and the discrete integration by part, i.e., let , then one gets that the log of the HL-constant is equal to . The latter is finite since is bounded above and below eventually by $0.2 n / \log n$ and $5 n / \log n$.
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By: John
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-470788
Wed, 20 Jul 2016 01:24:02 +0000http://terrytao.wordpress.com/?p=6728#comment-470788Why is obvious that the Hardy-Littlewood infinite product does not converge to 0?
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By: Where has the Time Gone? | Sublime Illusions
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-467574
Wed, 06 Apr 2016 05:38:37 +0000http://terrytao.wordpress.com/?p=6728#comment-467574[…] Tao explains concepts wonderfully, especially if you want a summary of a long and intricate paper (like the explanation of Zhang Yitang’s work on the Twin Prime Conjecture and related backgroun…). Beware, not for the faint of […]
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By: Terence Tao
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-461131
Fri, 30 Oct 2015 15:32:43 +0000http://terrytao.wordpress.com/?p=6728#comment-461131This can be found for instance in the texts of Montgomery-Vaughan or Iwaniec-Kowalski, or any book on sieve theory (e.g. Friedlander-Iwaniec). I also have it as an application of the Selberg sieve in Exercise 34 of these notes.
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By: Alexey
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-461113
Fri, 30 Oct 2015 04:06:11 +0000http://terrytao.wordpress.com/?p=6728#comment-461113I have a question about an upper bound of conjecture 2 (about k-tuple of primes). Where can I find a proof of this upper bound? How C_{k_0} depends from k_0?

Thank you

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By: The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang | fragments
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-456596
Wed, 17 Jun 2015 15:49:20 +0000http://terrytao.wordpress.com/?p=6728#comment-456596[…] for instance, if Conjecture 2 holds, then the number of twin primes less than should equal , where is the twin prime […]
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By: wanglaoxinr
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-455907
Sat, 16 May 2015 08:13:26 +0000http://terrytao.wordpress.com/?p=6728#comment-455907for Cramér Conjecture ，change Pn to x, will that hold?
Pn means n th prime, Pn<Pn+1<=x, Max(Pn+1-Pn) means the maximum gap between two neighboring primes in (0,x], when x tending to infinity, Max(Pn+1-Pn)/(lnx)^2 tending to 1.
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By: Together and Alone, Closing the Prime Gap | For a better Vietnam
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-401506
Thu, 31 Jul 2014 16:33:25 +0000http://terrytao.wordpress.com/?p=6728#comment-401506[…] work. At the time, though, this advance was one of the most fruitful ones. When the team filled in this piece of the puzzle on June 5, the bound on prime gaps dropped from about 4.6 million to […]
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By: Polymath8b: Bounded intervals with many primes, after Maynard | What's new
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-251694
Wed, 20 Nov 2013 06:50:21 +0000http://terrytao.wordpress.com/?p=6728#comment-251694[…] this claim was already essentially established back in this Polymath8a post (or see Proposition 4.1 and Lemma 4.7 of the Polymath8a paper for essentially these bounds). In […]
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By: Stijn Hanson
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-247237
Fri, 04 Oct 2013 20:43:55 +0000http://terrytao.wordpress.com/?p=6728#comment-247237I assumed it had something to do with that w restriction but I don’t see where comes from. The way I understand it we’re setting and then is some real number such that the term dips below whenever but, obviously, I must be missing something.
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