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Let $p_n$ be the $n$-th prime number, as usual: $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.

For $k=1,2,3,\ldots$, define $$ g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n). $$ Thus the twin prime conjecture asserts $g_1 = 2$.

Zhang's theorem (= weak twin prime conjecture) asserts $g_1 < \infty$.

The prime $m$-tuple conjecture asserts $g_2 = 6$ (infinitely many prime triplets), $g_3 = 8$ (infinitely many prime quadruplets), "etcetera" (with $m=k+1$).

Can Zhang's method be adapted or extended to prove $g_k < \infty$ for any (all) $k>1$?


Added a day later: Thanks for all the informative comments and answers! To summarize and update (I hope I'm attributing correctly):

0) [Eric Naslund] The question was already raised in the Goldston-Pintz-Yıldırım paper. See Question 3 on page 3:

Assuming the Elliott-Halberstam conjecture, can it be proved that there are three or more primes in admissible $k$-tuples with large enough $k$? Even under the strongest assumptions, our method fails to prove anything about more than two primes in a given tuple.

1) [several respondents] As things stand now, it does not seem that Zhang's technique or any other known method can prove finiteness of $g_k$ for $k > 1$. The main novelty of Zhang's proof is a cleverly weakened estimate a la Elliott-Halberstam, which is well short of "the strongest assumptions" mentioned by G-P-Y.

2) [GH] For $k>1$, the state of the art remains for now as it was pre-Zhang, giving nontrivial bounds not on $g_k$ but on $$ \Delta_k := \liminf_{n \rightarrow \infty} \frac{p_{n+k} - p_n}{\log n}. $$ The Prime Number Theorem (or even Čebyšev's technique) trivially yields $\Delta_k \leq k$ for all $k$; anything less than that is nontrivial. Bombieri and Davenport obtained $\Delta_k \leq k - \frac12$; the current record is $\Delta_k \leq e^{-\gamma} (k^{1/2}-1)^2$. This is positive for $k>1$ (though quite small for $k=2$ and $k=3$, at about $0.1$ and $0.3$), and for $k \rightarrow \infty$ is asymptotic to $e^{-\gamma} k$ with $e^{-\gamma} \approx 0.56146$.

3) [Nick Gill, David Roberts] Some other relevant links:

Terry Tao's June 3 exposition of Zhang's result and the work leading up to it;

The "Secret Blogging Seminar" entry and thread that has already brought the bound on $g_1$ from Zhang's original $7 \cdot 10^7$ down to below $5 \cdot 10^6$;

A PolyMath page that's keeping track of these improvements with links to the original arguments, supporting computer code, etc.;

A Polymath proposal that includes the sub-project of achieving further such improvements.

4) [Johan Andersson] A warning: phrases such as "large prime tuples in a given [length] interval" (from the Polymath proposal) refer not to configurations that we can prove arise in the primes but to admissible configurations, i.e. patterns of integers that could all be prime (and should all be prime infinitely often, according to the generalized prime $m$-tuple [a.k.a. weak Hardy-Littlewood] conjecture, which we don't seem to be close to proving yet). Despite appearances, such phrasings do not bear on a proof of $g_k < \infty$ for $k>1$, at least not yet.

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Prior to Zhang's paper, even on the full Elliott-Halberstam conjecture no one was able to get bounded intervals with three primes. Note that Goldstone-Pintz-Yildirm did investigate this in arxiv.org/abs/math/0508185. Zhang's contribution is largely a specialized case of the EH conjecture, so I doubt it directly sheds any new light on this problem. On the other hand, perhaps the connection with Elliot-Halberstam was previously under investigated considering that any result would have been conditional. –  Mark Lewko Jun 4 '13 at 19:17
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"On the other hand, perhaps the connection with Elliot-Halberstam was previously under investigated considering that any result would have been conditional." $$ $$ I seriously dispute with this comment, as I think the whole field was pretty much discerned. GPY in particular made extensive effort to apply to $p_{n+\nu}-p_n$. GPY were also quite extensive in their results from partial EH, and indeed the intersection. (See Theorem 3 in Annals paper, one gets $(\sqrt\nu-\sqrt{2\theta})^2$ where $\theta$ is the distro level, and an extra factor of $e^{-\gamma}$ from Maier matrix later). –  v08ltu Jun 4 '13 at 20:16
    
@v08ltu, As I pointed out in my comment GPY did investigate this problem in their original prime tuple paper (and, subsequently returned to the topic in follow-up papers). Certainly their work was a major breakthrough on a problem that was stuck for a long time. By "possibly under investigated" I really meant something like "I don't see any reason to believe the current best results in this direction are fundamentally the limitation of the ideas involved, or that I would be shocked to see further improvements in this direction the way I would be to, say, see twin primes proved by the method" –  Mark Lewko Jun 4 '13 at 20:56
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Fresh and relevant: arxiv.org/abs/1306.0948 –  GH from MO Jun 6 '13 at 6:24
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Now this seems to have been decisively answered! See arxiv.org/pdf/1311.4600.pdf –  Lucia Nov 20 '13 at 2:06

6 Answers 6

Edit (20/11/2013) : Yesterday James Maynard posted the paper Small gaps between primes on the arxiv in which he shows that for any $m$ there exists a constant $C_m$ such that $$ p_{n+m}-p_n\leq C_m$$ infinitely often. More about this result can be found on Terence Tao's blog, or in this expository article by Andrew Granville.


In Goldston, Pintz, and Yildirim paper Primes in tuples I, they show that under the assumption of the Elliott Halberstam Conjecture,

$$\liminf_{n\rightarrow\infty}p_{n+1}-p_n \leq 16$$

and they leave the following question on page 3:

Question 3. Assuming the Elliott-Halberstam conjecture, can it be proved that there are three or more primes in admissible k-tuples with large enough k? Even under the strongest assumptions, our method fails to prove anything about more than two primes in a given tuple.

From what I understand, the issue is increasing a coefficient from $1$ to $2$.
Let $\mathcal{H}=\left\{ 1,\dots,h_{k}\right\}$ be our admissible set, and suppose that $\max_{i}h_{i}\leq x.$ The approach is to look at the sum

$$\sum_{x<n\leq 2x}\left(\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)-\log(3x)\right)W(n),$$

where $\vartheta(n)=1_{\mathcal{P}}(n)\log n$, $1_{\mathcal{P}}(n)$ is the indicator function for the primes, and $W(n)$ is a positive weight function. If this sum is positive, then one of the terms must be positive, so for some $x<n\leq2x$ we have

$$\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)>\log(3x),$$

and since $\log(n+h_{i})\leq\log(3x)$ for all $n$ in our range, it follows that there are at least two indices $i\neq j$ such that

$$\vartheta(n+h_{i}),\ \vartheta(n+h_{j})\neq0.$$

Selberg advocated that in general for ease of calculation one should take a positive weight function to be a square, $W(n)=\lambda(n)^{2},$ so the goal is to prove the inequality $$\sum_{x<n\leq2x}\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)\lambda(n)^{2}>\log(3x)\sum_{x<n\leq2x}\lambda(n)^{2}$$

for some choice of $\lambda(n).$ In Goldston, Pintz, and Yildirim's paper, they choose

$$\lambda(n)=\frac{1}{\left(k+l\right)!}\sum_{\begin{array}{c} d|P(n)\\ d\leq R \end{array}}\mu(d)\log\left(\frac{R}{d}\right)^{k+l}$$

where $P(n)=\prod_{j=1}^{k}\left(n+h_{j}\right)$, and $R$ depends on $x$. To use the same approach for $3$ terms, we would need to examine the sum

$$\sum_{x<n\leq2x}\left(\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)-2\log(3x)\right)\lambda(n)^{2},$$

and show that

$$\sum_{x<n\leq2x}\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)\lambda(n)^{2}>2\log(3x)\sum_{x<n\leq2x}\lambda(n)^{2},$$

for a suitable choice of $\lambda(n)$. Increasing the coefficient to a $2$ seems to be a fundamental issue, and hopefully an expert can explain why this is the case.

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And nothing more can be expected (in fact something less such as a larger constant than 16) to be gained from Zhang's approach than Goldston-Yildirim-Pintz + Elliot-Halberstam! The only reason to possibly expect such progress is that Zhang's paper has resulted in more mathematicians learning the field and trying to improve stuff. But it should not be something simple. Rather another breakthrough is needed. –  Johan Andersson Jun 4 '13 at 16:13

This has now achieved by James Maynard "Small gaps between primes."

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It is probably worth noting that Maynard's method is significantly different from Zhang's (and indeed, is more elementary). Of course this is all explained in the well-written paper. –  Sam Hopkins Nov 20 '13 at 15:24

This is just another comment: in this paper http://arxiv.org/abs/1306.0948 James Maynard generalizes the previous result of Pintz and gives explicit bound on the number of divisors of an almost prime. More precisely, there era infinitely many $n,$ such that the interval $[n, n+10^8]$ contains two primes and an almost prime with at most $31$ prime divisors. Both bounds $31$ and $10^8$ can be improved by using recent improvement of the Polymath project.

Also, in the recent talk, he mentioned that by modifying the weights $\lambda (n)$ in GPY and Zhang and assuming Elliott-Halberstam one can bring the bound $p_{n+1}-p_n\le 16$ to $p_{n+1}-p_n\le 12.$

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Good question! Perhaps it is worthwhile to note that Goldston-Pintz-Yildirim asked a similar question in their original paper (Primes in tuples I, Question 3, Page 822). As of now it is not even known if $$ \liminf_{n \rightarrow \infty} \frac{p_{n+2} - p_n}{\log n}=0.$$

Added. To answer Noam's question addressed in his comment below, let us denote $$ \Delta_\nu:= \liminf_{n \rightarrow \infty} \frac{p_{n+\nu} - p_n}{\log n}, $$ then Bombieri-Davenport (1965) proved $\Delta_v\leq\nu-\frac{1}{2}$, which was improved by Huxley, Maier, Goldston-Yildirim in several papers. The current best result, as far as I know, appears in Goldston-Pintz-Yildirim: Primes in tuples III, Funct. Approx. Comment. Math. 35 (2006), 79–89.), namely $$ \Delta_\nu\leq e^{-\gamma}(\sqrt{\nu}-1)^2. $$

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Thanks for this chapter-and-verse reference (yes, it must be "worthwhile"). So is it known that $\liminf_{n \rightarrow \infty} \frac{p_{n+2} - p_n}{\log n} < 2$? Even this does not follow logically from the G-P-Y or Zhang results. –  Noam D. Elkies Jun 4 '13 at 19:13
    
@Noam: I added some information to my response. –  GH from MO Jun 4 '13 at 19:32
    
Thanks. This last formula $\Delta_\nu \leq e^{-\gamma} (\nu-1)^2$ can't be right, though: it's too strong for $\nu=1$, and too weak for $\nu \geq 4$ (since it then exceeds $\nu$ itself). Did you mean something else? –  Noam D. Elkies Jun 4 '13 at 19:38
    
@Noam: Sorry, I had a typo, $\nu-1$ should be $\sqrt{\nu}-1$. For $\nu=1$ the result is the original breakthrough of Goldston-Pintz-Yildirim (very small gaps between primes). –  GH from MO Jun 4 '13 at 19:43
    
I see, that makes sense now, thanks. –  Noam D. Elkies Jun 4 '13 at 19:57

Pintz paper from arxiv last week

"Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture" http://arxiv.org/abs/1305.6289,

proves some results in this direction. His result is in a sense much stronger than Zhang's as he proves in his Main Theorem that if $ \mathcal H= \{ h_1,\ldots,h_{k} \} $ is an admissable set with $k>k_0=3.5 \cdot 10^6$ (this constant $k_0$ seems to have been improved recently. See Terence Tao's website) we can find infinitely many (and he also gives a lower bound for how many less than $x$) $n$'s such such that the $k-$tuples $n+\mathcal H$ have two primes but also all other elements are almost primes (a bounded number $c$ of prime factors, where the bound $c=c(k)$ depends on $k$). Thus he proves infinitely many "two primes + any fixed number of almost primes" in a bounded range.

Of course he uses Zhang's method of proof (as well as some of his previous results/methods).

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$k_0 = 341640$ at present, see michaelnielsen.org/polymath1/… for progress and references. –  David Roberts Jun 4 '13 at 22:46
    
$k_0=34429$ at the moment –  GH from MO Jun 6 '13 at 22:42

This is slightly too long for a comment....

Terry Tao has just written a long post on Zhang's result and the generalization to tuples. He also links, at the bottom, to a polymath proposal for improving Zhang's bounds, and for extending the result to tuples, as per the OP.

It seems to me that those two sites are the best places to go for a description of the state-of-the-art on this question.

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Thank you for these links. I already saw Tao's post, but didn't see that it claimed results on prime tuples, only on admissible tuples (which are a key tool in Zhang's proof but aren't themselves known to correspond to prime tuples). The polymath proposal does explicitly target "Finding narrow prime tuples of a given cardinality (or, dually, finding large prime tuples in a given interval)" as part of Part 1 of the project, but doesn't seem to say whether it's known that Zhang's techniques can give such a result. –  Noam D. Elkies Jun 4 '13 at 14:56
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@Noam, Tao's formulation of the polymath proposal suggests to me that (as far as he can see) Zhang's techniques may well yield results for prime tuples, but that some work needs to be done to establish this for certain... –  Nick Gill Jun 4 '13 at 15:10
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Nick, It sounds like that but I think it might be a misunderstanding. Finding the narrow prime tuples that Tao talks about I think seems related to given a small k0 finding an optimal admissible set H. (that means just finding one such set, not infinitely many) to get as small gap as possible. On the link he gives, such matters are discussed (finding smallest k0 and optimal admissible set). –  Johan Andersson Jun 4 '13 at 15:27
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As for the original question, I do not think that anyone has any ideas how to treat more than two primes (I might not be correct of course) –  Johan Andersson Jun 4 '13 at 15:28

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