Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Question

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) $k>1$?

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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.

Answer 1

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.

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In Goldston, Pintz, and Yıldırım 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 Yıldırım'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.

Answer 2

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

Answer 3

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.

Answer 4

Good question! Perhaps it is worthwhile to note that Goldston–Pintz–Yıldırım 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–Yıldırım in several papers. The current best result, as far as I know, appears in Goldston–Pintz–Yıldırım: Primes in tuples III, Funct. Approx. Comment. Math. 35 (2006), 79–89.), namely $$ \Delta_\nu\leq e^{-\gamma}(\sqrt{\nu}-1)^2. $$

Answer 5

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).

Answer 6

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 $p_{n+1}-p_n\le 12.$