In a recent paper, Yitang Zhang has proven the following theorem:

Theorem 1 (Bounded gaps between primes) There exists a natural number ${H}$ such that there are infinitely many pairs of distinct primes ${p,q}$ with ${|p-q| \leq H}$.

Zhang obtained the explicit value of ${70,000,000}$ for ${H}$. A polymath project has been proposed to lower this value and also to improve the understanding of Zhang’s results; as of this time of writing, the current “world record” is ${H = 4,802,222}$ (and the link given should stay updated with the most recent progress.

Zhang’s argument naturally divides into three steps, which we describe in reverse order. The last step, which is the most elementary, is to deduce the above theorem from the following weak version of the Dickson-Hardy-Littlewood (DHL) conjecture for some ${k_0}$:

Theorem 2 (${DHL[k_0,2]}$) Let ${{\mathcal H}}$ be an admissible ${k_0}$-tuple, that is to say a tuple of ${k_0}$ distinct integers which avoids at least one residue class mod ${p}$ for every prime ${p}$. Then there are infinitely many translates of ${{\mathcal H}}$ that contain at least two primes.

Zhang obtained ${DHL[k_0,2]}$ for ${k_0 = 3,500,000}$. The current best value of ${k_0}$ is ${341,640}$, as discussed in this previous blog post. To get from ${DHL[k_0,2]}$ to Theorem 1, one has to exhibit an admissible ${k_0}$-tuple of diameter at most ${H}$. For instance, with ${k_0 = 341,640}$, the narrowest admissible ${k_0}$-tuple that we can construct has diameter ${4,802,222}$, which explains the current world record. There is an active discussion on trying to improve the constructions of admissible tuples at this blog post; it is conceivable that some combination of computer search and clever combinatorial constructions could obtain slightly better values of ${H}$ for a given value of ${k_0}$. The relationship between ${H}$ and ${k_0}$ is approximately of the form ${H \approx k_0 \log k_0}$ (and a classical estimate of Montgomery and Vaughan tells us that we cannot make ${H}$ much narrower than ${\frac{1}{2} k_0 \log k_0}$, see this previous post for some related discussion).

The second step in Zhang’s argument, which is somewhat less elementary (relying primarily on the sieve theory of Goldston, Yildirim, Pintz, and Motohashi), is to deduce ${DHL[k_0,2]}$ from a certain conjecture ${MPZ[\varpi,\delta]}$ for some ${\varpi,\delta > 0}$. Here is one formulation of the conjecture, more or less as (implicitly) stated in Zhang’s paper:

Conjecture 3 (${MPZ[\varpi,\delta]}$) Let ${{\mathcal H}}$ be an admissible tuple, let ${h_i}$ be an element of ${{\mathcal H}}$, let ${x}$ be a large parameter, and define

$\displaystyle D := x^{1/4+\varpi},$

$\displaystyle {\mathcal P} := \prod_{p: p < x^{\delta}} p,$

$\displaystyle P(n) := \prod_{h \in {\mathcal H}} (n+h),$

$\displaystyle C_i(d) := \{ c \in {\bf Z}/d{\bf Z}: (c,d) = 1; P(c-h_i) = 0 \hbox{ mod } d \}$

for any natural number ${d}$, and

$\displaystyle \Delta(\gamma;d,c) = \sum_{x \leq n \leq 2x: n = c \hbox{ mod } d} \gamma(n) - \frac{1}{\varphi(d)} \sum_{x \leq n \leq 2x: (n,d) = 1} \gamma(n)$

for any function ${\gamma: {\bf N} \rightarrow {\bf C}}$. Let ${\theta(n)}$ equal ${\log p}$ when ${n}$ is a prime ${p}$, and ${\theta(n)=0}$ otherwise. Then one has

$\displaystyle \sum_{d < D^2; d|{\mathcal P}} \sum_{c \in C_i(d)} |\Delta(\theta; d, c )| \ll x \log^{-A} x$

for any fixed ${A > 0}$.

Note that this is slightly different from the formulation of ${MPZ[\varpi]}$ in the previous post; I have reverted to Zhang’s formulation here as the primary purpose of this post is to read through Zhang’s paper. However, I have distinguished two separate parameters here ${\varpi,\delta}$ instead of one, as it appears that there is some room to optimise by making these two parameters different.

In the previous post, I described how one can deduce ${DHL[k_0,2]}$ from ${MPZ[\varpi,\delta]}$. Ignoring an exponentially small error ${\kappa}$, it turns out that one can deduce ${DHL[k_0,2]}$ from ${MPZ[\varpi,\delta]}$ whenever one can find a smooth function ${g: [0,1] \rightarrow {\bf R}}$ vanishing to order at least ${k_0}$ at ${1}$ such that

$\displaystyle k_0 \int_0^1 g^{(k_0-1)}(x)^2 \frac{x^{k_0-2}}{(k_0-2)!}\ dx > \frac{4}{1+4\varpi} \int_0^1 g^{(k_0)}(x)^2 \frac{x^{k_0-1}}{(k_0-1)!}\ dx.$

By selecting ${g(x) := \frac{1}{(k_0+l_0)!} (1-x)^{k_0+l_0}}$ for a real parameter ${l_0>0}$ to optimise over, and ignoring the technical ${\kappa}$ term alluded to previously (which is the only quantity here that depends on ${\delta}$), this gives ${DHL[k_0,2]}$ from ${MPZ[\varpi,\delta]}$ whenever

$\displaystyle k_0 > (\sqrt{1+4\varpi} - 1)^{-2} \ \ \ \ \ (1)$

It may be possible to do better than this by choosing smarter choices for ${g}$, or performing some sort of numerical calculus of variations or spectral theory; people interested in this topic are invited to discuss it in the previous post.

The final, and deepest, part of Zhang’s work is the following theorem (Theorem 2 from Zhang’s paper, whose proof occupies Sections 6-13 of that paper, and is about 32 pages long):

Theorem 4 (Zhang) ${MPZ[\varpi,\varpi]}$ is true for all ${0 < \varpi \leq \frac{1}{1168}}$.

The significance of the fraction ${1/1168}$ is that Zhang’s argument proceeds for a general choice of ${\varpi > 0}$, but ultimately the argument only closes if one has

$\displaystyle \frac{31}{32} + 36 \varpi \leq 1 - \frac{\varpi}{2}$

(see page 53 of Zhang) which is equivalent to ${\varpi \leq 1/1168}$. Plugging in this choice of ${\varpi}$ into (1) then gives ${DHL[k_0,2]}$ with ${k_0 = 341,640}$ as stated previously.

Improving the value of ${\varpi}$ in Theorem 4 would lead to improvements in ${k_0}$ and then ${H}$ as discussed above. The purpose of this reading seminar is then twofold:

1. Going through Zhang’s argument in order to improve the value of ${\varpi}$ (perhaps by decreasing ${\delta}$); and
2. Gaining a more holistic understanding of Zhang’s argument (and perhaps to find some more “global” improvements to that argument), as well as related arguments such as the prior work of Bombieri, Fouvry, Friedlander, and Iwaniec that Zhang’s work is based on.

In addition to reading through Zhang’s paper, the following material is likely to be relevant:

I envisage a loose, unstructured format for the reading seminar. In the comments below, I am going to post my own impressions, questions, and remarks as I start going through the material, and I encourage other participants to do the same. The most obvious thing to do is to go through Zhang’s Sections 6-13 in linear order, but it may make sense for some participants to follow a different path. One obvious near-term goal is to carefully go through Zhang’s arguments for ${MPZ[\varpi,\delta]}$ instead of ${MPZ[1/1168,1/1168]}$, and record exactly how various exponents depend on ${\varpi,\delta}$, and what inequalities these parameters need to obey for the arguments to go through. It may be that this task can be done at a fairly superficial level without the need to carefully go through the analytic number theory estimates in that paper, though of course we should also be doing that as well. This may lead to some “cheap” optimisations of ${\varpi}$ which can then propagate to improved bounds on ${k_0}$ and ${H}$ thanks to the other parts of the Polymath project.

Everyone is welcome to participate in this project (as per the usual polymath rules); however I would request that “meta” comments about the project that are not directly related to the task of reading Zhang’s paper and related works be placed instead on the polymath proposal page. (Similarly, comments regarding the optimisation of ${k_0}$ given ${\varpi}$ and ${\delta}$ should be placed at this post, while comments on the optimisation of ${H}$ given ${k_0}$ should be given at this post. On the other hand, asking questions about Zhang’s paper, even (or especially!) “dumb” ones, would be very appropriate for this post and such questions are encouraged.