Update: I'm making most of the current version of the book publicly accessible. Comments and other feedback are much appreciated!
Just a few remarks so as to keep everybody informed. (I came across this page
by chance while looking for something else.)
As far as I know, nobody has found any serious issues with the proof.
(There was a rather annoying but non-threatening error that I found in section 11.2 and fixed myself, and of course some typos and slips here are there; none affect the overall strategy or the final result.)
A manuscript containing the full proof was accepted for publication at
Annals of Mathematics Studies back in 2015. I was asked to rewrite matters
fairly substantially for expository reasons, though the extent of the revisions
was left up to my discretion.
Publishing a lengthy proof (about 240 pages in its shortest complete version,
which was considered too terse by some) is never trivial. Publishing it in top
journals, where the backlog is often very large, is even more complicated.
(Many thanks are due to the editors of a top journal -- which does often
publish rather long articles -- for their candid description of complicated decisions in the
editorial process.) I was thus delighted when the manuscript was accepted for
publication in Annals of Mathematics Studies, which publishes book-length
research monographs.
A very detailed referee report was certainly helpful; it was as detailed as
one could reasonably ask from a single author. At the same time, I felt that
it would be best for everybody if there were a second round of refereeing,
with individual referees taking care of separate chapters. So, I asked the publishers for such a second round, and they graciously accepted.
One of the (first-round) referees had suggested that I treat the manuscript as a draft to be fairly thoroughly restructured, and that I add several introductory
chapters. While I found the request a little overwhelming at first, and while
the editors did not demand as much of me, I became convinced that the referee
was right, and set about the task.
What follows is a long, still not quite finished story of a process that took
longer than expected, in part due to my commitments to other projects, in part
perhaps due to a certain perfectionism on my part, in part due to publishing
mishaps that you definitely do not want to hear about, and above all because
it became clear to me, not only that the proof had had fairly few
thorough readers, but that it would be worthwhile for it to have a
substantially wider readership.
To expand on what has been said by other people who replied to or commented
on the original poster's question: knowing that ternary Goldbach holds for all
even integers $n\geq 4$ is not likely to have very many applications, though
it does have some. In that sense it may be seen as the end of a road. The
further use of the proof will reside mainly in the techniques that had to be
applied, developed and sharpened for its sake. For that matter, the same is
arguably true of Vinogradov's work -- it arguably brought the circle method to
its full maturity, after the foundational work of Hardy, Littlewood and
Ramanujan, besides showing the power that combinatorial identities can have in
work on the primes.
From that perspective, it makes sense for the proof to be published as a book
that, say, a graduate student, or a specialist in a neighboring field,
can read with profit. Of course it is still fair and necessary to assume that the reader has taken the equivalent of a first graduate course in analytic number theory.
In the current version, the first hundred pages are taken by an
introduction and by chapters on what can be called the basics of analytic
number theory from an explicit and computational viewpoint. Then come 40 pages on further
groundwork on the estimation of common sums in analytic number theory
- sums over primes, sums of $\mu(n)$, sums of $\mu^2(n)/\phi(n)$, etc. (I should
single out the contributions of O. Ramaré to the explicit understanding of
sums of $\mu(n)/n$ and $\mu^2(n)/\phi(n)$ as invaluable.) Then there
are close to 120 pages on improvements or generalizations on various versions
of the large sieve, their connection to the circle method, and also
on an upper-bound quadratic sieve. (This last subject got a little too
interesting at some point; I am glad my treatment is done!) Then comes an
explicit treatment of exponential sums, in some sense the core of the proof.
(The smoothing function used here has been changed from that in the original
version.)
Then comes the truly complex-analytic part. I am editing that part a little
so that people who are not interested mainly in ternary Goldbach
will be able to take what they need on parabolic cylinder functions, the
saddle-point method or explicit formulas (explicit explicit formulas?). Then
comes the part where different smoothing functions have to be chosen - again,
I am currently editing so that others can readily pick up ideas that probably
have wider applicability. The calculations that are needed for the ternary
Goldbach problem and no other purpose take fewer than 20 pages at the end.
I believe I can say the heavy part is mostly over; I am currently
doing some editing on the second half (or rather the last two fifths) of the
book while waiting to hear from several of the second-round referees I
requested myself. Of course I am also working on other things as well.
All being said, I would not necessarily recommend any non-masochist friend to
write a book-length monograph in the future -- though some other people seem to
manage -- not just because the time things take seems to be quadratic on the
length of the text, which itself increases monotonically, but also because it
is frustrating that it is hard to post periodic updates (certainly
harder than for independent papers), in that always some part of the whole is
undergoing construction. At the same time, I hope to be happy with the end
result.
Best Answer
Zhang proved that any admissible set of $k_0=3\,500\,000$ (or more) numbers contains some $a,b$ such that $n+a$ and $n+b$ are both prime infinitely often. There is an admissible set of this size with values between 0 and 70,000,000 thus the statement that there are infinitely many prime gaps at most 70 million.
The best value proved for $k_0$ so far is 50, which leads to the gap of 246 via the admissible tuple (0, 4, 6, 16, 30, 34, 36, 46, 48, 58, 60, 64, 70, 78, 84, 88, 90, 94, 100, 106, 108, 114, 118, 126, 130, 136, 144, 148, 150, 156, 160, 168, 174, 178, 184, 190, 196, 198, 204, 210, 214, 216, 220, 226, 228, 234, 238, 240, 244, 246).
But if you wanted you could choose a different tuple which showed, for example, that there are infinitely many prime gaps in a different range. For example, the admissible 50-tuple (0, 4, 10, 16, 22, 30, 34, 42, 46, 52, 60, 64, 70, 76, 84, 90, 94, 100, 106, 112, 126, 130, 136, 142, 150, 154, 160, 172, 184, 192, 202, 210, 214, 220, 226, 232, 240, 244, 252, 262, 270, 276, 280, 286, 294, 312, 316, 324, 330, 336) proves that there are infinitely many prime gaps of length between 4 and 336 (inclusive).*
So if the current methods were extended to prove the twin prime conjecture it would automatically prove Polignac's conjecture. Now that might be too much to expect -- the Polymath project has already changed its methodology in significant ways in the course of its several months of operation. But it does serve to show that Polignac's conjecture is not far from the twin prime conjecture.
A reasonable question, then, is "can Zhang's method be so extended?". At the moment the answer seems to be "no": even on the assumption of the generalized Elliott-Halberstam conjecture, the best that has been achieved is $k_0=3$ which means (via the 3-tuple (0, 2, 6)) that at least one of twin primes, cousin primes, and sexy primes have infinitely many members. But even with that high-powered assumption we can't narrow it down further.
* Similarly I can show that there are infinitely many prime gaps between 6 and 378, between 8 and 502, between 10 and 616, between 12 and 678, and so forth. On GEH the best you can do is $g$ to $2g$ if $3|g$ or $g$ to $2g+2$ otherwise.