The significance and acceptance of Helfgott’s proof of the weak Goldbach Conjecture
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.
Harald's CV has the entry,
Expository monographs – pure mathematics
M2. The ternary Goldbach problem, to appear in Ann. of Math. Studies.
But, it looks like he has not updated this CV since 2015. Also, I don't see it at the Annals of Math Studies site.
EDIT: There was a panel discussion of machine-assisted proofs at the ICM in Rio, August 2018. Harald was on the panel, and on page 9 he writes, concerning his proof of ternary Goldbach, "the version to be published is in preparation."
As for the question of "stir", Zhang was happy to find some $n$ such that there are infinitely many prime gaps no bigger than $n$; he found it was possible to take $n=70,000,000$, and didn't try to make the sharpest estimates. This left the field wide open for others to try to bring that value of $n$ down, and they did. For quite a while it seemed there were improvements reported every day, even every hour, and the work took place in public, on the polymath blog. And of course, there's still work to do. The current value of $n$, if I'm not mistaken, is $246$, where it's conjectured that $n=2$ will do. So, there has been a lot to keep people interested.
Harald's work, on the other hand, completely solved ternary Goldbach. There was nothing left to do (except, of course, to solve Goldbach proper, but [and I hesitate to write the following, since I'm out of my depth here, and could be way wrong] Harald's work doesn't seem to show the way to do that). So far as I know (and, again, I could be badly misinformed), nothing has come out of ternary Goldbach at all. That's not Harald's fault, and his work was a stunning achievement, but maybe it goes some way toward answering the question about the "stir".