And so it came to pass, that an almost millenial quest found a safe resting place…
Like all analytic number theorists, I’ve been amazed to learn that Yitang Zhang has proved that there exist infinitely many pairs of prime numbers with bounded by an absolute constant .
So, how did he do it?
Well, since the paper just became available, I don’t have anything intelligent to say yet on the new ideas that he introduced (but I certainly hope to come back to this!). However, one can easily list those previously-known tools that he uses, which involve some of the deepest and most clever results in analytic number theory of the last 30 to 35 years.
(1) At the core, the proof is based on the method discovered about ten years ago by Goldston, Pintz and Yıldırım to show that
As I discussed a while back, this remarkable result — besides its intrinsic interest — was notable for being the first to bring the problem of bounded gaps between primes within a circle of well-studied and widely believed conjectures on primes in arithmetic progressions to large moduli. Precisely, Goldston, Pintz and Yıldırım had derived the statement above, after many ingenious steps, by applying the Bombieri-Vinogradov Theorem, and they showed that any progress beyond it towards the so-called Elliott-Halberstam Conjecture would imply the bounded gap property. However, in my former blog post, I discussed why it seemed extremely difficult to go in that direction…
(2) … despite the existence of some results going beyond the Bombieri-Vinogradov theorem, due first to Fouvry-Iwaniec and later improved by Bombieri-Friedlander-Iwaniec; but Zhang uses indeed some of the ideas behind these results…
(3) … results which themselves depend crucially on two big ideas: the well-factorable weights of the linear sieve, due to Iwaniec, and the development and applications of the Kuznetsov formula and other results concerning the spectral theory of automorphic forms and estimates for sums of Kloosterman sums, the outcome of the work of Deshouillers and Iwaniec (actually, at first glance, it seems that Zhang does not explicitly use those results arising from the Kuznetsov formula; he does reach sums with incomplete Kloosterman sums which the spectral methods are designed to handle, but he can deal with them with the Weil bound only; this might be a place where the result can be improved…)
(4) … but furthermore, Zhang uses also an estimate for a certain character sum over finite fields which had appeared in the work of Friedlander and Iwaniec on the exponent of distribution for the ternary divisor function; this sum is a three-variable additive character sum, and its estimation (with square-root cancellation), proved by Bombieri and Birch in an Appendix to the paper of Friedlander and Iwaniec, depends crucially on the Riemann Hypothesis over finite fields of Deligne.
Here are some references to surveys or explanations of some of these tools. Amusingly, I have written something on most of them…
- There have been many surveys of the work of Goldston, Pintz and Yıldırım, and in particular I wrote a Bourbaki report on it, which may be interesting to those who read French;
- Concerning the automorphic Kloostermania that comes into the Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec circle of ideas (although it is apparently not needed for Zhang’s proof…), I happened to write a few years ago, for a book on Poincaré’s mathematical work, an account of the applications of Poincaré series to analytic number theory, which are used to prove the Kuznetsov formula;
- Fouvry has written a survey Cinquante ans de théorie analytique des nombres from the point of view of sieve methods, which discusses the philosophy of extending the ranges of exponents of distribution for important sequences, as well as the well-factorable weights of Iwaniec;
- Fans of trace functions may remember that I noticed in a previous post (see the very end) that the exponential sum of Friedlander-Iwaniec, estimated by Birch and Bombieri, is (for prime moduli) just a special case of the general “correlation sums” that appeared in my recent work with Fouvry and Ph. Michel — in particular, our arguments (based on the sheaf-theoretic Fourier transform of Deligne, Laumon, Katz and others) give a conceptually simple proof of that estimate (I just wrote it down in a short separate note);
And although it doesn’t seem that Zhang uses it directly, I’d like to mention that the result of Friedlander and Iwaniec concerning the exponent of distribution of was improved by Heath-Brown a few years later, and that Fouvry, Michel and I very recently improved it quite a bit further (for prime moduli; the second part of that paper involves another application of the Bombieri-Friedlander-Iwaniec techniques to improve the exponent of distribution on average…)
And a philosophical preliminary conclusion, before diving into the work of Zhang: it is thrilling to see this result, and I particularly like that it comes completely unexpectedly, and yet uses all these beautiful ideas and methods from this analytic number theory that I love!