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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
liminf frac{p_{n+1}-p_n}{log n}=0.
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 Yı
ldırım, 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 d_3 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!