Update (July 22, ’14). The polymath8b paper “Variants of the Selberg sieve, and bounded intervals containing many primes“, is now on the arXiv. See also this post on Terry Tao’s blog. Since the last update, we also had here at HUJI a beautiful learning seminar on small gaps between primes. James Maynard gave a series of three lectures and additional lectures were given by Zeev Rudnick and Tamar Ziegler.
Update (Jan 9, ’14, corrected Jan 10): Polymath8b have just led to an impressive progress: Goldston, Pintz, and Yıldırım showed that conditioned on the Elliott-Halberstam conjecture (EHC) there are infinitely many primes of bounded gap below 16. Maynard improved it to 12. Polymath8b have just improved it based on a generalized form of the EHC (proposed in 1986 by Bombieri, Friedlander, and Iwaniec) further to 8. [Further update: 6 and there are reasons so suspect that further improvement requires major breakthrough - namely getting over the "parity problem".] The unconditional bound for gaps stands now on 270.
Update: A paper by James Maynard entitled “Small gaps between primes” proved that for every k there are infinitely many intervals of length f(k) each containing at least k primes. He also reduced the gap between infinitely many pairs of primes to 600. The method is also (said to be) much simpler. Amazing! Similar results were obtained independently by Terry Tao.
Terry Tao launched a followup polymath8b to improve the bounds for gaps between primes based on Maynard’s results.
Zhang’s breakthrough and Polymath8
The main objectives of the polymath8 project, initiated by Terry Tao back in June, were “to understand the recent breakthrough paper of Yitang Zhang establishing an infinite number of prime gaps bounded by a fixed constant , and then to lower that value of as much as possible.”
Polymath8 was a remarkable success! Within two months the best value of H that was 70,000,000 in Zhang’s proof was reduced to 5,414. Moreover, the polymath setting looked advantageous for this project, compared to traditional ways of doing mathematics.
The polymath project gave opportunity to a number of researchers to understand Zhang’s proof and the earlier breakthrough by Daniel Goldston, János Pintz, and Cem Yıldırım. It also gave an opportunity to a larger number of mathematicians to get some feeling about the involved mathematics.
Twin primes are two primes p and p+2. The ancient twin prime conjecture asserts that there are infinitely many twin primes. The prime number theorem asserts that there are (asymptotically) n/log n primes whose value is smaller than a positive integer n, and this implies that we can find arbitrary large pairs of consecutive primes p and q such that q-p is at most (log p). Until a few years ago nothing asymptotically better was known. Goldston, Pintz, and Yıldırım (GPY), showed in 2005 that there infinitely many pairs of primes p and q such that q-p is . A crucial idea was to derive information on gaps of primes from the distribution of primes in arithmetic progressions. GPY showed that conditioned on the Elliott-Halberstam conjecture (EHC) there are infinitely many primes of bounded gaps (going all the way to 16, depending on a certain parameter in the conjecture, but not to 2). Yitang Zhang did not prove the EHC but based on further understanding of the situation found a way to shortcut the conjecture and to prove that there are infinitely many primes of with bounded gaps unconditionally!
Here is a very nice 2007 survey article by Kannan Soundararajan on this general area of research and the GPY breakthrough. (One thing I recently learned is that Soundararajan is called by friends and colleagues “Sound”. ) This article starts with a very thoughtful and endearing answer to the quastion: “Why do we care at all? After all primes were meant to be multiplied, not subtracted (or added).”
Here is a short list of thoughts (things I learned, things I wish to understand better…) from following (from distance) Polymath8 and related Internet activity.
1) How information on primes in arithmetic progressions leads to information on gaps between primes?
I do not really understand why the information on primes in arithmetic progressions e.g. the Elliott-Halberstam conjecture lead to the conclusion regarding primes with bounded gaps. I would be very happy to get a feeling for it.
2) The three-primes barrier.
Already GPY tried to extend their methods to show the existence of three primes in a bounded interval of integers. So far, it is not known how to show that intervals of the form [n,n+o(log n)] contain triples of primes infinitely often. Perhaps, to actually solve the twin prime conjecture we will need to get a breakthrough for triples of primes, but maybe not. See also this MO question asked by Noam Elkies.
Update: Here is another interesting MO question Quantitative lower bounds related to Zhang’s theorem on bounded gaps, asked by Eric Naslund. Eric asks: what can be say based on Zhang’s work about the smallest value of a pair of primes of distance k apart?
3) Cauchy-Schwarz everywhere;
This may sound silly but the way Cauchy-Schwarz (C-S) inequality is used again and again make you wonder again why C-S is it so useful, and why it is mainly C-S which is so useful.
4) Can detailed statistical understanding of primes in sets other than AP be useful?
In recent years there was much activity (and I also was interested) in Mobius randomness and analogs of the prime number theorem for various more complicated subsets of integers. (E.g., subsets defined by various properties of the digital expansion.) Can understanding of this kind also be used for the prime-gaps questions?
5) Usefulness of Deligne’s work on Riemann’s hypothesis for functions fields for questions in analytic number theory.
I knew, of course that Deligne famously proved analogs of the Riemann hypothesis for function fields in great generality but I was not aware that these results have applications to “ordinary” analytic number theory. Again, this is something I would be happy to know a little more about. There is a nice recent post on the Riemann hypothesis in various settings on “What’s new”.
6) Parity problem. (Added Nov 27) There is a difficult “parity problem” which seems to be a difficult obstacle for getting the gap to two. (And for various related goals). Terry Tao wrote about it in 2007 in this post. In polymath8b VII an attempt to cross the “parity barrier” was made but (as people expected) it turned out that the parity barrier indeed shows up causes this attempt to fail. (Update July 14:) This is further explained in this new post over Tao’s blog.
7) (Added Nov 27) One thing I am curious about is the following. Consider a random subset of primes (taking every prime with probability p, independently, and say p=1/2). Now consider only integers involving these primes. I think that it is known that this system of “integers” satisfies (almost surely) PNT but not at all RH. We can consider the properties BV (Bombieri Vinogradov), or more generally EH(θ) and the quantities . For such systems does BV typically hold? or it is rare like RH. Is Meynard’s implication applies in this generality? Nicely here we can hope even for infinite consecutive primes. Update: after thinking about it further and a little discussion over polymath8b it looks that current sieve methods, and some of the involved statements, rely very strongly on both the multiplicative and additive structure of the integers and do not allow extensions to other systems of “integers.”
Update (August 23): Before moving to small gaps, Sound’s 2007 survey briefly describes the situation for large gaps. The Cramer probabilistic heuristic suggests that there are consecutive primes in [1,n] which are apart, but not apart where c and C are some small and large positive constants. It follows from the prime number theorem that there is a gap of at least . And there were a few improvements in the 30s ending with a remarkable result by Rankin who showed that there is a gap as large as times . Last week Kevin Ford, Ben Green, Sergei Konyagin, and Terry Tao and independently James Maynard were able to improve Rankin’s estimate by a function that goes to infinity with n. See this post on “What’s new.”