The bounded gaps between primes polymath projecta retrospective. Two weeks ago, yitang zhang announced his result establishing that bounded gaps between primes occur infinitely often, with the explicit upper bound of 70,000,000 given for this gap. Yitang zhang proves landmark theorem in distribution of. This has recently been spectacularly extended by zhang. For instance, the twin prime conjecture, which asserts that the gap between primes can equal 2 infinitely often, remains open. I think the argument for 246 shares more with maynards proof of bounded gaps which doesnt use the same innovations as zhang and only uses a little bit from zhangs proof. The proof is essentially elementary, relying only on the bombierivinogradov theorem. Theorem zhang for m 2, maynardtao for m 2 for each integer m 2, there is a. In other words, that the gap between one prime and the next is bounded by 70,000,000 in nitely often thus, the bounded gaps conjecture.
Small and large gaps between the primes there are many questions about the gaps between consecutive prime numbers which are not completely solved, even after decades of effort. On rst glance, this might seem a miraculous phenomenon. Without communicating with the fields experts, zhang started thinking about the problem. Bounded gaps between primes in hecke equidistribution problems 3 where p n is the nth prime. Bounded gaps between primes andrew granville abstract. Bounded gaps between primes the most general conjecture which i know along these lines is schinzels hypothesis h.
It comes as no surprise then that a seminar given yesterday afternoon at harvard by yitang zhang of the university of new hampshire reporting on his new paper bounded gaps between primes attracted a diverse audience. The prime number theorem says that for any 1tuple h. Since then there has been a flurry of activity in reducing this bound, with the current record being 4,802,222 but likely to improve at least by a little bit in the near future. Zhang read the gpy paper, and in particular the sentence referring to the hairs breadth between gpy and bounded prime gaps. The techniques of zhang and goldston, pintz, and y ld r m have subsequently been signi cantly expanded upon by maynard 11, tao, and the polymath project 14, so that the best known bound on gaps between primes, at least at the time of writing, is 252. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime. Bounded gaps between primes by yitang zhang abstract it is proved that liminf n. In 2005, investigating this relaxed version of the ktuples conjecture, goldston, pintz and yildirim designed a sieve method i. Maynards paper used tools that he developed last year to understand small gaps between primes. Bounded gaps between primes in chebotarev sets springer for. A recent breakthrough in prime n umber theory is on the bounded prime gaps. Expanding upon these techniques, zhang 18 proved the amazing result that liminf n. The gpy method produces bounded gaps between primes assuming that. Yitang zhang, genannt tom zhang, chinesisch, pinyin zhang yitang.
In 3, zhang proved that there are infinitely many pairs of distinct primes p1,p2 satisfying p1. Bounded gaps between primes clay mathematics institute. Sutherland mit new bounds on gaps between primes october 17, 20 3 28. Bounded gaps between primes in multidimensional hecke equidistribution problems jesse thorner abstract. And so it came to pass, that an almost millenial quest found a safe resting place like all analytic number theorists, ive been amazed to learn that yitang zhang has proved that there exist infinitely many pairs of prime numbers with bounded by an absolute constant. Their work was substantially improved upon by zhang 24, who proved for the very rst time that there exist in nitely many bounded. Download fulltext pdf bounded gaps between primes of a special form article pdf available in international mathematics research notices 201523 march 2014 with 61 reads. May 22, 20 all of us believed the bounded gaps conjecture before zhangs big reveal, and we all believe the twin primes conjecture even though it remains unproven. The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. Zhangs original bound here was 70 million, but it has since been cut down to 246 thanks to the efforts of james maynard and an online collaborative polymath project. Following 20s amazing bounded gaps between primes result, mathematician yitang zhang has gone from an unknown maths lecturer to a mathematical celebrity. Recently, zhang has proved the existence of a nite hsuch that there exist in nitely many pairs pn, pn 1 of consecutive primes whose di erence pn 1 pn does not exceed h. In, zhang proved that there are infinitely many pairs of distinct primes p 1,p 2 satisfying p 1p 2.
The sequence g n of prime gaps has been extensively studied. This paper describes the authors joint research on small gaps between primes in the last 5 decade and how their methods were developed further independently by zhang, maynard, and tao to 6 prove stunning new results on primes. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime numbers that had previously seemed intractable. In the present paper, the author makes signi cant progress in the bounds of gaps between primes. This was a signi cant advance because previously no such bounded gap was known. Moreover, this refinement produces bounded gaps between primes with any level of distribution \\theta 0\. Indeed, zhang provided the explicit value of h 70000000 for this theorem.
Using dukes large sieve inequality for hecke gr ossencharaktere and the new sieve methods of maynard and tao, we prove a general result on gaps between primes in the context of multidimensional hecke equidistribution. Last year, in a breakthrough work of yitang zhang, it was shown that there were infinitely many gaps between primes of bounded size. Our method is a re nement of the recent work of goldston, pintz and y ld r m on the small gaps between consecutive primes. Zhang just proved is that there are infinitely many pairs of primes that differ by at most 70,000,000. A major ingredient of the proof is a stronger version of the bombierivinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.
Yitang zhang bounded gaps between primes 2014 youtube. Introduction the primary purpose of this note is to collect a few hitherto unnoticed or unpublished results concerning gaps between powers of consecutive primes. In this paper, we extend a result of maynard and tao concerning small gaps between primes to intervals of this length. Hongze li, hao pan, bounded gaps between primes of a special form, international mathematics research notices, volume 2015. A chinese immigrant who received his doctorate from purdue university, he had always been interested in number theory, even though it wasnt the subject of his dissertation. There is only one pair of consecutive gaps having length 2. Bounded gaps between primes by yitang zhang now available. May 29, 20 to complete the story started as a rumour report in primes gotta stick together and confirmed in primes really do stick together, here we report that annals of mathematics has posted the pdf of bounded gaps between primes by yitang zhang on its to appear in forthcoming issues page. Key input provided by an analogue of bombierivinogradov proved by murtymurty. In late 20, james maynard found a simpli ed method. Zhangs work is inspiring but seems difficult to adapt to other settings. Stony brook mathematics colloquium video bounded gaps between primes yitang zhang university of new hampshireias date.
Monday, september 22, 2014 to friday, september 26, 2014. Recently, yitang zhang proved the existence of a finite bound b such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn b. Bounded gaps between primes in chebotarev sets springer. Due to further advances of maynard and tao and the collaborative polymath project, 70. This suggests that the maynardtao sieve can be adapted to study gaps between primes in special subsets. It is based on the prior sievetheoretic arguments of goldstonpintzy ld r m 38 and motohashi. The recent sensational news on bounded gaps between primes made me wonder. Young in the spring of 20, yitang zhang proved that there are in nitely many pairs of primes that di er by at most 70. The techniques of zhang and goldston, pintz, and y ld r m have subsequently been signi. New equidistribution estimates of zhang type, and bounded gaps between primes.
Feb 12, 2014 in his paper, titled bounded gaps between primes and bearing his name alone, zhang attacked the problem by proving that the number of primes that are less than 70 million units apart is infinite. In a recent mathematical breakthrough, mathematician tom yitang zhang proved. In a spectacular breakthrough, yitang zhang proved that there are infinitely many pairs of primes differing by at most 70 million. In his paper, titled bounded gaps between primes and bearing his name alone, zhang attacked the problem by proving that the number of primes that are less than 70 million units apart is. Terence tao and in collaboration with a team of top mathematicians, was launched to optimize the records of the bounded gaps between primes based on the breakthrough work of bounded gaps between primes by dr. A major ingredient of the proof is a stronger version of the bombierivinogradov theorem that. Bounded gaps between primes annals of mathematics princeton. Yitang zhangs preprint on landausiegel zeros mathoverflow. In any case, you are certainly right that 246 is the result of multiple obscure calculations. A major ingredient of the proof is a stronger version of the. In other words, that the gap between one prime and the next is bounded by 70,000,000 infinitely oftenthus, the bounded gaps conjecture. Naturally, if we assume that the primes have a higher level of distribution,then we can obtain stronger results. Bounded gaps between primes yitang zhang abstract it is proved that liminf n. The five researchers have now joined together to refine their new bound, and plan to release a preprint within a week or two which, tao feels, pushes rankins basic method as far as possible using.
The study of gaps between primes has attracted many mathematicians and led to many deep realizations in number theory. A prime gap is the difference between two successive prime numbers. One of the oldest open problems in analytic number theory. The mathematical research sciences institute at berkeley has put together a film telling the story of zhangs proof, and his life before and after the announcement. Zhang, of the university of new hampshire, showed for the first time that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes that are a bounded distance apart within 70 million, he proved. In particular, thorner extended the methods in to show that there exist bounded gaps between primes in chebotarev sets. Indeed they conditionally proved for the rst time the existence of a nite. If this result is correct, then in my opinion it is even bigger news for analytic number theory. Zhang to establish the existence of bounded gaps between primes. May 19, 20 meanwhile, zhang was working in solitude to try to bridge the gap between the gpy result and the bounded prime gaps conjecture. These are notes on zhangs work and subsequent developments produced in preparation for 5 hours of talks for a general.
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