BOUNDED GAPS BETWEEN PRIMES


 Sydney Gardner
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1 BOUNDED GAPS BETWEEN PRIMES ANDREW GRANVILLE Abstract. Recetly, Yitag Zhag proved the existece of a fiite boud B such that there are ifiitely may pairs p, p of cosecutive primes for which p p B. This ca be see as a massive breakthrough o the subject of twi primes ad other delicate questios about prime umbers that had previously seemed itractable. I this article we will discuss Zhag s extraordiary work, puttig it i its cotext i aalytic umber theory, ad sketch a proof of his theorem. Zhag eve proved the result with B A cooperative team, polymath8, collaboratig oly olie, has bee able to lower the value of B to 4680, ad it seems plausible that these techiques ca be pushed somewhat further, though the limit of these methods seem, for ow, to be B 2. Cotets. Itroductio 2 2. The distributio of primes, divisors ad prime ktuplets 8 3. Uiformity i arithmetic progressios 5 4. GoldstoPitzYıldırım s argumet Distributio i arithmetic progressios Prelimiary reductios Complete expoetial sums Icomplete expoetial sums The Grad Fiale Weaker hypotheses Mathematics Subject Classificatio. P32. To Yiliag Zhag, for showig that oe ca, o matter what.
2 2 ANDREW GRANVILLE sec:itro. Itroductio.. Itriguig questios about primes. Early o i our mathematical educatio we get used to the two basic rules of arithmetic, additio ad multiplicatio. Whe we defie a prime umber, simply i terms of the umber s multiplicative properties, we discover a strage ad magical sequece of umbers. O the oe had, so easily defied, o the other, so difficult to get a firm grasp of, sice they are defied i terms of what they are ot (i.e. that they caot be factored ito two smaller itegers)). Whe oe writes dow the sequece of prime umbers: 2, 3, 5, 7,, 3, 7, 9, 23, 29, 3, 37, 4, 43, 47, 53, 59, 6,... oe sees that they occur frequetly, but it took a rather clever costructio of the aciet Greeks to eve establish that there really are ifiitely may. Lookig further at a list of primes, some patters begi to emerge; for example, oe sees that they ofte come i pairs: 3 ad 5, 5 ad 7, ad 3, 7 ad 9, 29 ad 3, 4 ad 43, 59 ad 6,... Oe might guess that there are ifiitely may such prime pairs. But this is a ope, elusive questio, the twi prime cojecture. Util recetly there was little theoretical evidece for it. All that oe could say is that there was a eormous amout of computatioal evidece that these pairs ever quit; ad that this cojecture (ad various more refied versios) fit ito a eormous etwork of cojecture, which build a beautiful elegat structure of all sorts of prime patters; ad if the twi prime cojecture were to be false the the whole edifice would crumble. The twi prime cojecture is certaily itriguig to both amateur ad professioal mathematicias alike, though oe might argue that it is a artificial questio, sice it asks for a very delicate additive property of a sequece defied by its multiplicative properties. Ideed, umber theorists had struggled, util very recetly, to idetify a approach to this questio that seemed likely to make ay sigificat headway. I this article we will discuss these latest shockig developmets. I the first few sectios we will take a leisurely stroll through the historical ad mathematical backgroud, so as to give the reader a sese of the great theorem that has bee recetly proved, ad also from a perspective that will prepare the reader for the details of the proof..2. Other patters. Lookig at the list of primes above we see other patters that begi to emerge, for example, oe ca fid four primes which have all the same digits, except the last oe:, 3, 7 ad 9, which is repeated with 0, 03, 07 ad 09, ad oe ca fid may more such examples are there ifiitely may? More simply how about prime pairs with differece 4, 3 ad 7, 7 ad, 3 ad 7, 9 ad 23, 37 ad 4, 43 ad 47, 67 ad 7,... ; or differece 0, 3 ad 3, 7 ad 7, 3 ad 23, 9 ad 29, 3 ad 4, 37 ad 47, 43 ad 53,...?
3 BOUNDED GAPS BETWEEN PRIMES 3 Are there ifiitely may such pairs? Such questios were probably asked back to atiquity, but the first clear metio of twi primes i the literature appears i a paper of de Poligac from 849. I his hoour we ow call ay iteger h, for which there are ifiitely may prime pairs p, p h, a de Poligac umber. The there are the Sophie Germai pairs, primes p ad q : 2p, which prove useful i several simple algebraic costructios: 2 2 ad 5, 3 ad 7, 5 ad, ad 23, 23 ad 47, 29 ad 59, 4 ad 83,... ; Now we have spotted all sorts of patters, we eed to ask ourselves whether there is a way of predictig which patters ca occur ad which do ot. Let s start by lookig at the possible differeces betwee primes: It is obvious that there are ot ifiitely may prime pairs of differece, because oe of ay two cosecutive itegers must be eve, ad hece ca oly be prime if it equals 2. Thus there is just the oe pair, 2 ad 3, of primes with differece. Oe ca make a similar argumet for prime pairs with odd differece. Hece if h is a iteger for which there are ifiitely may prime pairs of the form p, q p h the h must be eve. We have see may examples, above, for each of h 2, h 4 ad h 0, ad the reader ca similarly costruct lists of examples for h 6 ad for h 8, ad ideed for ay other eve h that takes her or his facy. This leads us to bet o the geeralized twi prime cojecture, which states that for ay eve iteger 2k there are ifiitely may prime pairs p, q p 2k. What about prime triples? or quadruples? We saw two examples of prime quadruples of the form 0, 0 3, 0 7, 0 9, ad believe that there are ifiitely may. What about other patters? Evidetly ay patter that icludes a odd differece caot succeed. Are there ay other obstructios? The simplest patter that avoids a odd differece is, 2, 4. Oe fids the oe example 3, 5, 7 of such a prime triple, but o others. Further examiatio makes it clear why ot: Oe of the three umbers is always divisible by 3. This is very similar to what happeed with, ; ad oe ca verify that, similarly, oe of, 6, 2, 8, 24 is always divisible by 5. The geeral obstructio ca be described as follows: For a give set of distict itegers a a 2... a k we say that prime p is a obstructio if p divides at least oe of a,..., a k, for every iteger. I other words, p divides Ppq p a qp a 2 q... p a k q for every iteger ; which ca be classified by the coditio that the set a, a 2,..., a k pmod pq icludes all of the residue classes mod p. If o prime is a obstructio the we say that x a,..., x a k is a admissible set of forms. 3. Pitz makes a slightly defiitio: That is, that p ad p h should be cosecutive primes. 2 These are useful because, i this case, the group of reduced residues mod q is a cyclic group of order q 2p, ad therefore isomorphic to C 2 C p if p 2. Therefore every elemet i the group has order (that is, ), 2 (that is, ), p (the squares mod q) or 2p q. Hece g geerates the group of reduced residues if ad oly if g is ot a square mod q ad g. 3 Notice that a, a 2,..., a k pmod pq ca occupy o more tha k residue classes mod p ad so, if p k the p caot be a obstructio.
4 4 ANDREW GRANVILLE Number theorists have log made the optimistic cojecture if there is o such obvious obstructio to a set of liear forms beig ifiitely ofte prime, the they are ifiitely ofte simultaeously prime. That is: Cojecture: If x a,..., x a k is a admissible set of forms the there are ifiitely may itegers such that a,..., a k are all prime umbers. I this case, we call a,..., a k a ktuple of prime umbers. To date, this has ot bee prove for ay k though, followig Zhag s work, we are startig to get close for k 2. Ideed, Zhag proves a weak variat of this cojecture, as we shall see. The above cojecture ca be exteded, as is, to all sets of k liear forms with iteger coefficiets i oe variable, so log as we exted the otio of admissibility to also exclude the obstructio that two of the liear forms have differet sigs for all, but fiitely may,, sice a egative iteger caot be prime (for example, ad 2 ); some people call this the obstructio at the prime,. We ca also exted the cojecture to more tha oe variable (for example the set of forms m, m, m 4): The prime ktuplets cojecture: If a set of k liear forms i variables is admissible the there are ifiitely may sets of itegers such that whe we substitute these itegers ito the forms we get a ktuple of prime umbers. There has bee substatial recet progress o this cojecture. The famous breakthrough was Gree ad Tao s theorem for the ktuple of liear forms i the two variables a ad d: a, a d, a 2d,..., a pk qd. Alog with Ziegler, they wet o to prove the prime ktuplets cojecture for ay admissible set of liear forms, provided o two satisfy a liear equatio over the itegers. What a remarkable theorem! Ufortuately these exceptios iclude may of the questios we are most iterested i; for example, p, q p 2 satisfy the liear equatio q p 2; ad p, q 2p satisfy the liear equatio q 2p ). Fially, we also believe that the cojecture holds if we cosider ay admissible set of k irreducible polyomials with iteger coefficiets, with ay umber of variables. For example we believe that 2 is ifiitely ofte prime, ad that there are ifiitely may prime triples m,, m We will ed this sectio by statig Zhag s mai theorem ad a few of the more beguilig cosequeces: Zhag s mai theorem: There exists a iteger k such that the followig is true: If x a,..., x a k is a admissible set of forms the there are ifiitely may itegers such that at least two of a,..., a k are prime umbers.
5 BOUNDED GAPS BETWEEN PRIMES 5 Note that the result states that oly two of the a i are prime, ot all (as would be required i the prime ktuplets cojecture). Zhag proved this result for a fairly large value of k, that is k , which has bee reduced to k 632 by the polymath8 team. Of course if oe could take k 2 the we would have the twi prime cojecture, but the most optimistic pla at the momet, alog the lies of Zhag s proof, would yield k 5. To deduce that there are bouded gaps betwee primes from Zhag s Theorem we eed oly show the existece of a admissible set with k elemets. This is ot difficult, simply by lettig the a i be the first k primes k. 4 Hece we have proved: Corollary: [Bouded gaps betwee primes] There exists a boud B such that there are ifiitely may itegers pairs of prime umbers p q p B. Fidig the smallest B for a give k is a challegig questio. The prime umber theorem together with our costructio above suggests that B kplog k Cq for some costat C, but it is iterestig to get better bouds. Our Corollary further implies Corollary: There is a iteger h, 0 of primes p, p h. h B such that there are ifiitely may pairs That is, some positive iteger B is a de Poligac umber. I fact oe ca go a little further usig Zhag s mai theorem: Corollary: Let k be as i Zhag s Theorem, ad let A be ay admissible set of k itegers. There is a iteger h P pa Aq : ta b : a b P Au such that there are ifiitely may pairs of primes p, p h. Fially we ca deduce from this Corollary: A positive proportio of itegers are de Poligac umbers Proof. If A t0,..., Bu is a admissible set the ma : tma : a P Au is admissible for every iteger m. Give large x let M rx{bs. By Zhag s Theorem there exists a pair a m b m P A such that mpb m a m q is a de Polgac umber. Sice there are at most B{2 differeces d b a with a b P A there must be some differece which is the value of b m a m for at least 2M{B values of m M. This gives rise to 2M{B x{b 2 distict de Poligac umbers of the form md x. Our costructio above implies that the proportio is at least {k 2 plog k Cq 2. 4 This is admissible sice oe of the a i is 0 pmod pq for ay p k, ad the p k were hadled i the previous footote.
6 6 ANDREW GRANVILLE.3. The simplest aalytic approach. There are 4 odd primes up to 50, that is 4 out of the 25 odd itegers up to 50, so oe ca deduce that several pairs differ by 2. We might hope to take this kid of desity approach more geerally: If A is a sequece of itegers of desity {2 (i all of the itegers) the we ca easily deduce that there are may pairs of elemets of A that differ by o more tha 2. Oe might guess that there are pairs that differ by exactly 2, but this is by o meas guarateed, as the example A : t P Z : or 2 pmod 4qu shows. Moreover, to use this kid of reasoig to hut for twi primes, we presumably eed a lower boud o the desity of primes as oe looks at larger ad larger primes. This was somethig that itrigued the youg Gauss who, by examiig Cherik s table of primes up to oe millio, surmised that the desity of primes at aroud x is roughly { log x (ad this was subsequetly verified, as a cosequece of the prime umber theorem). Therefore we are guarateed that there are ifiitely may pairs of primes p q with q p log p, which is ot quite as small a gap as we are hopig for! Noetheless this raises the questio: Fix c 0. Ca we prove that There are ifiitely may pairs of primes p q with q p c log p? This follows for all c by the prime umber theorem, but it is ot easy to prove such a result for ay particular value of c. The first such results were proved coditioally assumig the Geeralized Riema Hypothesis. This is, i itself, surprisig: The Geeralized Riema Hypothesis was formulated to better uderstad the distributio of primes i arithmetic progressios, so why would it appear i a argumet about short gaps betwee primes? It is far from obvious by the argumet used, ad yet this coectio has deepeed ad broadeed as the literature developed. We will discuss primes i arithmetic progressios i detail i the ext sectio. The first ucoditioal (though iexplicit) such result, boudig gaps betwee primes, was proved by Erdős i 940 usig the small sieve (we will obtai ay c e γ by such a method i sectio 3.2 MaierTrick ). I 966, Bombieri ad Daveport [2] bomdav substituted the BombieriViogradov theorem for the Geeralized Riema Hypothesis i earlier, coditioal argumets, to prove this ucoditioally for ay c ; ad i 988 Maier maier 2 [25] observed that oe ca easily modify this to obtai ay c 2 e γ. The Bombieri Viogradov Theorem is also a result about primes i arithmetic progressios, as we will discuss later. Maier further improved this, by combiig the approaches of Erdős ad of Bombieri ad Daveport, to some boud a little smaller tha, with substatial 4 effort. The first big breakthrough occurred i 2005 whe Goldsto, Pitz ad Yildirim [5] gpy were able to show that there are ifiitely may pairs of primes p q with q p c log p, for ay give c 0. Ideed they exteded their methods to show that, for ay ɛ 0, there are ifiitely may pairs of primes p q for which q p plog pq {2 ɛ. It is their method which forms the basis of the discussio i this paper. Like Bombieri ad Daveport, they showed that oe ca could better uderstad small gaps betwee primes, by obtaiig strog estimates o primes i arithmetic progressios, as i the
7 BOUNDED GAPS BETWEEN PRIMES 7 BombieriViogradov Theorem. Eve more, if oe assumes a strog, but widely believed, cojecture about the equidistributio of primes i arithmetic progressios, which exteds the BombieriViogradov Theorem, the oe ca show that there are ifiitely may pairs of primes p q which differ by o more tha 6 (that is, p q p 6)! What a extraordiary statemet, ad oe that we will briefly discuss: We kow that if p q p 6 the q p 2, 4, 6, 8, 0, 2, 4 or 6, ad so at least oe of these differece occurs ifiitely ofte. That is, there exists a positive, eve iteger 2k 6 such that there are ifiitely pairs of primes p, p 2k. Very recetly this has bee refied further by James Mayard, improvig the upper boud to 2, by a variat of the origial argumet. After Goldsto, Pitz ad Yildirim, most of the experts tried ad failed to obtai eough of a improvemet of the BombieriViogradov Theorem to deduce the existece of some fiite boud B such that there are ifiitely may pairs of primes that differ by o more tha B. To improve the BombieriViogradov Theorem is o mea feat ad people have loged discussed barriers to obtaiig such improvemets. I fact a techique had bee developed by Fouvry [0], fouvry ad by Bombieri, Friedlader ad Iwaiec [3], bfi but these were either powerful eough or geeral eough to work i this circumstace. Eter Yitag Zhag, a ulikely figure to go so much further tha the experts, ad to fid exactly the right improvemet ad refiemet of the BombieriViogradov Theorem to establish the existece of the elusive boud B such that there are ifiitely may pairs of primes that differ by o more tha B. By all accouts, Zhag was a brilliat studet i Beijig from 978 to the mid80s, fiishig with a master s degree, ad the workig o the Jacobia cojecture for his Ph.D. at Purdue, graduatig i 992. He did ot proceed to a job i academia, workig i odd jobs, such as i a sadwich shop, at a motel ad as a delivery worker. Fially i 999 he got a job at the Uiversity of New Hampshire as a lecturer, with a high teachig load, workig with may of the less qualified udergraduate studets. From timetotime a lecturer devotes their eergy to workig o provig great results, but few have doe so with such aplomb as Zhag. Not oly did he prove a great result, but he did so by improvig techically o the experts, havig importat key ideas that they missed ad developig a highly igeious ad elegat costructio cocerig expoetial sums. The, so as ot to be rejected out of had, he wrote his difficult paper up i such a clear maer that it could ot be deied. Albert Eistei worked i a patet office, Yitag Zhag i a Subway sadwich shop; both foud time, despite the urelated calls o their time ad eergy, to thik the deepest thoughts i sciece. Moreover Zhag did so at the relatively advaced age of 50 (or more). Truly extraordiary.
8 8 ANDREW GRANVILLE 2. The distributio of primes, divisors ad prime ktuplets veheuristic 2.. The prime umber theorem. As we metioed i the previous sectio, Gauss observed, at the age of 6, that the desity of primes at aroud x is roughly { log x, which leads quite aturally to the cojecture that #tprimes p xu» x 2 dt log t x log x as x Ñ 8. (We use the symbol Apxq Bpxq for two fuctios A ad B of x, to mea that Apxq{Bpxq Ñ as x Ñ 8.) This was proved i 896, the prime umber theorem, ad the itegral provides a cosiderably more precise approximatio to the umber of primes x, tha x{ log x. However, this itegral is rather cumbersome to work with, ad so it is atural to istead weight each prime with log p; that is we work with θpxq : p prime px ad the prime umber theorem implies 5 that log p θpxq x as x Ñ 8. (2.) pt A sievig heuristic to guess at the prime umber theorem. How may itegers up to x have o prime factors y? If y? x the this couts ad all of the primes betwee y ad x, so a accurate aswer would yield the prime umber theorem. The usual heuristic is to start by observig that there are x{2 Opq itegers up to x that are ot divisible by 2. A proportio 2 rds of these remaiig itegers are ot 3 divisible by 3; the a proportio 4 ths of the remaiig itegers are ot divisible by 5, 5 etc. Hece we guess that the umber of itegers x which are free of prime factors y, is roughly ¹ x. p py Evaluatig the product here is tricky but was accomplished by Mertes: If y Ñ 8 the ¹ e γ p log y. py Here γ is the EulerMascheroi costat, defied as lim NÑ8... log N. 2 N There is o obvious explaatio as to why this costat, defied i a very differet cotext, appears here. If? x y opxq (that is, for ay fixed ɛ 0 we have y ɛx oce x is sufficietly large) the we kow from the prime umber theorem that there are x{ log x itegers left usieved, whereas the predictio from our heuristic varies cosiderably as y varies i this rage. This shows that the heuristic is wrog for large y. Takig y? x it 5 This is really statig thigs backwards sice, i provig the prime umber theorem, it is sigificatly easier to iclude the log p weight, ad the deduce estimates for the umber of primes by partial summatio.
9 BOUNDED GAPS BETWEEN PRIMES 9 predicts too may primes by a factor of 2e γ ; takig y x{ log x it predicts too few primes by a factor of e γ. I fact this heuristic gives a accurate estimate provided y x opq. We will exploit the differece betwee this heuristic ad the correct cout, to show that there are smaller tha average gaps betwee primes i sectio 3.2. MaierTrick 2.3. The prime umber theorem for arithmetic progressios, I. Ay prime divisor of pa, qq is a obstructio to the primality of values of the polyomial qx a, ad these are the oly such obstructios. The prime ktuplets cojecture therefore implies that if pa, qq the there are ifiitely may primes of the form q a. This was first proved by Dirichlet i 837. Oce proved oe might ask for a more quatitative result. If we look at the primes i the arithmetic progressios pmod 0q:, 3, 4, 6, 7, 0 3, 3, 23, 43, 53, 73, 83, 03 7, 7, 37, 47, 67, 97, 07 9, 29, 59, 79, 89, 09 the there seem to be roughly equal umbers i each, ad this patter persists as we look further out. Let φpqq deote the umber of a for which pa, qq, ad so we expect that θpx; q, aq : log p x as x Ñ 8. φpqq p prime px pa This is the prime umber theorem for arithmetic progressios ad was first proved by suitably modifyig the proof of the prime umber theorem. The fuctio φpqq was studied by Euler, who showed that it is multiplicative, that is φpqq ¹ p e }q φpp e q (where p e }q meas that p e is the highest power of prime p dividig q) ad that φpp e q p e p e for all e Dirichlet s divisor trick. Aother multiplicative fuctio of importace is the divisor fuctio τpq : where the sum is over the positive itegers d that divide. It is ot difficult to verify that τpp e q e. If is squarefree ad has k prime factors the τpq 2 k, so we see that τpq varies greatly depedig o the arithmetic structure of. Noetheless oe might ask for the average of τpq, that is the average umber of divisors of a positive iteger x. A first d
10 0 ANDREW GRANVILLE approach yields that x τpq x d d x d dx x d sice the positive itegers up to x that are divisible by d ca be writte as dm with m x{d, ad so there are rx{ds such itegers, where rts deotes the largest iteger t. It evidet that rts t Opq, where Opq sigifies that there is a correctio here of at most a bouded multiple of. If we substitute this approximatio i above, we obtai τpq x Opq O x x d d x x dx ³ dx dx Oe ca approximate dx by x dt{t log x. Ideed the differece teds to a limit, d the EulerMascheroi costat γ : lim NÑ8... log N. Hece we have 2 N proved that the itegers up to x have log x Opq divisors, o average, which is quite remarkable for such a wildly fluctuatig fuctio. Dirichlet studied this argumet ad oticed that whe we approximate rx{ds by x{d Opq for large d, say for those d i px{2, xs, the this is ot really a very good approximatio, ad gives a large cumulative error term, Opxq. However we kow that rx{ds exactly, for each of these d, ad so we ca estimate this sum by x{2 Opq, which is much more precise. Dirichlet realized that the correct way to formulate this observatio is to write dm, where d ad m are itegers. Whe d is small the we should fix d, ad cout the umber of such m, with m x{d (as we did above); but whe m is small, the we should fix m, ad cout the umber of d with d x{m. I this way our sums are all over log itervals, which allows us to get a accurate approximatio of their value. I fact we ca exploit the symmetry here to simply break the sum at x {2. Hece Dirichlet proceeded as follows: τpq x x dm d? x x d d? x x d Opq m m? x x m? x x m Opq, d? x m? x x Op? xq. Oe ca do eve better with these sums tha above, showig that N { log N γ Op{Nq. Hece we ca deduce that x x τpq log x 2γ O a extraordiary improvemet upo the earlier error term.?x, I the calculatios i this article, this same idea is essetial. We will take some fuctios, that are difficult to sum, ad rewrite them as a sum of products of other fuctios, that are easier to sum, ad fid a way to sum them over log eough itervals for our methods to take effect. So we should defie the covolutio of two fuctios f ad g as f g where pf gqpq : fpaqgpbq, ab
11 BOUNDED GAPS BETWEEN PRIMES for every iteger, where the sum is over all pairs of positive itegers a, b whose product is. Hece τ, where is the fuctio with pq for every. Let δ pq if, ad δ pq 0 otherwise. Aother importat multiplicative fuctio is the Mobius fuctio µpq, sice µ δ. From this oe ca verify that µppq ad µpp e q 0 for all e 2, for all primes p. We defie Lpq : log, ad we let Λpq log p if is a power of prime p, ad Λpq 0 otherwise. By factorig, we see that L Λ. We therefore deduce that Λ pµ q Λ µ p Λq µ L; that is # log p if p m, where p is prime, m ; Λpq µpaq log b. (2.2) VMidetity 0 otherwise. ab We ca approach the prime umber theorem via this idetity by summig over all x to get Λpq µpaq log b. x abx The lefthad side equals θpxq plus a cotributio from prime powers p e with e 2, ad it is easily show that this cotributio is small (i fact Op? xq). The right had side is the covolutio of a awkward fuctio µ ad somethig very smooth ad easy to sum, L. Ideed, it is easy to see that bb log b log B! ad we ca estimate this very precisely usig Stirlig s formula. Oe ca ifer (see [8] GS for details) that the prime umber theorem is equivalet to provig that x x µpq Ñ 0 as x Ñ 8. I our work here we will eed a more covoluted idetity that ( 2.2) VMidetity to prove our estimates for primes i arithmetic progressios. There are several possible suitable idetities, the simplest of which is due to Vaugha [35]: vaugha Vaugha s idetity : Λ V µ U L µ U Λ V µ U Λ V (2.3) Vaughidet where g W pq gpq if W ad gpq 0 otherwise; ad g g W g W. To verify this idetity, we maipulate the algebra of covolutios: rimektuples Λ V Λ Λ V pµ Lq Λ V p µq µ U L µ U L µ U Λ V µ U Λ V µ U L µ U Λ V µ U pλ Λ V q, 2.5. A quatitative prime ktuplets cojecture. We are goig to develop a heuristic to guesstimate the umber of pairs of twi primes p, p 2 up to x. We start with Gauss s statemet that the desity of primes at aroud x is roughly { log x. Hece the probability that p is prime is { log x, ad the probability that p 2 is prime is { log x so, assumig that these evets are idepedet, the probability that p ad p 2
12 2 ANDREW GRANVILLE are simultaeously prime is log x log x plog xq 2 ; ad so we might expect about x{plog xq 2 pairs of twi primes p, p 2 x. But there is a problem with this reasoig, sice we are implicitly assumig that the evets p is prime for a arbitrary iteger p x, ad p 2 is prime for a arbitrary iteger p x, ca be cosidered to be idepedet. This is obviously false sice, for example, if p is eve the p 2 must also be. 6 So, to correct for the oidepedece, we cosider the ratio of the probability that both p ad p 2 are ot divisible by q, to the probabiliity that p ad p are ot divisible by q, for each small prime q. Now the probability that q divides a arbitrary iteger p is {q; ad hece the probability that p is ot divisible by q is {q. Therefore the probability that both of two idepedetly chose itegers are ot divisible by q, is p {qq 2. The probability that q does ot divide either p or p 2, equals the probability that p 0 or 2. If q 2 the p ca be i ay oe of q 2 residue classes mod q, which occurs, for a radomly chose p, with probability 2{q. If q 2 the p ca be i ay just oe residue class mod 2, which occurs with probability {2. Hece the correctio factor for divisibility by 2 is p 2 q p 2, q2 2 whereas the correctio factor for divisibility by ay prime q 2 is p 2 q q p q q2. Now divisibility by differet small primes i idepedet, as we vary over values of, by the Chiese Remaider Theorem, ad so we might expect to multiply together all of these correctio factors, correspodig to each small prime q. The questio the becomes, what does small mea? I fact, it does t matter much because the product of the correctio factors over larger primes is very close to, ad hece we ca simply exted the correctio to be a product over all primes q. (More precisely, the ifiite product over all q, coverges.) Hece we defie the twi prime costat to be C : 2 ¹ q prime q 3 p 2 q q p , q2 q ad we cojecture that the umber of prime pairs p, p x C plog xq. 2 2 x is 6 Also ote that the same reasoig would tell us that there are x{plog xq 2 prime pairs p, p x.
13 BOUNDED GAPS BETWEEN PRIMES 3 Computatioal evidece suggests that this is a pretty good guess. The aalogous argumet implies the cojecture that the umber of prime pairs p, p 2k x is ¹ p x C p 2 plog xq. 2 p k p 3 This argumet is easily modified to make a aalogous predictio for ay ktuple: Give a,..., a k, let Ωppq be the set of distict residues give by a,..., a k pmod pq, ad the let ωppq Ωppq. Noe of the a i is divisible by p if ad oly if is i ay oe of p ωppq residue classes mod p, ad therefore the correctio factor for prime p is p ωppq p q p. p qk Hece we predict that the umber of prime ktuplets a,..., a k x is, x ¹ Cpaq where Cpaq : plog xq k p p ωppq p q p. p qk Recogktuple A aalogous cojecture, via similar reasoig, ca be made for the frequecy of prime ktuplets of polyomial values i several variables. What is remarkable is that computatioal evidece suggests that these cojectures do approach the truth, though this rests o a rather shaky theoretical framework. A more covicig theoretical framework (though rather more difficult) was give by Hardy ad Littlewood [9] hardy see sectio 3.3. HLheuristic 2.6. Recogizig prime ktuples. The idetity ( 2.2) VMidetity allows us to distiguish prime powers from composite umbers i a arithmetic way. Such idetities ot oly recogize primes, but ca be used to idetify itegers with o more tha k prime factors. For example I geeral Λ 2 pq : µpdqplog {dq 2 d $ '& '% p2m qplog pq 2 if p m ; 2 log p log q if p a q b, p q; 0 otherwise. Λ k pq : µpdqplog {dq k d equals 0 if νpq k (where νpmq deotes the umber of distict prime factors of m). We will be workig with (a variat of) the expressio Λ k pppqq. We have see that if this is ozero the Ppq has k distict prime factors. We will ext show that if 0 a... a k ad a... a k the Ppq must have exactly k distict prime factors. I that case if the k prime factors of Ppq are p,..., p k, the Λ k pppqq k!plog p q... plog p k q.
14 4 ANDREW GRANVILLE Now, suppose that Ppq has r k distict prime factors, call them p,..., p r. For each p i select j jpiq for which the power of p i dividig a j is maximized. Evidetly there exists some J, J k which is ot a jpiq. Therefore if p e i i } a J the Hece p e i i p a Jq p a jpiq q pa J a jpiq q, which divides a J lcm i p e i i divides ¹ jk jj pa J a j q, ¹ jk jj ad so a J ± j a j, by hypothesis, which is impossible. The expressio for Λpq i ( 2.2) VMidetity ca be rewritte as Λpq µpdq log {d, ad eve d d µpdq log R{d, for ay R, provided. Selberg has show that the trucatio d dr µpdq log R{d pa J a j q. is also sesitive to primes ; ad ca be cosiderably easier to work with i various aalytic argumets. I our case, we will work with the fuctio d Ppq dr µpdqplog R{dq k, which is aalogously sesitive to prime ktuplets, ad easier to work with tha the full sum for Λ k pppqq.
15 BOUNDED GAPS BETWEEN PRIMES 5 3. Uiformity i arithmetic progressios 3.. Whe primes are first equidistributed i arithmetic progressios. There is a importat further issue whe cosiderig primes i arithmetic progressios: I may applicatios it is importat to kow whe we are first guarateed that the primes are moreorless equidistributed amogst the arithmetic progressios a with pa, qq ; that is θpx; q, aq x for all pa, qq. (3.) PNTaps φpqq To be clear, here we wat this to hold whe x is a fuctio of q, as q Ñ 8. If oe does extesive calculatios the oe fids that, for ay ɛ 0, if q is sufficietly large ad x q ɛ the the primes up to x are equidistributed amogst the arithmetic progressios a with pa, qq, that is ( 3.) PNTaps holds. This is ot oly uproved at the momet, also o oe really has a plausible pla of how to show such a result. However the slightly weaker statemet that ( 3.) PNTaps holds for ay x q 2 ɛ, ca be show to be true, assumig the Geeralized Riema Hypothesis. This gives us a clear pla for provig such a result, but oe which has see little progress i the last cetury! The best ucoditioal results kow are much weaker tha we have hoped for, equidistributio oly beig proved oce x e qɛ. This is the SiegelWalfisz Theorem, ad it ca be stated i several (equivalet) ways with a error term: For ay B 0 we have θpx; q, aq x φpqq O x plog xq B Or: for ay A 0 there exists B 0 such that if q " * θpx; q, aq x O φpqq plog xq B for all pa, qq. (3.2) SW plog xq A the for all pa, qq. (3.3) SW2 That x eeds to be so large compared to q limited the umber of applicatios of this result. The great breakthough of the secodhalf of the twetieth cetury came i appreciatig that for may applicatios, it is ot so importat that we kow that equidistributio holds for every a with pa, qq, ad every q up to some Q, but rather that this holds for most such q (with Q x {2 ɛ ). It takes some jugglig of variables to state the BombieriViogradov Theorem: We are iterested, for each modulus q, i the size of the largest error term max x a mod q θpx; q, aq φpqq, or eve max yx pa,qq max a mod q pa,qq y θpy; q, aq φpqq. The bouds 0 θpx; q, aq! x log x are trivial, the upper boud obtaied by boudig q the possible cotributio from each term of the arithmetic progressio. (Throughout the symbol!, as i fpxq! gpxq meas there exists a costat c 0 such that
16 6 ANDREW GRANVILLE fpxq cgpxq. ) We would like to improve o the trivial upper boud, perhaps by a power of log x, but we are uable to do so for all q. However, it turs out that we ca prove that if there are exceptioal q, the they are few ad far betwee, ad the BombieriViogradov Theorem expresses this i a useful form. The first thig we do is add up the above quatities over all q Q! qq x. The trivial upper boud is the x q log x! xplog xq2. The BombieriViogradov states that we ca beat this trivial boud by a arbitrary power of log x, provided Q is a little smaller tha? x: The BombieriViogradov Theorem. For ay give A 0 there exists a costat B BpAq, such that qq where Q x {2 {plog xq B. max a mod q pa,qq x θpx; q, aq φpqq! A x plog xq A I fact oe ca take B 2A 5; ad oe ca also replace the summad here by the expressio above with the extra sum over y (though we will ot eed to do this here). It is believed that this kid of estimate holds with Q sigificatly larger tha? x; ideed Elliott ad Halberstam cojectured elliott [8] that oe ca take Q x c for ay costat c : The ElliottHalberstam cojecture For ay give A 0 ad η, 0 have max x a mod q θpx; q, aq φpqq! x plog xq A where Q x {2 η. qq pa,qq However, it was show i fg [3] that oe caot go so far as to take Q x{plog xq B. η, we 2 gpythm This cojecture was the startig poit for the work of Goldsto, Pitz ad Yıldırım gpy [5], as well as of Zhag [38]. zhag This startig poit was a beautiful argumet from [5], gpy that we will spell out i the ext sectio, which yields the followig result. gpy Theorem 3. (GoldstoPitzYıldırım). [5] Let k 2, l be itegers, ad 0 η {2, such that 2η 2l 2l k. (3.4) thetal If the ElliottHalberstam cojecture holds with Q x {2 η the the followig is true: If x a,..., x a k is a admissible set of forms the there are ifiitely may itegers such that at least two of a,..., a k are prime umbers. The coclusio here is exactly the statemet of Zhag s mai theorem.
17 BOUNDED GAPS BETWEEN PRIMES 7 For ow the ElliottHalberstam cojecture seems too difficult to prove, but progress has bee made whe restrictig to oe particular residue class: Fix iteger a 0. We believe that for ay fixed η, 0 η, oe has qq pq,aq 2 x θpx; q, aq φpqq! x plog xq A where Q x {2 η. The key to progress has bee to otice that if oe ca factor the key terms here ito a sum of covolutios the it is easier to make progress, much as we saw with Dirichlet ad the divisor problem. I this case the key covolutio is ( 2.2) VMidetity ad Vaugha s idetity ( 2.3). Vaughidetity A secod type of factorizatio that takes place cocers the modulus: it is much easier to proceed if we ca factor the modulus q as, say dr where d ad r are roughly some prespecified sizes. The simplest class of itegers q for which this sort of thig is true is the ysmooth itegers, those itegers whose prime factors are all y. For example if we are give a ysmooth iteger q ad we wat q dr with d ot much smaller tha D, the we select d to be the largest divisor of q that is D ad we see that D{y d D. This is precisely the class of moduli that Zhag cosidered: 7 MaierTrick Yitag Zhag s Theorem There exist costats η, δ 0 such that for ay give iteger a, we have qq pq,aq q is y smooth q squarefree where Q x {2 η ad y x δ. x θpx; q, aq φpqq! A x plog xq A (3.5) EHsmooth Zhag [38] zhag proved his Theorem for η{2 δ, ad the argumet works provided 68 44η 72δ. We will prove this result, by a somewhat simpler proof, provided 62η 90δ. We expect this estimate holds for every η P r0, {2s ad every δ P p0, s, but just provig it for ay positive pair η, δ 0 is a extraordiary breakthrough that has a eormous effect o umber theory, sice it is such a applicable result (ad techique). This is the techical result that truly lies at the heart of Zhag s result about bouded gaps betwee primes, ad sketchig a proof of this is the focus of the secod half of this article. startig sectio 5. GeeralBV 3.2. A first result o gaps betwee primes. We will ow exploit the differece betwee the heuristic, preseted i sectio 2.2, SieveHeuristic for the prime umber theorem, ad the correct cout. ± Let m py p, N m2 ad x mn, so that y log m log x by the prime 3 umber theorem, ( 2.). pt2 We cosider the primes i the short itervals rm, m Js for N 2N 7 We will prove this with ψpx; q, aq : x, a Λpq i place of θpx; q, aq. It is ot difficult to show that the differece betwee the two sums is! x {2 opq.
18 8 ANDREW GRANVILLE HLheuristic with J y log y. Note that all of the short itervals are px, 2xs. The total umber of primes i all of these short itervals is 2Ņ N πpm Jq πpm q J j πp2x; m, jq πpx; m, jq jj pj,mq assumig ( PNTaps 3.). Hece, sice the maximum is always at least the average, J max πpm Jq πpm q PpN,2Ns log x #t j J : pj, mq u pφpmq{mqj x φpmq log x e γ J log x. usig the prime umber theorem, ad Merte s Theorem, as i sectio 2.2. SieveHeuristic Therefore we have proved that there is i a iterval of legth J, betwee x ad 2x, which has at J least primes, ad so there must be two that differ by e γ log x À e γ log x Hardy ad Littlewood s heuristic for the twi prime cojecture. The rather elegat ad atural heuristic for the quatitative twi prime cojecture, which we described i sectio 2.5, Primektuples was ot the origial way i which Hardy ad Littlewood made this extraordiary predictio. The geesis of their techique lies i the circle method., that they developed together with Ramauja. The idea is that oe ca distiguish the iteger 0 from all other itegers, sice» # if 0; eptqdt (3.6) expitegra 0 otherwise, 0 where, for ay real umber t, we write eptq : e 2πit. Notice that this is literally a itegral aroud the uit circle. Therefore to determie whether the two give primes p ad q differ by 2, we simply determie» 0 eppp q 2qtq dt. If we sum this up over all p, q x, we fid that the umber of twi primes p, p 2 x equals, exactly,»» eppp q 2qtq dt P ptq 2 ep 2tq dt, where P ptq : epptq. p,qx p,q primes 0 0 px p prime I the circle method oe ext distiguishes betwee those parts of the itegral which are large (the major arcs), ad those that are small (the mior arcs). Typically the major arcs are small arcs aroud those t that are ratioals with small deomiators. Here the width of the arc is about {x, ad we wish to uderstad the cotributio at t a{m, where pa, mq. Note the that P pa{mq b pmod mq pb,mq e m pabqπpx; m, bq.
19 BOUNDED GAPS BETWEEN PRIMES 9 where e m pbq ep b m q e2πib{m. We ote the easily proved idetity r pmod mq, pr,mq e m prkq φppk, mqqµpm{pm, kqq. Assumig the prime umber theorem for arithmetic progressios with a good error term we therefore see that x P pa{mq e m pabq µpmq x φpmq log x φpmq log x. b pmod mq pb,mq Hece i total we predict that the umber of prime pairs p, p x mm a: pa,mq x plog xq 2 e m p 2aq µpmq x φpmq log x φp2q ¹ p 2 2 φppq 2 x plog xq 2 m x C plog xq, 2 2 x is roughly µpmq 2 φpp2, mqqµpm{p2, mqq φpmq2 as i sectio quatprimektuples??. Moreover the aalogous argumet yields the more geeral cojecture for prime pairs p, p h. Why does t this argumet lead to a proof of the twi prime cojecture? For the momet we have little idea how to show that the mior arcs cotribute very little. Give that we do ot kow how to fid cacelatio amogst the mior arcs, we would eed to show that the itegrad is typically very small o the mior arcs, meaig that there is usually a lot of cacelatio i the sums P ptq. For ow this is a importat ope problem. Noetheless, it is this kid of argumet that has led to Helfgott s recet proof HH [2] that every odd iteger 3 is the sum of o more tha three primes.
20 20 ANDREW GRANVILLE gpysec 4. GoldstoPitzYıldırım s argumet We ow give a versio of the combiatorial argumet of GoldstoPitzYıldırım gpy [5], which was the ispiratio for provig that there are bouded gaps betwee primes: 4.. The set up. Let H pa a 2... a k q be a admissible ktuple, ad take x a k. Our goal is to select a fuctio ν for which νpq 0 for all, such that x 2x νpqp If we ca do this the there must exist a iteger such that νpqp ķ i ķ i θp a i q log 3xq 0. (4.) gpy θp a i q log 3xq 0. I that case νpq 0 so that νpq 0, ad therefore ķ i θp a i q log 3x. However each a i 2x a k 2x x ad so each θp a i q log 3x. This implies that at least two of the θp a i q are ozero, that is, at least two of a,..., a k are prime. A simple idea, but the difficulty comes i selectig the fuctio νpq with these properties for which we ca evaluate the sum. I [5] gpy they had the further idea that they could select νpq so that it would be sesitive to whe each a i is prime, or almost prime, ad so they relied o the type of costructio that we discussed i sectio 2.6. Recogktuple I order that νpq 0 oe ca simply take it to be a square. Hece we select νpq : d Ppq where λpdq : µpdq m log R{d m! log R whe d P D, ad λpdq 0 otherwise, for some positive iteger m k l, where D is a subset of the squarefree itegers i t,..., Ru, ad we select R x {3. I the argumet of [5], gpy D icludes all of the squarefree itegers i t,..., Ru, whereas Zhag uses oly the ysmooth oes. Our formulatio works i both cases. λpdq Evaluatig the sums, I. Now, expadig the above sum gives d,d 2 D:rd,d 2 s λpd qλpd 2 q ķ i x 2x D Ppq θp a i q log 3x x 2x D Ppq. (4.2) gpy2
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