On twin primes associated with the Hawkins random sieve

Transcription

1 Joural of Nuber Theory wwwelsevierco/locate/jt O twi pries associated with the Hawkis rado sieve HM Bui,JPKeatig School of Matheatics, Uiversity of Bristol, Bristol, BS8 TW, UK Received 3 July 005; revised 7 July 005 Available olie 4 Jauary 006 Couicated by J Bria Corey Abstract We establish a asyptotic forula for the uber of k-differece twi pries associated with the Hawkis rado sieve, which is a probabilistic odel of the Eratosthees sieve The forula for k = was obtaied by MC Wuderlich [A probabilistic settig for prie uber theory, Acta Arith ] We here eted this to k ad geeralize it to all l-tuples of Hawkis pries 005 Published by Elsevier Ic Itroductio The rado sieve was itroduced by Hawkis [4,5] as follows Let S ={, 3, 4, 5,} Put P = i S Every eleet of the set S \{P } is the sieved out, idepedetly of the others, with probability /P, ad S is the set of the survivig eleets I geeral, at the th step, defie P = i S We the use /P as the probability with which to delete the ubers i S \{P } The set reaiig is deoted by S + The Hawkis sieve is essetially a probabilistic aalogue of the sieve of Eratosthees The sequeces {P,P,,P,} of Hawkis pries iic the pries i the sese that their statistical distributio is epected to be like that of the pries The pries theselves correspod to oe realizatio of the process A great deal is kow about the Hawkis pries For istace, the aalogues of the prie uber theore [5,6,9], Mertes theore [6,9] ad the Riea hypothesis [7,8] are true with * Correspodig author E-ail address: hbui@bristolacuk HM Bui 00-34X/$ see frot atter 005 Published by Elsevier Ic doi:006/jjt00505

2 HM Bui, JP Keatig / Joural of Nuber Theory probability We here cocer ourselves with the desity of k-differece Hawkis twi pries ad its geeralizatio to other l-tuples Istead of a sequece of probability spaces, as cosidered by Hawkis, Wuderlich [9] siplified the process i a sigle probability space Let X be the space of all sequeces of itegers greater tha, ie, X cosists of all fiite ad ifiite sequeces The class of all sets of those sequeces is Ω Forα X, we deote by α the set of eleets of α which are less tha, ie, α = α {, 3, 4,, } ad α = α \ α Defiitio A eleet E Ω is called a eleetary set if there eists a sequece {a,a,,a k } X ad a iteger >a k such that E cosists of all the sequeces α such that α ={a,a,,a k } E is deoted by {a,a,,a k ; }, ad if k = 0, E ={ ;} is the set of all sequeces whose eleets are ot less tha The probability fuctio is ow defied recursively o the class of eleetary sets Defiitio Defie a o-egative real-valued fuctio μ o the class of eleetary sets as follows: i μ{ ; } =, ii μ{a,,a k,; + } = k i= a i μ{a,,a k ; }, iii μ{a,,a k ; + } = k i= a i μ{a,,a k ; } For ay α X, the aalogue of the k-differece twi prie coutig fuctio is defied as Π X,X+k ; α = #{j : j α ad j + k α} Wuderlich [9] showed that Π X,X+ /log alost surely, which is a aalogue of Hardy ad Littlewood s faous cojecture cocerig the distributio of the twi pries [3] The absece of the twi prie costat factor here is due to the drawback of the probabilistic settig of the rado sieve that it cotais little arithetical iforatio about the pries Though the result is ot uepected, it is ot easy to establish, as it is, for eaple, i Craer s odel [], where every uber is idepedetly deleted with probability / log I Sectio, we follow the lies of Wuderlich [9] ad eted the result to k =, Theore Alost surely Π X,X+ log Theore requires rather ore work tha [9, Theore 4], but the idea is siilar ad straightforward Nevertheless, it is clear fro the proof for k = that as k icreases, the calculatios will becoe etreely coplicated, ad the proof for the geeral case usig Wuderlich s ethod is likely to be etreely essy I Sectio 3, we therefore develop a differet approach ad establish the followig theore

3 86 HM Bui, JP Keatig / Joural of Nuber Theory Theore Alost surely, for ay fied iteger k,as, Π X,X+k log As we ote i Sectio 3, our approach eteds straightforwardly to l-tuples of Hawkis pries to yield: Theore 3 Let 0 <k <k < <k l ad deote by Π X,X+k,,X+k l ; α the uber of such that the set {, + k,,+ k l } α The as, alost surely Π X,X+k,,X+k l A iediate corollary of this theore is: log l Corollary For ay positive itegers d, l, ad l, alost surely, as, Π X,X+d,,X+l d log l, which is reiiscet of a recet theore of Gree ad Tao [] o the eistece of arbitrarily log arithetic progressios i the pries, proved usig powerful techiques fro aalytic uber theory, cobiatorics ad ergodic theory Proof of Theore We begi the proof by statig a lea fro [9] Lea For r, s, t o-egative itegers, r t, defie The k M k = + j j= k s M r k= k = s+ s + M r + c,rs+ M r+ = s+ t r s + r + j= cj,r + + s+ s + M r+j ct r,rs+ M t s+ M t, s+ M t where cj,r = rr + r + j /s + j+ As i [9], we defie y α = j<,j α j

4 HM Bui, JP Keatig / Joural of Nuber Theory The P α = y α, ad if we let C be the set of all sequeces cotaiig, μc = Ey = y dμ Wuderlich the obtaied the asyptotic forula for the kth oet of y, which is a aalogue of Mertes theore, Soe siple calculatios give E y k = k k+ P α, + α = y α y 3 + α Defie the auiliary fuctio Π X,X+ ; α =, α + α y α y + α I what follows, we write Π; α for Π X,X+ ; α, ad if f : R R, we defie the usual differece operator applied to f by f := f+ fwehave { Π; α = y+ α + + y + α if + α, + 3 α, 0 otherwise Hece E Π+ E Π = Ey + + Thus E Π = + = + M Ey + = M 3 M M 3 =3 We ow wish to estiate the variace of Π It is easy to see fro that E Π = E y + Π 3 +

5 88 HM Bui, JP Keatig / Joural of Nuber Theory It is ecessary to fid aother recursio for y i + α Π; α Wehave Sice y+ i α Π+ ; α + = iy i + Π; α + y + α + if + α, + 3 α, iy i + + Π; α if + α, + 3 / α, y+ i Π; α if + / α P + α, + 3 α = + y + α P + α, + 3 / α = y + α + P + / α = y + α, we easily obtai + + y + α + y 3 + α, y + α + + y 3 + α, E y+ i Π = i+ E y i+ + + i E y i+ + Π + 4 Takig i = 3, suig fro to, ad usig Eyk+ 4 Πk= OkEyk+ 4, wehave E y+ 3 Π = O E y 4 = O 4 = O 4 Lettig i = i4, E y+ Π+ E y+ Π = 3 E y 3 + E y Π Suig fro to, we obtai E y + Π = = 3 E y = 3 We are ow ready to fid Ey + Π Lettig i = i4, M 4 4 E y + Π+ E y + Π = E y E y+ Π

6 HM Bui, JP Keatig / Joural of Nuber Theory So E y + Π = = M E y M 3 E y Π M 4 = Substitutig this ito 3, we have E Π = = = + 3 Fro ad 5 we deduce that M 4 M 4 E Π= M + Var Π= O M M, 3 Theore i [9] the iplies that Π log Now we defie { if α, + α, r α = 0 otherwise The Π X,X+ ; α = r α = r α y α + y α y y α + α = a αb α, where a α = r α y α + y α

7 90 HM Bui, JP Keatig / Joural of Nuber Theory ad b α = y α y + α Let A 0 α = 0 ad A α = j= a j α Usig Abel suatio, Π X,X+ ; α = a αb α = A αb α < A α b + α b α Sice A α = Π; α, A b / log / log /log The result follows if we ca show that A b + b =o log Firstly, < A α = = 3 j, j α j+ α j, j α j+ α y j α j+ j y j α y j α, j, j α j+ α y j α 3 y j α which is O/log fro [9, Theore 4] Secodly, b+ α b α Sice = we obtai So y + α + y + + α y α y + α { y + α = y α = y α if α, y α otherwise, b+ α b α { y α y α if α, = ++ y α otherwise b + α b α { y α if α, otherwise + y α

8 HM Bui, JP Keatig / Joural of Nuber Theory Hece Thus < A α b + α b α = < α A α b + α b α + A α b + α b α < / α < α A α b + α b α = O < = O A αy α + < / α < α < α log < + A αy α / α log, log 3 which is easily see to be O/log 3 fro the aalogue of prie uber theore for Hawkis rado sieve The result follows 3 Proof of Theores ad 3 I this sectio, we take [i,i,,i l ] α, where i <i < <i l, to ea + {i,i,,i l } α, ad + h/ α for all h [i,i l ]\{i,i,,i l } Lea Give a o-egative iteger l ad 0 = i 0 <i <i < <i l <i l+ = k, defie T [0,i,i,,i l,k] := [0,i,i,,i l,k] α The T [0,i,i,,i l,k] /log l+ alost surely Proof We siply write T for T [0,i,i,,i l,k] LetA be the evet α ad B be the copleet of A, ie, B = A c We the have P = P + [0,i,i,,i l,k] α = PA +k+ B +k B ++il A ++il B + A +

9 9 HM Bui, JP Keatig / Joural of Nuber Theory By the chai rule P = PA + or, i short, P = = y + l j=0 PB + A + PB +i B +i B + A + PA ++i B +i B + A + PA +k+ B +k B ++il A ++il B + A + + y + i + y i l + + i j l+ j y l+ + j=0 l l j + + i l h=0 y +, i h y + il+ j i l j Sice we have l l j il+ j i l j y i h j=0 = h=0 l i l+ j i l j y + j=0 = k l y + y+ y+ y +, + y l+3 P = y l+ + k l yl+3 + y+ y l+4 + Fro the defiitio of T, { if + [0,i,i T+ T=,,i l,k] α, 0 otherwise

10 HM Bui, JP Keatig / Joural of Nuber Theory Hece E T+ E T = E y l+ + k l E y l+3 Ey l = M l+ + = M l+ + Suig fro to yields E T = M l+ k l M l+3 + k l M l = l+ l + M l+3 k l 3 M l+3 M l+4 + M l+4 + k l l+3 M l+4 M l+3 + M l+4 E y l The et step is to estiate the variace of T As i the case of the previous theore, we eed to establish a recursio for y+ i T For this we have Sice y i + we deduce that T+ = y+ i T if + / α, + iy i + T + if + [0,i,i,,i l,k] α, + iy i + T otherwise { P + / α = y+, P + [0,i,i,,i l,k] α = P, E y+ i T+ = E y+ i T y + + or, equivaletly, + + i E y+ i + T y + P, i E y+ i T+ P E y+ i T = i E y+ i + P i E y i+ + + T 7 Lettig i = l + 4, ad recallig that P = y l+ + k l yl+3 + y+ l+4,

11 94 HM Bui, JP Keatig / Joural of Nuber Theory we obtai So E y l+4 + T = O E y l+6 + = O M l+6 + E y l+4 + T = O M l+6 = O l+6 Lettig i = l + 3 i 7, ad suig fro to, we have E y l+3 + T = M l+5 Fially, substitutig i = l + i7, M l+6 = l+5 E y l+ + T = l+ E y l+ + + P = M l+4 + k + M l+5 + M l M l+6 l+ E y l+3 + T Thus E y l+ + T = = M l+4 = M l+4 M l+4 k + + l + 4 M l+5 k l 3 M l+5 k + M l+5 M l+5 l+6 M l+6 M l+6 8 Now, back to the variace of T, Therefore { T T + T+ if + [0,i + =,i,,i l,k] α, T otherwise E T + = E T P + E T + T+ P = E T + E TP + EP

12 HM Bui, JP Keatig / Joural of Nuber Theory Ad hece Fro 8, we obtai So E T = E y l+ + T k l E y l+3 T E y l+4 + T E y+ l+ E T = l+4 E T = = l+4 M l+4 = k l 3 k 3l 4 M l+5 M l+4 = l+4 + M l+5 k 3l 4 l + l+5 k 4l 6 l+5 Cobiig 6 with 9, we have ET = l+ VarT = O k l M l+6 M l+5 k 3l 4 l+6 k l 3 l+3 M l+6 l+5 M l+5 M l+6 l+6 M l+6 9, l+4 Theore i [9] agai yields T /log l+ as asserted The proof of Theore ow follows iediately fro Lea by otig that k Π X,X+k = l=0 0<i <i < <i l <k k = T [0,k] + T [0,i,i,,i l,k] l= 0<i <i < <i l <k = + o log log 3 T [0,i,i,,i l,k] Siilarly for Theore 3, Π X,X+k,,X+k l = T [0,k,k,,k l ] log l+ = + o log l log l+

13 96 HM Bui, JP Keatig / Joural of Nuber Theory Ackowledget JPK is supported by a EPSRC Seior Research Fellowship Refereces [] H Craer, O the order of agitude of the differeces betwee cosecutive prie ubers, Acta Arith [] B Gree, T Tao, The pries cotai arbitrarily log arithetic progressios, athnt/ [3] GH Hardy, JE Littlewood, Soe probles i Partitio Nueroru III: O the epressio of a uber as a su of pries, Acta Math [4] D Hawkis, The rado sieve, Math Mag 3 957/958 3 [5] D Hawkis, Rado sieves II, J Nuber Theory [6] CC Heyde, O asyptotic behaviour for the Hawkis rado sieve, Proc Aer Math Soc [7] CC Heyde, A log log iproveet to the Riea hypothesis for the Hawkis rado sieve, A Probab [8] W Neudecker, D Willias, The Riea hypothesis for the Hawkis rado sieve, Copos Math [9] MC Wuderlich, A probabilistic settig for prie uber theory, Acta Arith Further readig [0] W Neudecker, O twi pries ad gaps betwee successive pries for the Hawkis rado sieve, Math Proc Cabridge Philos Soc [] P Ribeboi, The Book of Prie Nuber Records, secod ed, Spriger, New York, 989 [] MC Wuderlich, The prie uber theore for the rado sequeces, J Nuber Theory