Probabilistic approach to the distribution of primes and to the proof of Legendre and Elliott-Halberstam conjectures VICTOR VOLFSON

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1 Probblstc pproch to the dstrbuto of prmes d to the proof of Legedre d Ellott-Hlberstm cojectures VICTOR VOLFSON ABSTRACT. Probblstc models for the dstrbuto of prmes the turl umbers re costructed the rtcle. The uthor foud d proved the probblstc estmtes of the devto R() ( ) L( ). The uthor hs lyzed the probblstc models of the dstrbuto of prmes the turl umbers d ffrmed the vldty of the probblstc estmtes of proved devtos R() stroger th the estmtes mde uder the ssumpto of Rem cojecture. Legedre's cojecture ws proved ths pper wth probblty rbtrrly close to 1 bsed o the probblty estmtes. Probblstc models for the dstrbuto of prmes the rthmetc progresso k l, ( kl, ) 1re lso bult ths pper. The uthor hs proved the probblty estmtes for the devto R(,k,l) (, k, l) L( ) / ( k). The uthor hs lyzed the probblty models of the dstrbuto of prmes the rthmetc progresso d ffrmed the vldty of probblstc estmtes of proved devtos R(,k,l) stroger th the estmtes mde uder the ssumpto of the eteded Rem cojecture. Ellott-Hlberstm A cojecture { m( kl, ) 1 R(,k,l) } C / l ( ) ws proved ths pper wth probblty 1k rbtrrly close to 1 for ll 0 1 bsed o the probblty estmtes. 1. INTRODUCTION The probblstc methods hve recetly bee used to model the dstrbuto of prme umbers. Hrdy, Lttlewoods d Krmer were the frst to mke t ther cojectures. These cojectures showed tht whe modelg the dstrbuto of prmes probblstc methods provde stroger results th the theory of fuctos of comple vrble d therefore cot be uderestmted. The tsk of studyg the dstrbuto of vlues of rel rthmetc fuctos c be ssocted wth the theory of summto of rdom vrbles some cses [1]. It turs out tht the questo of the symptotc behvor of these fuctos c be reduced to the kow lmt theorems of the probblty theory. Keywords: probblstc model, lmt dstrbuto of the rdom vrble, orml dstrbuto, probblty estmtes, prme umbers, Rem cojecture, Legedre cojecture, Crmer cojecture, Ellott-Hlberstm cojecture. 1

2 It s terestg to see the probblstc ssessmet of rel rthmetc fuctos of the umber of prmes the turl umbers d rthmetc progresso. Ths questo s dscussed ths pper. Ths rtcle cosders severl probblty spces such tht the ctul rthmetc fucto of the umber of prmes the turl umbers d rthmetc progresso re reltvely smple sums of rdom vrbles. The symptotc lw of prme umbers proved for the frst tme usg the theory of fuctos of comple vrble s well kow the followg form: ( ) ~ / l( ), (1.1) where ( ) s the umber of prmes ot eceedg postve teger. The lytcl pproch of Chebyshev ws erly ttempt to estmte the vlue: R( ) ( ) / l( ). (1.) I [] the estmte: / l( ) ( ) b / l( ) (1.3) ws show to be vld. Chebyshev showed tht 0. 91db Vlues more close to 1 were obted subsequet ppers. From (1.3) t follows: ( 1) / l( ) ( ) / l( ) ( b 1) / l( ). (1.4) However, estmtes of type (1.4) re ccurte: R( ) O( / l( )). (1.5) The symptotc lw of the prme umbers gves more ccurte vlue ( ) usg the tegrl logrthm: ( ) ~ L( ). (1.6) The ccurcy of the formul (1.6) s gve by: 1/ ( ) L( ) O( l( )), (1.7) f the Rem hypothess s true. There s lso Legedre s formul:

3 ( ) ~ / l( ) B. (1.8) But the formul (1.8) s less ccurte th equto (1.6). Ths rses the questo whether t possble to get better probblstc ssessmet of the ccurcy of the formul: ( ) / l( ) o(1/ l( )). (1.9) To swer ths questo, we cosder probblstc models of the dstrbuto of prmes the turl umbers.. PROBABILISTIC MODELS OF THE DISRIBUTION OF PRIMES IN THE NATURAL NUMBERS Cosder the frst probblstc model. Let there be blls dstgushble to the touch. Number them by cosecutve turl umbers from 1 to d put them ll bo. Select oe bll from the bo t rdom. If ts umber belogs to pre-selected teger postve jectve sequece, the we cosder the evet s success, d f the umber does ot belog to the selected sequece, the we cosder ths evet s flure. We ssume tht the probblty of the successful evet s p. Accordgly, the probblty of flure evets wll be equl to1 p. We troduce the rdom vrble I 1 dctor of the success of the evet. The vlue of the rdom vrble s equl to I1 1, f t s success d vlue s equl to I1 0, f t s flure. Retur the frst bll the bo, toss blls the bo d choose the secod bll from the bo t rdom. If the bll umber belogs to the selected sequece, the ssg to the rdom vrble the vlue I 1. If the bll umber does ot belog to the selected sequece, the ssg to the rdom vrble the vlue I 0. The retur the secod bll the bo, too. Repet t tmes. Sce the blls re ech tme retured to the bo d every et bll s resmpled the sme codtos, the rdom vrbles I re depedet. Thus, we obt sequece of depedet rdom vrbles dctors of success of the evet: I1, I,... I. The epectto of rdom vrble I s equl to: M( I ) p1 (1 p) 0 p. (.1) The vrce of rdom vrble I s equl to: D I p p p p p p. (.) ( ) (1 ) (1 ) (1 ) 3

4 Let us cosder the rdom vrble equl to the sum: I( ) I. (.3) 1 Defe the chrcterstcs of the rdom vrble I( ). The epectto of the rdom vrble I( ) bsed o (.1) d (.) d the lerty of epecttos s equl to: M ( I( X )) M ( I ) M ( I ) p. (.4) 1 1 Bsed o (.), (.3) d depedecy of the rdom vrbles I, the vrce of the rdom vrble I( ) s equl to;. (.5) 1 1 D( I( )) D( I ) D( I ) p(1 p) Thus, we hve the depedet, detclly dstrbuted rdom vrbles I1, I,... I wth bouded dsperso. Therefore, the rdom vrble I( ) I hs boml dstrbuto. The lmtg dstrbuto for the rdom vrble I( ) for the fed probblty p s the orml dstrbuto. Bsed o Movre-Lplce theorem [3], the followg relto s held: lm { P( I( ) M ( I( )) C D( I( ))} F( C), (.6) where P() the probblty of the evet s specfed pretheses d FC ( ) s the fucto of the module of the stdrd orml dstrbuto t pot C. Let us substtute the chrcterstcs of the rdom vrble I( ) from (.4) d (.5) to the epresso (.6) d obt: 1 lm { P( I( ) p C p(1 p) )} F( C). (.7) The vlue FC ( ) teds rpdly to 1 wth cresg C, bsed o the propertes of the orml dstrbuto. Thus, we c choose such vluec tht the probblty of the evet: I( ) p C p(1 p) (.8) s rbtrrly close to 1. 4

5 Ths s dvtge of probblstc ssessmets s t gves the opportuty to select prtculr vlue C. For emple, formul (.8) hs the beeft, s compred to the formul (1.7), whch the ccurcy cot be evluted. Deote by ( f, A, B) the umber of members of the teger, postve, jectve sequece f( ) o the tervl [ AB)., It ws proved [4] tht the desty of the sequece f( ) of the turl umbers wth the tervl [ AB), defed by the formul: ( f, A, B) / ( B A) (.9) s fte probblty mesure. codtos. The sequece of prme umbers s teger, postve, jectve,.e. stsfes the bove O the bss of the symptotc lw of prme umbers d (.9), t ws lso show [4] tht the probblty of choosg prme wth the tervl of the turl umbers [, ) s equl to: p 1/ l( ) o(1/ l( )). (.10) The rdom vrble I( ), ths cse, c be cosdered s the umber of prmes ot eceedg turl umber - ( ) [1]. Thus the formul (.7) for the sequece of prmes d lrge my be wrtte o the bss of (.10): P( ( ) (1/ l( ) o(1/ l( )) < C (1/ l( ) o(1/ l( ))(1 1/ l( ) o(1/ l( ))) F( C). (.11) of evet: O the bss of (.11), for lrge vlues we c choose vlue such tht the probblty ( ) (1/ l( ) o(1/ l( )) C (1/ l( ) o(1/ l( ))(1 1/ l( ) o(1/ l( ))) (.1) s rbtrrly close to 1. Let us lyze formul (.1). It gves the upper boud ( ) for the devto of the umber of prmes ot eceedg from the vlue / l( ) o(1/ l( )),.e. the estmte of ccurcy for the formul (1.9). 5

6 Cosder ths probblstc model the cse whe o 1/ l s the fucto: f ( ) ( 1) / l ( ) L( ) / 1/ l( ). (.13) 1 Substtute fucto (.13) to the formul for the vrce of the rdom vrble I( ) the frst probblstc model d obt: dt ( ) dt l( t) D1 ( I( )) L( )(1 L( ) / ) L( ) L ( ) /. (.14) l( t) By substtutg (.14) to (.1) we fd tht the probblty of the evet: ( ) ( ) ( ) / L C L L (.15) s rbtrrly close to 1. Let us lyze ths probblstc model. After smplg ( ths model) the bll returs to the bo g. Therefore, ths model t s possble to choose the sme bll severl tmes. Ths does ot hppe whe we clculte the umber of members of the sequece the rge of the turl umbers from 1 to the rel stuto. Cosder the secod probblstc model of the dstrbuto of prmes, whch s free from ths drwbck. It wll be bsed o the of Crmer s model. Let ( P ) be fte seres of urs cotg blck d whte blls? The chce of drwg whte bll from U beg p, whle the composto of U1 d U my be rbtrry chose. We ow ssume tht oe bll s drw from ech ur, so tht fte seres of ltertely blck d whte blls s obted. If P deotes the umber of the ur from whch the -th whte bll the seres s drw? The umber PP,,... 1 wll from cresg sequeces of tegers, Ad we shll cosder the clss C of ll possble sequeces. Obvously the sequece S of ordry prme umbers (p ) belogs to ths clss. We shll deote by I( ) the umber of those P, whch re. We deote by I the rdom vrble tht tkes the vlue 1 f whte bll gets from the -th ur d the vlue 0 otherwse. Thus we hve I( ) I. 1 6

7 Thus, the probblty model s free from the bove drwbcks. We shll fd the chrcterstcs of rdom vrbles of the secod probblstc model. The epectto of the rdom vrble I s equl to: M( I) 1 p 0 (1 p ) p. (.16) The vrce of the rdom vrble I s equl to: D I p p p p p p. (.17) ( ) (1 ) ( ) (1 ) (1 ) Defe chrcterstcs I( ). Bsed o (.16) the epectto I( ) s equl to:. (.18) M ( I( X )) M ( I ) M ( I ) p The vrce of the rdom vrble I( ) (bsed o depedet rdom vrbles I d formuls (.17) s equl to:. (.19) D( I( )) D( I ) D( I ) p (1 p ) p ( p ) The lmtg dstrbuto of the rdom vrble I( ) s orml, bsed o Lypuov theorem [3]. Therefore, bsed o formuls (.6), (.17) d (.19), we obt: lm ( ( ) ( ) } 1 1 1, (.0) { P I p C p p F( C) where FC ( ) s the vlue of the module of the fucto of the stdrd orml dstrbuto t pot C. Formul (.0) (for lrge vlues ) c be wrtte s: ( ( ) ( ) ) ( ) (.1) P I p C p p F C The probblty teds rpdly to 1 wth cresg, bsed o the propertes of the orml dstrbuto. Thus, we c choose such vlue C, tht probblty (.1) s rbtrrly close to 1. We shll fd the chrcterstcs of the probblstc model for the sequece of prme umbers. Let us mke heurstc ssumpto tht the probblty of turl umber to be prme s equl to1/ l( ), the the probblty of choosg the whte bll the Krmer s model 7

8 s p 1/ l() for. Bsed o Krmer rdom vrble I( ) c be regrded s the umber of prmes ot eceedg turl umber - ( ). The epectto of the rdom vrble I( ) for the sequece of the prme umbers, bsed o formul (.18) d the heurstc ssumpto, s equl to: M ( I( )) 1/ l( ) dt l( t). (.) The vrce of the rdom vrble I( ) for the sequece of prme umbers, bsed o formul (.19) d the ssumpto, s equl to: l( t) l ( t) D( I( )) 1/ l( ) (1/ l( )) dt. (.3) dt Bsed o (.1) we obt (for lrge ): dt dt dt P( I( ) C ) F( C), (.4) l( t) l( t) l ( t) where FC ( ) s the vlue of the fucto of the module of the stdrd orml dstrbuto t pot C. Thus, we c choose such vlue C, tht probblty (.4) s rbtrrly close to 1. The questo rses cocerg the ppromto sg formuls (.) d (.3). There re ssertos 1,, 3 for prtl summto [5] bsed o the Euler-McLur formul. Asserto 1. Assume tht the fucto F( ) s cotuously dfferetble d mootoclly decresg o the tervl [ A, ) d lm F( X) 0. The F( A ) F( ) d C O( F( )), where 0 A C lm[ F( A ) F( ) d]. 0 A Asserto. Suppose tht there ests the fucto F o the tervl [ A, ) wth the followg propertes: 1. lm F ( ) 0.. It hs dervtves of the desred order. 3. (k F 1) ( ) 0. (k1) B k ( K)! F ( A) 4. 1, (k1) B (k )! F ( A) k where B s - Beroull umber. The C F( A) / F( A) /1, wherec lm[ F( A ) F( ) d] 8. 0 A

9 k Asserto 3. The followg estmte s stsfed for the fucto F( ) 1/ l ( ) o the tervl [ A, ): C 0.60F( k 1), wherec lm[ F( A ) F( ) d]. 0 We obt the followg estmte bsed o sserto 1: A 1/ l( ) dt / l( t) C O(1/ l( )). 1 1/ l ( ) dt / l ( t) C O(1/ l ( )). From ssertos d 3 t follows tht C , C These vlues re eglgble for lrge. The fucto 1/ l( ) 1/ l ( ) decreses mootoclly oly for vlues 7, d therefore sserto 1 s pplcble to ths tervl. Let us clculte seprtely the sum d the tegrl o the tervl from to 7. The sum s equl to d the tegrl to I vew of the dfferece betwee the sum d the tegrl, overll estmte of the costt s equl to C for vrce,.e. t s lso eglgble. 3. ANALYSIS OF THE PROBABILISTIC MODELS OF THE DISTRIBUTION OF THE PRIME NUMBERS Compre the vrce of the frst probblstc model defed by formul (.14) wth the vrce of the secod probblstc model defed by formul (.3). We use specl cse of Cuchy-Schwrz equlty for comprso: ( ( ) ( ) ) ( ) ( ) u v d u d v d. (3.1) The equto (3.1) s wrtte s f u ( ) 1 d v ( ) 0. We hve: ( v( ) d) v ( ) d, (3.) dt ( ) l( t) dt, (3.3) l ( t) 9

10 bsed o (3.) for lrge vlues. From (3.3) we obt: D ( I( )) D ( I( )). (3.4) 1 The bsed o (3.4), we hve the followg upper boud: D ( I( )) D ( I( )) L( ), (3.5) 1 dt where L( ). l( t) The questo turlly rses bout the vldty of these probblty estmtes s the probblty estmtes for devtos r( ) ( ) L( ) re stroger th estmte (1.7) mde uder the ssumpto of the Rem cojecture. To cofrm these probblty estmtes we hve to fd lredy proved lower boud for dt dt the fucto Rtht ( ) hs the order less th D ( I( )) l( t) l ( t) cert vlue. strtg from The pper [6] cots such ssessmet. Lttlewoods showed tht the vlue Rcot ( ) hve the order less th: lll( ) l( ) (3.6) s teds to fty. Therefore, t suffces to compre the order of the squre of the fucto dt dt (3.6) wth the vlue D ( I( )) l( t) l ( t). Let us fd the lmt of the vlue: l (ll( )) A lm { }. (3.7) dt dt l ( )( ) l( t) l ( t) We fd tht the vlue of (3.7) s equl to: l (l( )) lll( ) { { l( ) (ll( ))l ( ) A lm } lm } 0, (3.8) 10

11 (usg L'Hosptl's Rule d the dervtve o the upper lmt of tegrto), sce both epressos the sum ted to 0. Thus, the fucto (3.6) hs the order of mgtude smller dt dt th D ( I( )) l( t) l ( t) probblstc models.. It vldtes estmtes mde bsed o the frst d secod The secod probblty model s more ccurte th the frst probblty model, s t does ot hve the dsdvtges of the frst oe. O the other hd, bsed o (3.5), dsperso for the frst probblstc model (.14) s greter th tht for the secod probblstc model (.3), so t c be used for the upper estmte (.15). Thus, the frst probblstc model s strctly proved (wthout ssumg the secod probblstc model) d equlty (.15) (for the frst probblstc model) s rgorously proved sserto. 9 Numercl clcultos show tht the equlty s t lest stsfed for vlues 10. However, Lttlewoods, s metoed erler, proved the theorem [6] d showed tht ths cse s ot lwys held. But ths theorem does ot fd such vlue tht the equlty ( ) L( ) s stsfed. Skewers hve lredy swered ths questo. He foud such vlue ep ep ep ep(7.705) [7] d cofrmed ths fct. The probblstc models of the dstrbuto of prmes the turl umbers lsted bove correspod to the specfed Lttlewoods theorem d prove tht the equlty sg c be chged. It hs bee coclusvely estblshed tht these devtos ccel ech other [8]. Ths s cosstet wth the results obted o the stdrd orml dstrbuto of the devto. Therefore, the term of the dstrbuto of prme umbers s well fouded the termology of the theory of probblty. 4. APPLICATION OF THE PROBABILISTIC MODELS OF THE DISTRIBUTION OF PRIMES IN THE NATURAL NUMBERS Suppose tht the Rem cojecture s true. The (for suffcetly lrge umber A 0 d ) there lwys ests t lest oe prme betwee the umbers [9]: d A 1/ l( ). (4.1) vlues: Smlrly, from (.14) t follows tht there lwys ests t lest oe prme betwee the d C L( ) L ( ) / (4.) 11

12 wth probblty rbtrrly close to 1. Bsed o (4.6), ote tht the codto: L( ) L ( ) / L( ). (4.3) s held. Thus, codto (4.) s stroger th (4.1). umbers: Recll tht the Krmer cojecture ssumes tht there lwys est prmes betwee the d B l ( ). (4.4) for lrge vlues d pproprte B 0.Therefore, from (4.4) we hve: 1/ A C L L B l( ) ( ) ( ) / l ( ). (4.5) Thus, the codto (4.4) s stroger th (4.), but (4.4) s oly the cojecture. Legedre cojecture, tht there ests t lest oe prme betwee the squres of cosecutve postve tegers re well kow. Asserto 4. Legedre cojecture s held wth probblty rbtrrly close to 1. Proof. It s kow [9], tht the Legedre cojecture wll be proved, f we show, tht there lwys ests t lest oe prme betwee some lrge vlues d bsed o the strctly prove frst probblstc model d relto (4.).. We shll prove t, Thus, we must prove, tht: L( ) L ( ) /. (4.6) It s kow, tht: L( ) / l( ) C / l ( ) (1/ l( ) C / l ( )), (4.7) where 1C. If e d 1 C, the: 1/ l( ) C / l ( ) 1. (4.8) 1

13 Therefore, for the vlue e from (4.7) d (4.8), t follows thtl( ) (1/ l( ) C / l ( )). So, for the vlues e we hve tht L( ) L ( ) / L( ). Q.E.D. 5. PROBABILISTIC MODELS OF THE DISTRIBUTION OF PRIMES IN THE ARITHMETIC PROGRESSIONS Let us cosder the thrd probblstc model. Suppose there s rthmetc progresso f () k l where ( kl, ) 1. Wrte the vlues f (0) l, f (1) k l, f () k l,... f ( ) k l o dfferet blls, dstgushble to the touch, put them bo d m. Choose t rdom the frst bll from the bo d ssg the vlue I1 1to the rdom vrble of the dctor of success, f the frst bll umber s prme or the vlue I1 0 to the rdom vrble, f ot. The retur the bll the bo, m the blls d tke out the secod bll from the bo. Assg the vlue I 1 to the rdom vrble (the dctor of success), f the secod bll umber s prme or the vlue I 0to the rdom vrble f ot. Repet t g / ktmes. Sce the blls re retured to the bo d the et bll s selected uder the sme codtos, the rdom vrbles I re depedet d the probblty to choose prme ech test p s equl to p p. We hve lredy cosdered the chrcterstcs of the rdom vrble: M ( I ) D( I ) p(1 p). the formul: p, The desty of the sequece g, ( ) s proporto of the sequece f( ) s determed by P( g / f,1, ) ( g,1, ) / ( f,1, ), (5.1) where ( g,1, ), ( f,1, ) re the umbers of members of the sequece gd ( ) f( ), respectvely, the rge of the turl umbers [1, ). 13

14 Bsed o [4], the desty P( g / f,1, ) s fte probblty mesure the rge of the turl umbers[1, ). Therefore, p P( g / f,1, ) s stsfed. The probblty of rdomly selected bll umber to be prme s equl to: p k / ( k)(1/ l( ) o(1/ l( ))), (5.) where ( k) s the Euler fucto (bsed o Drchlet's theorem bout the umber of the prmes rthmetc progresso d (5.1)). Let us cosder the rdom vrble: k / I(, k) ( I ). (5.3) 1 From the lerty of the epectto d (5.) we obt tht: M( I(, k)) ( p) / k (1/ l( ) o(1/ l( ))) / ( k). (5.4) Bsed o the depedece of rdom vrbles I, the vrce of the rdom vrble I(, k) s equl to: k D( I(, k)) p(1 p) / k (1/ l( ) o(1/ l( )))[1 ( (1/ l( ) o(1/ l( )))] ( k) ( k).(5.5) The rdom vrble I(, k) the probblstc model c be cosdered, followg [1], s the umber of members of the rthmetc progresso f ( ) k l, ( kl, ) 1, whch re the prmes. We cosder the thrd probblstc model for the cse, whe o 1/ l s the fucto f ( ) ( 1) / l ( ) L( ) / 1/ l( ). Substtute the fucto f( ) to the 1 formul for the epectto (5.4) of the rdom vrble I(, k) d obt: M 1 3( I (, k )) ( ) ( k) L. (5.6) Now we substtute the fucto f( ) the formul for the vrce (5.5) of rdom vrble I(, k ) d get: 1 kl( ) L( ) kl ( ) 3( (, )) L( )[1 ] D I k ( k) ( k) ( k) ( k). (5.7) 14

15 The rdom vrble I(, k) s the sum of the depedet rdom vrbles I wth fte vrce d thus, t hs the boml dstrbuto. Bsed o Movre-Lplce theorem [3] the lmt dstrbuto for I(, k) s orml, therefore the epresso s true for lrge vlues : P( I(, k) M ( I(, k)) C D( I(, k) ) F( C), (5.8) where FC ( ) s the fucto of the module of the stdrd orml dstrbuto t pot C. Substtute the chrcterstcs of the rdom vrble I(, k) to the epresso (5.8) defed by the formuls (5.6) d (5.7) d obt: 1 L( ) kl ( ) ( k) ( k) ( k) P( I(, k) L( ) C ) F( C). (5.9) Thus, bsed o the epresso (5.9) we c select the vlue such tht the probblty of the evet 1 L( ) kl ( ) ( k) ( k) ( k) I(, k) L( ) C s rbtrrly close to 1. Note tht formuls (5.8) d (5.9) re vld for lrge vlues / k. Let us lyze the thrd probblstc model. I ths model, s the frst oe, the bll s retured to the bo fter t ws chose. Therefore, there s the possblty to choose the sme bll few tmes ths model. Such evet does ot hppe the rel stuto, whe oe clcultes the umber of prmes of the rthmetcl progresso p k / ( k)l( ) wth the tervl of the turl umbers from 1 to. Therefore, ths probblstc model requres clrfcto. Cosder the fourth probblstc model tht s free from the drwbck of the thrd probblstc model. We wll be bsed o Crmer s model dscussed bove. Let mke heurstc ssumpto, tht the probblty of member of the rthmetc progresso k l,( k, l) 1, k l to be prme ( Krmer s model the probblty to choose from -th ur whte bll) s equl to: p k / ( k)l( ). (5.10) The sequece of prmes the rthmetc progresso s specfed teger, o-egtve, strctly cresg, therefore belogs to the clss C Krmer s model. Cosder the rdom vrble I( ) ( I) 1 15, where the rdom vrble I tkes the vlue 1 f we get whte bll

16 from the -th ur d the vlue 0 otherwse. Summrzg Krmer rdom vrble I ( ) c be regrded s the umber of members of the rthmetc progresso re the prmes. model: f (t) kt l, ( kl, ) 1, whch We hve lredy determed the chrcterstcs of the rdom the secod probblstc. M ( I( )) ( p ), D( I( )) ( p )(1 p ) 1 1 Bsed o (5.10) we fd the epectto of the rdom vrble I: ( ) 1 M ( I( )) k / ( k) k / ( k) dt 1 1 l( ) t k l l( kt l). (5.11) The ppromto sg formul (5.11) should be cosdered s formuls (.) d (.3), therefore the dfferece betwee the sum d the tegrl s eglgble (for lrge vlues t ). Chge the vrbles (5.11) u kt l d obt: dt du du M ( I( )) k / ( k) k / k( k) 1/ ( k) l( kt l) l( u) l( u) t 1 u kl. (5.1) ukl Equto (5.1) c be wrtte s: du kl du M ( I( )) 1/ ( k) 1/ ( k)[ L( ) ]. (5.13) ukl l( u) l( u) kl du The vlue of the tegrl (5.13) s bouded by ( k l ) / l(), d the l( u) followg relto s held for lrge vlues : M( I( )) L( ) / ( k). (5.14) The vrce of the rdom vrble Is ( ) equl to: 1 1 du du D( I( )) k / ( k) k / ( k) 1/ ( k) k / ( k) l( k l) l ( k l) l( u) l ( u) kl kl 1 1.(5.15) The ppromto sg formul (5.15) should be cosdered the sme sese s (5.11). As showed erler, the dfferece betwee the sum d the tegrl s eglgble for lrge 16

17 vlues.the vlue of the tegrl (5.15) s bouded by kl du l ( u) ( k l ) / l () followg formul s held for the vrce of the rdom vrble I(for ( ) lrge vlues ): d the du D I k k k du kl ( ( )) 1/ ( ) / ( ) l( u) l ( u). (5.16) The rdom vrble Is ( ) the sum of mutully depedet rdom vrbles I. Bsed o Lypuov theorem the lmt dstrbuto for the rdom vrble Is ( ) orml, therefore the epresso (for lrge vlues ) s true: P( I( ) M( I( )) C D( I( )) F( C), (5.17) where FC ( ) s the vlue of the fucto of the module of the stdrd orml dstrbuto t pot C. Substtute the chrcterstcs obted for the rdom vrble I ( ) (5.14), (5.16) to the epresso (5.17) d obt: k du P( I( ) L( ) / ( k) C L( ) / ( k) ) F( C) ( k). (5.18) l ( u) Thus, bsed o (5.18), you c select such vluec tht the probblty of the evet k du I( ) L( ) / ( k) C L( ) / ( k) ( k) s rbtrrly close to 1. l ( u) 6. ANALISYS OF THE PROBABILIISTIC MODELS OF THE DISTRIBUTION OF PRIMES IN THE ARITHMETIC PROGRESSION We hve the followg dsperso relto for probblstc models of the dstrbuto of the prmes rthmetc progressos: D( I( )) D ( I(, k)) L( ) / ( k), (6.1) 3 where the vlue D 3 ( I(, k )) s defed by (5.7) for the thrd probblstc model d D( I( )) s defed by (5.16) for the fourth probblstc model. Relto (6.1) s proved by the prtculr cse of Cuchy-Schwrz equlty usg formul (3.1). The geerlzed Rem cojecture s equvlet to the formul: (, k, l) L( ) / ( k) C l( ), (6.) 1 17

18 where (, k, l) s the umber of prmes ot eceedg the umber the rthmetcl progresso k l,( k, l) 1. Comprg (5.9) d (5.18) for the probblstc models of dstrbuto of prmes the rthmetc progresso wth formul (6.) we see tht these estmtes re stroger. However, s show erler, bsed o Lttlewoods s work, these estmtes re vld. O the other hd, formul (6.) s vld for ll k d formuls (5.9) d (5.18) re vld oly wth cert probblty. The ppromte clculto ofte use prctce d therefore formuls (5.9), (5.18) re coveet to use these cses. The vlue of the costt C these formuls s selected depedg o the desred ccurcy. The secod d fourth probblstc models re cojectures, sce they re bsed o ssumptos. The frst d the thrd probblstc models re the proved ssertos. As metoed erler, formuls (5.8) d (5.9) re vld for the vlue / ktedg to fty,.e. whe k o( ). For emple, the cse whe k ( 0 1 ). Ths correspods to Ttchmrsh theorem [10]: Let we hve 0 1 d1k. The there ests costt C C( ) such tht: (, k, l) C / ( k)l( ) for ll ( l, k) 1,0 l k d Theorem [11]: O the ssumpto of the prevous theorem, for ech k we c fd such c 1, c, tht there ests more th c1 ( k) dfferet vlues l, for whch: (, k, l) c / ( k)l( ). Cosder formul (5.9). The umber of prmes the rthmetc progresso kt l,( k, l) 1 ot eceedg depeds o the vlues k., The vlue L( ) / ( k) s the verge vlue over ll l d depeds oly o the vlues k., The vlue C D(, k) depeds o the ccurcy ( C) d vlues k., If we hve two rthmetcl progressos kt l1,( k, l1) 1d kt l,( k, l) 1, the from formul (5.9) we get: 1 L( ) kl ( ) ( k) ( k) ( k) I(, k) L( ) C (6.3) wth probblty rbtrrly close to 1. Smlrly to the bove theorem of Lttlewoods [6], t c be proved, tht (,4,1) (,4,3) C l l l( ) / l( ). (6.4) 4 Comprg formuls (6.3) d (6.4) we see tht (6.3) s more geerl th (6.4). 18

19 The fourth probblstc model s more ccurte th the thrd oe, s t s free of ts drwbcks. O the other hd, bsed o (6.1), the dsperso of thrd probblstc model (5.7) s greter th tht of the fourth probblstc model (5.16) d therefore t c be used to get upper-boud estmte the equlty (5.18). Sce the thrd probblstc model s rgorously proved d (5.18) s lso rgorously proved sserto. 7. APPLICATION OF THE PROBABILISTIC MODELS OF THE DISTRIBUTION OF PRIMES IN THE ARITHMETIC PROGRESSION Bsed o the thrd probblstc model, there ests t lest oe prme umber belogg to the rthmetc progresso k l,( k, l) 1 betwee the vlues: d D L( ) / k kl ( ) / ( ) (7.1) for the fed vlue k, lrge d pproprte D 0 wth the probblty close to 1. The codto (7.1) s stroger th the correspodg codtos resulted from the eteded Rem cojecture. Bsed o (5.18) we obt: m( kl, ) 1 I(, k) L( ) / ( k) C1 L( ) / ( k) kl ( ) / C1 L( ) / ( k), (7.) where the mmum s tke over ll l coprme wth k. Accordg to Ellott Hlberstm cojecture for ll 0 1d ll A 0 such C 0 tht s held: there ests A { m (, k, l) L( ) / ( k) } C / l ( ) (7.3) 1k ( kl, ) 1 for ll. Ths cojecture ws proved by Bomber d Vogrdov for vlues 1/. Asserto 5. Ellott-Hlberstm cojecture s held wth probblty rbtrrly close to 1 for ll 0 1. Proof. Bsed o the wek estmte (7.) we eed to prove tht: A C1 L( ) / ( k) C / l ( ). (7.4) 19

20 From L C d (7.4) we see tht we must prove the equlty: ( ) / l( ) / l ( ) C C k C. A 1 / l( ) / l ( ) 1/ ( ) / l ( ) 1k Bsed o the Ldu theorem [1]: ( ) C / l(l( )), (7.5) 3 f. Therefore, bsed o (7.5), we obt: 1/ ( ) l l( ) / C (7.6) 3 for.thus, bsed o (7.6), we obt: 1/ ( k) 1/ C l l( k) / k 1/ C l l( ) 1/ k (7.7) 3 3 1k k k sce (1) () 1. Bsed o sserto 3 we obt: dt y 1/ k C O(1/ y) y 3 C O(1/ y) O( y). (7.8) 3 1/ 3ky t Therefore, the equlty: / 1/ k C4 (7.9) 3k s stsfed. Substtute the estmte (7.7) (7.9) d obt: 1k 1/ ( k) C / C l l( ) / 4 3 1/ / ( k) C4 / C3 l l( ). (7.10) 1k Therefore, bsed o (7.10), we obt the estmte: C L( ) / ( k) C / l( ) C / l ( ) ( C / C l( ) l l( )), (7.11) / k where 0 1. From (7.11) t follows tht the estmte s held: (l l( )) C L k C C ( 1)/ 1/ ( 1)/ 1 ( ) / ( ) (7.1) 1/ 1k l( ) 0

21 for vlues 1 where 0 1.The followg equlty for vlues s held: (1 )/ A l ( ) (7.13) for y A 0 d 0 1. Therefore, bsed o (7.1) d (7.13) s held: ( 1)/ A C C / l ( ) (7.14) for vlues dc 0. Bsed o (7.), (7.1) d (7.14) we obt the estmte for the vlues m( 1, ) : A m { I(, k) L( ) / ( k) } C L( ) / ( k) C / l ( ). (7.15) ( kl, ) 1 1 1k Q.E.D. 8. CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK Ellott-Hlberstm cojecture wth the probblty close to 1 ws proved ths pper usg the wek estmte (7.). However, s show [13], from the vldty of Ellott-Hlberstm cojecture t follows, tht there s fte umber of the prme prs tht dffer by o more th 1. I hope tht usg the strog estmte (7.) t wll be ble to fd the probblstc pproch to the soluto of the bry Goldbch problem d to prove tht there s fte umber of prme tws. 9. ACKNOWLEDGEMENTS Thk you, everyoe, who hs cotrbuted to the dscusso of ths pper. Refereces 1. Kublus I. Probblstc methods umber theory, Vlus, p.. Buchstb A.A., Number Theory, Publshg House "Educto", Moscow, p. 3. Shryev A.N., Probblty-, Publsher MTsNMO, M, p. 1

22 4. Volfso V.L. Probblstc models d ssuptos of cojectures bout prme umbers, Appled fhyscs d mthemtcs, Volfso V.L., The dstrbuto of prme tuples the turl umbers, Appled Physcs d Mthemtcs, 013, Lttlewood J. E. Reserches the theory of the Rem zet fucto, Proc. Lod. Mth.Soc. (),0 (19), Skeweys S. O the dfferece ( ) l( ), Proc. Lod. Mth. Soc. (3), 5 (1955), Crmer H. Stude űber de Nullstelle der Remsche Zetfukkto, Mth.Z., 4 (1918), Crmer H. Some theorem cocerg prme umbers, Arkv mt., Astr. Och Fys., 15, 5 (190) 10. Ttchmrsh E.C. O the zeros of the Rem zet-fucto, Proc. Lod. Mth. Soc. (), 30 (199), Prchr K. The dstrbuto of prme umbers. Publsher "Mr", M, p. 1. Ldu E. Über de Verluf der zhletheoretsche Fukto, (), Arch. D. Mth u. Phys. (3), 5 (1903), Myrd J., Smll gps betwee prmes, upublshed t