The probability of Riemann's hypothesis being true is. equal to 1. Yuyang Zhu 1

Transcription

1 Th robablty of Ra's hyothss bg tru s ual to Yuyag Zhu Abstract Lt P b th st of all r ubrs P b th -th ( ) lt of P ascdg ordr of sz b ostv tgrs ad s a rutato of wth Th followg rsults ar gv ths ar: () Th followg ualty s tru () If ( ) l ( ) or l ( ) th l( ( )) Whr { } s a suc () Th robablty of Ra's hyothss bg tru s ual to (v) If l ( ) ad thr st ostv costat tgrs AK A K K th l( ( )) If l ( ) th thr st ostv tgrs K K Ks( s ) such that K K( s) ad l Whr K { K K K s } ( ) d ad s th Eulr costat d Kywords ubr Ra hyothss; o-trval zros; Rob s ualty; r Dartt of Math Ad Phys Hf Uvrsty Hf Cha 6 E-al: zhuyy@hfuuduc

2 AMS Subjct Classfcato M6 Itroducto ad a rsults Th Ra hyothss [] (RH) stats that th o-ral zros of th Ra zta fucto () s s ar o th l at () s / Rob [] otd out that f th RH s tru th ( ) hold for all 4 ad s Eulr s costat Covrsly If th RH s fals th thr st costats ad C such that ( ) for ft ubr of atural ubrs C ( ) I 7 Mar Wójtowcz [] rovd that thr s a subst W of asytotc dsty such that l ( ) whr stads for th st of W of ostv tgrs ad ( ) ( ) ( ) Zhu [4-] rovd th followg ualty: l ( H ) H ( ) Solé ad Zhu [6] rovd th followg cocluso: l f ( ) Furthr lt D( ) ( ) w hav th followg lts wh rags ovr Colossally Abudat Nubrs [6] If RH s fals th l f D ( ) If RH holds th thr l D ( ) or l f D ( ) s ft ad I fact accordg to ths cocluso ad th Rob crtro for RH th cssary ad suffct codto for RH bg tru s that l D( ) l( ( ))

3 I ths ar our uros s to furthr study th asytotc rorty of Rob s ualty ad rov that th robablty of RH bg tru s ual to Notatos For th covc of th statt w frstly gv so otatos Notato Dot a suc of r ubrs by ( ) wth Notato Dot by P th st of all r ubrs ad lt b th -th ( ) lt of P ascdg ordr of sz Ma rsults Th a rsults of th rst ar ar as follows Thor Lt b a ostv tgr ad s a rutato of wth th w hav () Not For ay suffctly larg atural ubr accordg to Thor f w rov ualty l( ) D( ) ( ( )) w oly d to rov ualty l( ) D( ) ( ( )) whr Accordg to th Rob crtro for RH ad th rsults [6] w oly d to study ualty

4 l D( ) l( ) Thor Lt { } b a suc ( ) P b th st of all r ubrs b th -th ( ) lt of P ascdg ordr of sz If l ( ) or l ( ) th l( ( )) () Accordg to Thor - w hav Ifrc Thr sts a ostv costat A such that for all atural ubrs wth f A th D ( ) Thor Th robablty of RH bg tru s ual to Thor 4 Lt { } b a suc ( ) P b th st of all r ubrs ad b th -th ( ) lt of P ascdg ordr of sz () If l ( ) ad thr st ostv costat tgrs AK such that A K K th l( ( )) () If l ( ) th thr st ostv tgrs K K Ks( s ) K such that K( s) ad l whr K { K K K s } Accordg to Thor ad () th Thor 4 w hav Ifrc Thr sts a ostv costat A for all atural ubrs f A th ( ) Cojctur Lt { } b a suc 4

5 ( ) P b th st of all r ubrs ad b th -th ( ) lt of P ascdg ordr of sz If l ( ) th l( ( )) If ths cojctur s rovd th RH s also rovd to b tru Orgazato of artcl W frst gv svral ortat las Scto Th roofs of a rsults ar rstd Scto Th roofs of las ar Scto 4 Las I th followg w wll gv s las as rlars for th roofs of Thor ~4 For th covc of statt w aot that th atural ubrs ar ozro ad dots th st of all atural ubrs La Lt f ( ) h h h h h h h h wth Th th u of f ( ) s ( ) f f ( ) f( ) La ([7 8]) Lt ( ) f 44 th ( ) () ( ) () Lt whr ( ) th ( ) ad

6 W hav th followg La : ( ) La Lt { } b a suc ad ( ) If l ( ) th thr sts costat a such that l ( ) a La 4 Th followg ualty holds: l () La ([9]) { O( )} La 6[6] W hav th followg lts wh rags ovr Colossally Abudat Nubrs If RH s fals th l f D ( ) If RH holds th thr l D ( ) or l f D ( ) s ft ad Proofs of Thors Proof of Thor To rov Thor th followg roostos ar rurd Proosto If a b th a b a b () Proof Sc a b () s uvalt to Lt a a b b () v ( ) ( ) 6

7 W wll rov v ( ) s a ootocally crasg fucto Sc v ( ) ad th roof of v( ) s uvalt to rovg that s o-gatv Lt w( ) Th w( ) ( ) Sc ad w( ) ( ) whch shows that w ( ) s a ootocally crasg fucto [ ) I addto w ( ) s cotuous at Thrfor w ca coclud that for w ( ) w() Hc Furthror v( ) v ( ) s a ootocally crasg fucto () ad () ar rovd Accordg to Proosto th followg ualty s obvous a b a b () wth ad a b Accordg to () aly ab a b ab a b a b a b a b a b (4) 7

8 Cosutly w hav Proosto Lt ( ) b a rutato of ad Th Proof For () accordg to (4) ths roosto s tru Assu th roosto holds for ( ) Th for lt a{ } r ( r ) W wll rov th roosto two cass: If th r r Accordg to (4) r r r r r r r r r r r (6) Accordg to th assuto of r r r (7) By (6) ad (7) If accordg to th assuto for By (8) (8) 8

9 To coclud th roosto s tru for It s rovd by th rcl of athatcal ducto Proosto If a b th Proof Fro a b ab ab ab ab Fro Proosto ad th rcl of athatcal ducto w ca rov th followg cocluso: Proosto 4 Lt ( ) b a rutato of ad Th (9) Proof s a trval cas For th roosto s tru accordg to Proosto Assu th roosto s tru for t Th for t lt t { } j t ( j t ) If j t accordg to Proosto j j t t t j t t j t j t t j t j t j t t t t t j t t j j t j t Usg th ducto hyothss wh t t j j t t (9) holds Thus Hc (9) holds for j t t t t t t t j = j t t t If j= t th t { } t accordg to th ducto hyothss t 9

10 wh t t t th t t t t t t t t t t t Sc t { } t t t t ad t t+ s a rutato of t t Hc (9) holds for j= t Th roosto s tru for t Fro th rcl of athatcal ducto th roosto s tru Now th roof of thor ca b gv as follows: Proof of Thor Accordg to (9) w hav Accordg to () By () ad () Accordg to La () () Thus () s tru by () ad () () () Proof of Thor

11 Proof of Thor Sc ( ) f s ft th thr sts N such that N Wh th ( ) (fro N ft Hc aly s ft) ad Sc N s a ostv costat s N l ( ) l l ( ) N l ( ( )) That s () s rovd udr th assuto of N Now w d to cosdr th cas Accordg to La thr sts costat a such that l ( ) I addto a ( ) ( ( )) aly ( )

12 ( ( )) Wh accordg to La ad 4 l ( ) l l ( ( )) l ( ( )) l ( a) ( a) That s l ( ( )) l ( a) Thus () holds I suary wh l ( ) or l ( ) th l ( ( )) Proof of frc sc ( ) l ( ) accordg to Thor ad Thor w hav l ( ( )) aly thr sts a ostv costat A such that for all atural ubr f A th D( ) ( ) Proof of Thor Proof of Thor Lt ( P ) by Thor w oly d to cosdr ( ) thrfor ( ) ad ( ) If

13 ( ) [] th for t t ( t ) t ( t ) t t t ( ) ( t) t ( ) [ t t ] Slarly f ( ) [ t t ] ( t ) th for t t ( ) t ( ) t [] sc t s arbtrary ( ) dstrbuts vly vry trval [ t t ]( t ) Accordg to Thor f l ( ) or l ( ) th l ( ) hc f l ( ) th l ( ) Accordg to l ( ) ad La 6 th robablty that RH bg tru s ual to Not : A vt wth a robablty ual to s ot cssarly a vtabl vt Accordg to Thor ad La 6 wh l ( ) () hold th RH s tru Thor 4 dscusss th asytotc rorty of Rob ualty wh l ( ) 4 Proof of Thor 4 () Lt R( ) []

14 R( ) K K K N K A K K A K K A K But l hc l K A K Naly l R( ) th thr sts a ostv tgr N wh N w hav R() R() Thus for N R() R() Lt th ( ) For a atural ubr sc ( ) by Thor l R() l ( ) 4

15 l( ) l R() l ( ) l( ) l R() Thus l ( ) aly l ( ) () Accordg to l ( ) thr st ostv tgrs K K Ks( s ) such that K K( s) ( f ot th j { } j j thus ( ) l ( ) but l ( ) whch s cotradctory ) K Lt K { K K K s } th K Wh K th l K l If K w d to rov that l K If l b th K b r (wth l r ) Sc Naly ( ) Tag arth o both sds ( ) K ( ) K K K K ( ) ( ) K K ( ) thus ( ) ( ) ( ) By () La wh K 44 K K K K ( K ) ( K ) ( ) ( ) ( ) ( ) ( ) ( K ) ( K ) Accordg to La ad (4) ( b r ) br ( ) (4) br

16 66788 ( ) { O } ( ) ( b r )( b r ){( b r ) (( b r ) ) ( b r ) O( )}( ) ( b r ) b r Dvdg both sds by ( ) { O } ( ) ( ) r r ( b )( b ){( b r) (( b r) ) But ( b r ) O }( ) ( b r ) b r l ( ) { O } ( ) ( ) Hc l ( ) r r l( b )( b ){( b r) (( b r) ) ( b r ) O( )}( ) ( b r ) b r r r l( b )( b ) b l ( ) { O } ( ) b ( ) K It s cotradctory Thus w rov that l Proof of frc sc ( ) l ( ) AK Accordg to Thor 4 th cocluso s corrct 6

17 4 Proof of las 4 Proof of La Sl calculato shows f h h h h h h h h h h h h h h h h ( ) h h h h h h h h h ( ) I th followg w frstly rov aly ( ) Lt g( ) ( ) Th g( ) ( )( ) Sc ad w hav g( ) ad g ( ) s a ooto dcrasg fucto of Thrfor g ( ) g() 7

18 Lt t( ) Th t ( ) By w hav t( ) ad t( ) f I suary g ( ) Hc ( ) Thus t() f ( ) has o tr ots D : ( ) h P h ad th u of f ( ) ca b foud at th boudary D Sc f ( ) Thrfor f ( ) s ooto-crasg wth rsct to ay Furthror thus f ( ) has ts u at th boudary ot aly ( ) ( ) ( ) f ( ) D 4 Proof of la f( ) f ( ) If l ( ) th l ( ) or l ( ) Wh l ( ) obvously thr sts costat a such that l ( ) a Lt s s s s( ) ( s ) I th cas of l ( ) w rov Sc for : 8

19 Th aly O th othr had Hc (4) Thus accordg to (4) Thrfor thus Sc Th by l accordg to th Cauchy covrgc crtro thr sts costat a such that l a La s rovd Not for ay gv ( ) ( ) w always hav ad ( ) [ ) For al lt ( )! th 9

20 !! ( )!! ( )! ( )! ( )!! ( ) 4 Wh ( ) th 4 Proof of La 4 Proosto 4 If ad c s a ostv costat th l ( ) c O whr O c s dfd as: thr st ostv costats A ad B for ay B such that O c A c Proof Accordg to th dfto of costats A B ad for all B such that O c thr sts ostv A c O c A c Accordg to L Hostal s rul l ( ) A c A A l l c c c( ) Lt :

21 l A c( ) c l A c( ) c Alyg L Hostal s rul gts l ( ) A c l A c( ) c l ( ) c ( ) A c A( ) th l c c ( ) If 4 If sc alyg L Hostal s rul aga gts 4 ( ) A( )( ) ( ) c c ( ) A l l c c Slarly c l ( )( A ) Accordg to th suz thor l ( ) c O Th roosto s rovd Th followg rsult s wll ow to us: Proosto 4([]) c O ( ) Hr s Eulr s costat c s a costat gratr tha ad s a ostv ubr that ca b arbtrarly sall To rov La 4 th followg roosto s rurd Proosto 4 For ay ( ) such that

22 c O (4) whr c s a costat gratr tha ad s a ostv that ca b arbtrarly sall Proof Sc ( ) Accordg to roosto 4 c O Thus (4) s tru c O Proosto 44[] For ay Proof Fro [8 ] ( ) O ( ) ( ) ( ) whr ( ) b th ubr of rs ot cdg Hc O th othr had Thus ( ) O ( ) ( ) O ( )

23 ( ) O Th roosto s rovd for Proof of la 4 Accordg to (4) c O Accordg to Proosto 44 ( ) O l l ( ) l ( ) c O l O c O l O c O l O ( ) c O

24 c ( ) O O l c O Sc s a arbtrarly sall ubr accordg to Proosto 4 c ( ) O O l c O l ( ) O l O c c O l () holds ad th La s rovd Acowldgts Th author thas Dr Ru Zhu at Graclad Uvrsty Dr L Wag Prof Yuwu Yao at Hf Uvrsty Dr Hogb Fag at Gorgtow Uvrsty ad Dr Wj Lv at X a Jaotog-lvrool Uvrsty for hlful dscussos ad traslato Rfrcs [] G F B Ra Ubr d Azahl dr Przal utr r ggb Gröss Moatsbr Aad Brl (89): [] G Rob Grads valurs d la fucto so ds dvsurs t hyothès d Ra J Math Purs Al 6(984): 84- [] S Nazardoyav S Yaubovch Surabudat ubrs thr subsucs ad th Ra hyothss arxv:47v [athnt] 6 Fb [4] Y Zhu O A Ra Hyothss Rlatd Iualty Joural of Hf Uvrsty Vol6 No (6): -8 [] Y ZHU Yuyag Study o a Ra Hyothss rlatd Iualty htt://wwwarduc/rlasar/cott/-4 [6] P Solé ad Y Zhu A Asytotc Rob Iualty INTEGERS #A8 6 (6) [7] P Dusart Itégaltés lcts our ( ) ( ) ( ) t ls obrs rrs C R Math R Acad Sc Caada (999) -9 [8] L Schofld Sharr bouds for th Chbyshv fuctos ϑ() adψ() II Math Co [9] P Dusart Th th r s gratr tha (l+ll-) for Math Co 68 4

25 (999) 4-4 [] A A Kaauyóa Basc aalytc ubr thory HAYKA ( Russa) [] G Tbau Itroducto to aalytc ad robablty ubr thory Cabrdg Uvrsty Prss 998 [] J B Rossr L Schofld Aroat forulas for so fuctos of r ubrs Illos J Math