Jiang Number Theory (JNT)

Transcription

1 Jiag Number Theory (JNT) Lauret Schadeck Abtract : Jiag Chu-Xua i a Chiee mathematicia who claim to have develoed ew umber theoretic tool coitig motly i the Jiag fuctio J( #) where # , deote the rimorial fuctio to olve fudametal roblem i Number Theory uch a the Goldbach Cojecture, the Twi Prime Cojecture, the k-tule Cojecture, et al. The fudametal motivatio of Jiag to develo a umber theory differet from the oe we are familiar with (we, umber theorit) come from hi recet claim (997) that the Riema Hyothei (RH) which lie at the foudatio of all rime umber theorie, i fale, that all calculatio doe to imrove it are fale, ad that the etire eculative theory doe through it (ee Coe, Bombieri, Zagier et al.) are obviouly fale. Our goal i thi aer will be to review Jiag achievemet from hi diroof of RH to hi etablihmet of the ew umber theory. ) Jiag 997 diroof of RH : The fuctio ( ) of the great mathematicia Riema i defied over the comlex umber by ( ) ad i claimed by Riema himelf [] to atify the followig fuctioal equatio ( ) ( ) t where ( ) e t dt deote the Euler Gamma fuctio. 0 The imlet form of Riema fuctioal equatio i ofte deoted by ( ) ( ) I their ideedet 896 roof of the rime umber theorem, Hadamard ad De La Vallée Poui tated baically that ( ) 0, ad it i baically evidet that ( ) ha o zero for =0.

2 Riema it hi eoch-makig 859 aer [] tated that all the otrivial root of hi fuctio lie i the critical tri [0,] ad made the followig: Riema Hyothei (RH) : ( ) 0 it where oe igore the trivial zero,-4, et al. RH ha become throughout the at decade the mot fudametal roblem i Aalytic Number Theory ad rime umber theory. I [] Jiag defied a ew fuctio ( ) that i the dual of Riema zeta-fuctio: ( ) ( ) where ( ) deote the Liouville fuctio ad the duality i exhibited by : ( ) ( ) ( ) ad the directly it it it Jiag motly roved i [] that the beta-fuctio i ot ifiite for real art equal to ½ ad the, followig the fudametal remark of Hadamard ad De la Vallée Poui, Riema zeta-fuctio caot have otrivial zero i the aid critical lie Re( ). Jiag tart with a amaizig exreio for both ( ) ad ( ) which he coi their exoetial formula. Thee formula, to the bet kowledge of the reet author, are ot foud i other RH book (i oe of them) ad are ufficiet to Jiag to follow hi etire diroof. Formula : ( ) co( t log ) e i( tlog ) ta co( tlog ) Formula :

3 ( ) co( t log ) e i( tlog ) ta co( tlog ) Jiag himelf doe ot give ay roof of thee beautiful idetitie ad the author himelf tried by may attemt to cotruct a roof of thee idetitie. Thee formula are atural if oe coider that Etry : ( ) Proof : The roof i obviou ( ) For all R, ( ) ( ) The ig of the fuctio i the give, whe the comlex are itroduced ito the formula ad thi jutifie the reece of the trigoometric fuctio. (A more comlete roof wo t be hort eough i thi brief moograh). However a imilar roof ha to be doe i reect to the Jiag beta fuctio to obtai the ecod idetity. (A exactly roof wo t be hort eough i thi brief moograh ad relatively uele give the firt). Further coideratio about Jiag roof are foud i [],[3],[4]. I [4] it i above all ee that the Frech mathematicia Atoie Bala [5] foud a reult about RH that i the exact ooite of thoe obtaied by Jiag. Therefore it eem to u that the fality of Jiag 997 tatemet i bet howed by howig that Bala i all right. But however Bala i ot a umber theorit, while Jiag i. Therefore the doubt comig from umber theorit have to be aiged with Bala work rather tha to Jiag. Jiag aer have bee wet worldwide to mathematicia of the tature of Alai Coe, Do Zagier et al. but rather tha coiderig Jiag cotributio i deth they imly igored it without readig a umber or a letter i Jiag calculu. Oe may alo imagie how ditateful it hould be to mathematicia to how them that the greatet mathematical cojecture ever, that eem rovide the umber theoretical foudatio 3

4 of mathematic. With the time ome ultimate beauty ha bee aiged with the true of RH. To differ from thi oit of view, Jiag quoted further Iwaiec : Aalytic umber theory i fortuate to have oe of the mot famou uolved roblem, the Riema hyothei. Not o fortuately, thi ut u i a defeive oitio, becaue outider who are ufamiliar with the deth of the roblem, i their uruit for the ultimate truth, ted to judge our abilitie rather harhly.i cocludig thitalk I wih to emhaize my advocacy for aalytic umber theory by ayig agai that the theory flourihe with or without the Riema hyothei. Actually, may brilliat idea have evolved while oe wa tryig to avoid the Riema hyothei, ad reult were foud which caot be derived from the Riema hyothei. So, do ot cry, there i healthy life without the Riema hyothei. I ca imagie a clever ero who rove the Riema hyothei, oly to be diaoited ot to fid ew imortat alicatio. Well, a award of oe millio dollar hould dry the tear ; o alicatio are required. [6] I order to follow the maitream rime cocetio, Jiag argue that: The ditributio of rime umber doe ot ivolve Quatum chao, radome et al. There i order i the equece of rime umber. [7] Thi view ha bee received with ethuiam by the great hilooher Stei Johae i [8]. Moreover Jiag work eem to move alog with the develomet of Hadroic Mechaic ioeered by Ruggero Maria Satilli, a ee i [3] ad articularly i Ioumber Theory. (If Jiag work i right the it i the foudatio of Ioumber Theory. I articular Satilli himelf claimed i [3] : I would like to exre my utmot areciatio to Profeor Chu-Xua Jiag for havig udertood the igificace of the ew io-, geo-, hyer-umber ad their iodual I idetified for a reolutio of the above roblem. The igificace of the ew umber had ecaed other cholar i umber theory i the at two decade ice their origial formulatio. I would like alo to cogratulate Profeor Jiag for the imly moumetal work he ha doe i thi moograh, work that, to my bet kowledge, ha o rior occurrece i the hitory of umber theory i regard to joit ovelty, dimeio, diverificatio, articulatio ad imlicatio. I have o doubt that Profeor Jiag' moograh create a ew era i umber theory which ecomae ad iclude a articular cae all recedig work i the field. More recetly Idia umber theorit Tribikram Pati claimed to have diroved RH i [9] By howig that RH i equivalet to : l a e a He maifeted furthermore iteret i Jiag work ad i readig [] ad [4] i [0]. Reultig correodece with Schadeck ad Pati, Jiag get i 008 the idea to write a fudametal aer [] : Riema Paer(859) I Fale, which i ot yet ublihed ad rejected i block by the umber theorit belogig to the maitream. 4

5 I thi mot atoihig aer (the mot imreive he ha ever writte) Jiag claim that the fuctioal equatio tated by Riema i reected by a fuctio ( ) that i ot the ame that ( ). I [] oe exlicitely foud that ( ) x x x e dx x dx 0 0 x Where ( x) : e i the Jacobi theta fuctio whoe fuctioal equatio i : ( ) ( ) x x x where the variable ha to be take oitive. From it, which i a mot BASIC well-kow by all umber theorit ad eve all real mathematicia Jiag claim that he obtai; ( x) ( ) ( x x ) ( ) dx ( ) ( ) ( ) The the roertie of thee ewly formed fuctio are :. ( ) ha o zero for ;. The oly ole of ( ) i at ; it ha reidue ad i imle; 3. ( ) ha trivial zero at, 4,... but ( ) ha o zero; 4. The otrivial zero lie iide the regio 0 ad are ymmetric about both the vertical lie ad from them Jiag claim that RH i exreed oly i term of the ew fuctio which we call here the eudozeta-fuctio ad the refer to hi diroof i [] ad ay : ( ) ad ( ) are the two differet fuctio. It i fale that ( ) relace ( ) He fiihe hi coideratio of RH ad Riema aer by givig brief coure about the ew umber theory he ugget to the ucomig geeratio of mathematicia, which maifet :. A good coectio ad a great comatibility (erha the greatet) with Satilli iomathematic which are Lie-admiible mathematic (ee [] for more iformatio 5

6 becaue it hould take hudred of age to itroduce it, that we caot doig here for evidet eed of brevity). The Prime ditributio maifet order rather tha radome 3. A great are of alicatio icludig ISOCRYPTOGRAPHY which may cotitute the greatet crytograhic ytem i the World (ee [3] ad iect the imreive lat chater) 4. Dee metamathematical ad hiloohical coequece a brilliatly ee i [8] by Johae, with coectio to Rowlad theory of Uiveral Patter ad the Fiboacci equece. 5. ad o o (the lit caot here be exhautive ad it i recommeded to the itereted reader to iect [3] for ome more detail. Jut like rime umber theory ad aalytic umber theory are roughly the tudy of Riema zeta fuctio, oe ha clearly to ay that Jiag Number Theory (the ew JNT) i EXCLUSIVELY the tudy of the cla of fuctio J( #) with reect to the idex iteger. The oe ha to tart with Satilli baic rule of iomathematic foud i [3], [] ad a larger ad larger literature, where we tart by recallig the Satilli iouit with related Satilli ioumber I, I 0 T ad Satilli ioroduct A I A A B AT B ad fially the fudametal idetity defie for the Satilli iouit through the Satilli ioroduct : I A T T A A I ATT A etc. More detail are foud i Jiag [3] ad Satilli []. Defiitio. (Jiag [3]): d iodivide ad we write d whe c d for ome c Similarly, d doe ot iodivide ad we write d whe c d for ome c. Oe ote that iodiviibility i imilar to covetioal diviibility with reect to Satilli ioumber ad Satilli ioroduct. From iodiviibility Jiag defie (ad it i rather itictive to defie) iocogruece by the followig: 6

7 Defiitio. (Jiag [3]): Give ioiteger a, b, m with m 0. We ay that â i iocogruet to b module m ad we write whe d. a b (mod m ) The iocogruece, jut a the iodiviibility, atifie all axiom of the covetioal cogruece (re. the covetioal diviibility). Here the term covetioal would refer to what i commoly coied covetioal mathematic, amely uitary re-satilli mathematic (the author rooed the term uimathematic becaue the uit i alway equal to : a ay Rowlad quoted i [8] the i already loaded.). The i [3] Jiag ivetigate a large umber of ioequatio coitig ito iocogruece ad defie the Jiag fuctio J ( ) through the followig theorem: Theorem.3 (Jiag [3]) : The equatio where P IP, P rime i x A (mod ) i ha exactly J ( ) ( ) olutio if d ad ha exactly J( ) whe d. The Defiitio A : (Fudametal defiitio i Jiag Number Theory) ( ) ( ) J( ) : Jiag doe ot give roof of Theorem.3 to the bet kowledge of the author becaue theorem.3 eem to be obtaied through the iotoic liftig of the correodig theorem i Number Theory ito uimathematic. Moreover the Jiag fuctio J( ) i ofte obtaied at the very begiig i [3] to cout the umber of olutio of uch baic ioequatio that ivolve iocogruece ad iorime ad iodiviibility. 7

8 A other mot geeral examle a i theorem.3, that ivolve multivalued fuctio of Satilli ioiteger i: Theorem.3 (Jiag [3]) : The equatio ha W olutio ad the W J O() f x,..., x A(mod ). Hudred of uch theorem which are baically obtaied by liftig the uitary oe ito uimathematic are foud i [3]. The fuctio J( ) i exteded to J( #) by the defiitio B till foud i [3] by Defiitio B : (Exteded fudametal defiitio i Jiag Number Theory) ( ) ( ) ( ) J ( #) : ( ) ( ) where N deote a Satilli ioiteger. 3 i / N The mot baic roerty of Jiag fuctio i that J( #) 0. Jiag claim that he get the idea to defie hi fuctio i 997 by makig ue the baic defiitio of Euler totiet fuctio udefied exlicitly but mot ueful i Arithmetic ad uable through a lit of imle roertie uch a:. ( ) whe i a rime. A A. ( ), a rime, that look like the roduct exreio of Riema zeta. 3. ( ) i ALWAYS EVEN for all ( ) / Uually Euler ( ) fuctio i take to cout the umber of iteger rime to a give iteger. Oe ha firtly to ote the reemblace betwee ( ) through it 4 th roerty ad the Defiitio B of Jiag fuctio. Moreover, give the formal defiitio of what the umber theorit ow call the twi rime cotat 8

9 ( ) 3 oe hould thik that it i deely coected to Jiag fuctio ad alo related to Riema zeta-fuctio, a the author could have how it i a aer that would erha be ublihed after the reet year. We would claim here that Defiitio C : (o-formal give the mathematical kowledge of our time) Euler ( ) fuctio i the arithmetical atter of Jiag J ( #) fuctio throughout Satilli Iomathematic, that i, Jiag J( #) fuctio i aturally geerated ito Iomathematic through Euler ( ) fuctio. Jiag ofte ay i [3] that Jiag J( #) fuctio i a geeralizatio i fact of Euler ( ) fuctio, that it cout the umber of olutio of baic ioequatio from which it defiitio follow, jut a Euler ( ) fuctio cout the umber of iteger that are rime to a give iteger ad le tha itelf. Jiag himelf ay : Let #=30, Euler fuctio (30)=i(-)=8,We have (30,j)= [ where (a,b) deote the gretet commo divior gcd ],where j=,7,,3,7,9,3,9. We have 8 equatio,(j)=30i+j, j=,7,,3,7,9,3,9.every ha ifiitely may rime olutio We tudy twi rime =+,J_(30)=i(-)=3.We have 3 twi rime ubequatio:=()+=(3),=(7)+=(9),=(9)+=().every ha ifiitely may twi rime olutio. We tudy 3=++, 8^=64.We have 64 equatio,j_3(30)=i[(-)^-x()]=i[^- 3+3]=39.We have 39 ubequatio:3=()+()+ write a 3=++,++,+7=,7++,+9+,9++,++,+7+,7++,+9+, 9++,+9+,9++,3+7+,7+3+,3+3+,3+3+,3+9+,9+3+,7+9 +,9+7+,7+3+,3+7+,7+9+,9+7+,9+3+,3+9+,9+9+,9+9+,3+ 3+,3+9+,9+3+,9+9+.Every ha ifiitely may rime olutio. We tudy 4=3+++.8^3=5,we have 5 equatio.j_4(30)=i[(-)^3-x()]=i[^3-4^+6-4}=55.we have 55 ubequatio of 4=3+++, 4=+7+7+,...,every ha ifiitely may rime olutio. [3] The fuctio J( #) of Jiag i how i [3] to exhibit a lot of amaizig fuctioal roertie which are exhautively: m m ( )( m ). J 9

10 . J() J( #) J() 3. J ( ab) J ( a) J ( b),( a, b) 4. m ( )( m ) J J ( ) 5. J ( ) J, k k d J( a) J( b) J( ab),( a, b) d J ( d) ( ) ( ) ( ) ( ) ( ) ( ) 8. J (, k ) J (, k ) m m k m k ( )( ) J( ) k m k m J ( ) ( ) ( ) ( ) ( ) ( ) ( ) J( ) J( ). a / b J ( a) / J ( b), However thee roertie eem to et u Jiag fuctio a the mot amaizig fuctio or aalytical toy ever built. Ufortuately o demotratio of the magic roertie are kow. Perha Jiag himelf will be able to give u them, becaue they are bet eeded to make covetioal umber theorit itereted about hi cotributio ad to imrove ome of hi tatemet. From the mot geeral J( #) Jiag defie a erie of articular fuctio uch a : Defiitio A. : J( ) = J ( #) = 0

11 Defiitio B. : J ( ) ( ) 3 i / J ( ) 0 J ( ) ( 4) 5 i J ( ) 6 ( 5) i J ( ) ( 6) 7 i J ( ) ( 7) i Note the mot reemblace with the twi rime cotat through it imlet exreio. Doze of differet exreio are foud i Jiag [3]. Defiitio B. : b J3( ) 3 3 ( ) 3 i N J3( ) ( O()) 3 3 / log i N N J ( ) ( ) ( ) 3 i / Defiitio B.3 : J ( ) 4 3 i 4 3 ( ) ( )

12 Defiitio B.4 : 5 3 i 4 3 J ( ) 3 Defiitio B.5 : 6 3 i J ( ) 3 Note that the defiitio of the twi rime cotat clearly aear i the right ide of the ecod exreio of the defiitio B.. A ifiitude of uch fuctio ca be built to raie umber theoretic roblem. The mot ueful are by far thoe reeted i def. B./B./B.3. Uig them Jiag claim to have roved the Goldbach Cojecture ad the Twi Prime Cojecture. Here oe reroduce hi claimed roof from [3]: The ecod oe i take biary ad alway uch imle

13 At the begiig of 008, Jiag cotacted the great britih umber theorit Marti Huxley, who i to the bet kowledge the firt i the to academic ititute to become itereted i Jiag work. Huxley the told Jiag [7] : To ay that omeoe ele' work i actually wrog, you have to be extremely certai that your ow calculatio are correct, ad that you have actually read ad ad udertood their work. ( ) If you have got a ew method, the Jiag Fuctio, which olve the famou roblem, the brig it ito the oe ad write a full exlaatio ad ed it to a Mathematic joural, Aal of Math or the Proceedig of the Lodo Math. Soc. or the Duke Math. Joural or uchlike. If it work, the mot eole will be hay to forget about the Riema Hyothei comletely ad ue your method itead. If you do't exlai your method, the everybody ele i etitled to be a rude about you a you are about them, or what i eve wore, to igore you comletely., which i what I myelf am likely to do, a I am et more aer tha I have time to tudy ayway. Objectio to JNT :. The Riema Hyothei might be true : Bala objectio [5] : Had we 0 ad 0 0, the by recallig the o-called rime zeta fuctio: oe would have ' fuctio to the rime zeta fuctio: ad the 0 ( ) : becaue of the well-kow idetity bridgig the zeta ( ) ( ) l 0 ' 0 ' ' ad 3

14 0 ' 0 ' ' The firt equality eem to how that ' by a recurrece (i Bala ow word) uoitio that the zero of the zeta fuctio lie o the critical lie whe Im Im. The it i foud i [5] that 0 Thi i aburd. Bala thu claim RH IS TRUE. [5] 0 ( ) 0 ( ) ( ) Fially, accordig to Bala i [5] he i able to et a geeralizatio of the fuctioal equatio for the zeta fuctio: ( ) k 0 a co k ka. The Jiag fuctio i robably ot the mot ueful ad ituitive geeralizatio of Euler totiet fuctio. Oe ha o the other ide Jorda totiet fuctio (referece are foud i [8]) : where iteretigly by defiig J k : k / k J k ( ) : J k k oe clearly ee that J ( ) ( k) a if o k 4

15 Rely to the objectio :. Had RH be true, JNT would have ot bee diroved, becaue Jiag claim that RH i ot true i ot take a a foudatio of JNT but oly a a motivatio ad a urget reao to ee further. Ad moreover Bala roof i much le ituitive that Jiag. The calculatio of the otrivial zero of the zeta fuctio, beig imroved, would ot cotradict JNT fudametally.. However Jorda fuctio i iteretigly ituitively idetified to the zeta fuctio for greater ad greater value, ad a bridgig betwee the zeta fuctio ad the Jiag fuctio ha to be how i the future. But thi defiitio, eem to u to be the exreio of Euler totiet fuctio the earet to the Jiag fuctio. The Jiag fuctio remai the mot ueful tool to rove the Twi Prime Cojecture ad the k- tule cojecture, jut a Jiag give a roof of the Prime Number Theorem uig Euler totiet fuctio i [3]: Followig it we are able to et the Jorda totiet fuctio a a quet to the Jiag fuctio which ha direct alicatio i imrovig the k-tule Cojecture. I the hoe that Jiag work, which, eve if it i fale, cotitute a formidable attemt to raie the larget umber of deeet umber theoretic roblem, will receive a echo ito the circle of mathematicia, the author would coclude hi coure o Jiag work by a lit of challege for the future. 5

16 ) Challege for the future:. To exted Jiag foudatio of Satilli ioumber theory to geomathematic ad hyermathematic []. To etablih the umber theoretical foudatio of iformatic through comutability theory which eem imlicitly coected to the iotoic formalim foud i [] 3. To dicover the exact ad comlete order behid the ditributio of rime, Satilli iorime, Satilli georime, Satilli hyerrime ad their reective iodual. 4. Hyerumber = equece of ordiary umber equece of bit = rogram what about ifiite equece? What about Number theoretic aect of the buildig of comuter rogram? What i ifiity ad what are thought that are comreed ito rogram? What i the lik betwee rogrammig ad LIFE (ice, a ee i [3] ad [] hyermathematic ha bee built to rereet coitetly biological ytem) 5. To exted iformatic to Hadroic Mathematic to which the bet itroductio eem to the author to be foud i Satilli latet work a i 008 [] with related oftware, rogram ad rogrammig. 6. To exted the formal defiitio of atter ditatefully evoked i thi aer to all mathematical cocet ad/or tructure. To cocetrate all ucomig idea ueful to olve thee roblem the author i tryig to geeralize Iformatio Theory ito Hadroic Iformatio Theory (HIT) with a aroriate hyermathematical formalim ad umber theoretic foudatio. The author would defie for itace HIT a the ematic embeddig of Hyerumber Theory thu eedig the rigorou etablihmet a recalled i the Challege above of the ew Satilli Hyerumber Theory (SHT) jut a the Jiag Satilli ioumber theory i which Jiag Number Theory (the great JNT) ha it kigdom. A erie of aer i to aear about HIT, followig [5] ad [6], i which iformatio will be udertood a rogram which are themelve udertood a equece of umber which themelve aear to be Satilli hyerumber. But the etablihmet of Satilli Hyerumber Theory hould take a log time. The tartig defiitio from SHT to HIT will be the defiitio of PATTERN. We have ee i [] that the atter for iomathematic i the Satilli iouit ad the atter for Jiag fuctio i Euler totiet fuctio. ( ) ( ) ( ) IS JIANG FUNCTION J( #) : 3 i / ( ) ( ) N UNIVERSAL PATTERN FOR SHT AND HIT? THE 6

17 Further imrovemet of JNT are eeded to et thi rigorouly. To the bet kowledge of the author, the greatet te doe for itace to defie atter of mathematical theorie ad the Uivere a a muic of article or a ytem are foud i Johae [8] ad Rowlad [4]. Ackowledgemet : I caot fid word to exre my gratitude ad eteem toward Profeor Jiag Chu-Xua, Che I Wa, Stei Johae, Tribikram Pati ad everal other for their helful coveratio ad feedback. Lauret Schadeck. 7

18 Referece : [] Chu-Xua Jiag, Diroof of Riema Hyothei, Algebra, Grou ad Geometrie, Vol., 004. [] B. Riema, Über die Azahl der Primzahle uter eier gegebee Groe, Mo. Not. Berli Akad (859). [3] Chu-Xua Jiag, Foudatio of Satilli ioumber theory with alicatio to ew crytogram, Fermat theorem ad Goldbach Cojecture, Iteratioal academic re, 00. [4] Lauret Schadeck, Remarque ur quelque tetative de démotratio Origiale de l Hyothèe de Riema et ur la oibilité De le rologer ver ue théorie de ombre remier coitate, uublihed, 007. [5] A. Bala, Formule our le ombre remier, Arxiv 05003, /05/0. [6]Che I Wa, Chu-Xua Jiag, rivate commuicatio, 4//07. [7] Chu-Xua Jiag, rivate commuicatio. [8] Stei Johae, Iitiatio to Hadroic Philoohy, the hiloohy uderlyig Hadroic Mechaic, Lecture at 8 th workho o hadroic mechaic, Uiverity of Karltad, Swede, /07/05. [9] Tribikram Pati, The Riema hyothei, arxiv: math/ v, 9/03/07. [0] Tribikram Pati, rivate commuicatio, 7//07. [] Chu-Xua Jiag, Riema Paer(859) I Fale, oo ublihed, Jauary 008. [] Ruggero Maria Satilli, HADRONIC MATHEMATICS, MECHANICS AND CHEMISTRY Volume I: Io-, Geo-, Hyer-Formulatio for Matter ad Their Iodual for Atimatter, July 007. [3] Chu-Xua Jiag, Private Commuicatio, 4/0/07. [4] Peter Rowlad, B. Diaz, A uiveral alhabet ad rewrite ytem, 003. htt://arxiv.org/ft/c/aer/009/00906.df [5] Stei Johae, rivate commuicatio, december 007. [6] Lauret Schadeck, Itroductio to HIT, to be writte. [7] Jiag, Marti Huxley, rivate commuicatio, /0/08. [8] htt://laetmath.org/ecycloedia/jordatotietfuctio.html 8