CORRECTIONS TO THE WU-SPRUNG POTENTIAL FOR THE RIEMANN ZEROS AND A NEW HAMILTONIAN WHOSE ENERGIES ARE THE PRIME NUMBERS Jos Javir Garcia Morta Graduat studt of Physics at th UPV/EHU (Uivrsity of Basqu coutry) I Solid Stat Physics Addrs: Practicats Ada y Grijalba 6 G P.O 644 489 Portugalt Vizcaya (Spai) Pho: () 34 685 77 6 53 E-mail: josgarc@yahoo.s MSC: 45-, 45G99, 6A33, 34A8, 7F6 ABSTRACT: W rviw th Wu-Sprug pottial addig a corrctio ivolvig a fractioal drivativ of Rima Zta fuctio, w study a global smiclassical aalysis i ordr to fit a Hamiltoia H=T+V fittig to th Rima zros ad aothr w Hamiltoia whos rgy lvls ar prcisly th prim umbrs, through ths papr w us th otatio log ( ) l( ) log( ) for th logarithm, also uls w spcify h( ) mas that w sum ovr ALL th imagiary parts of th otrivial zro o both th uppr ad lowr compl pla..wu-sprung POTENTIAL WITH OSCILLATING TERM From th poit of viw of Physics a asy proof to prov th clbrat Rima Hypothsis would b to fid a slf-adjoit oprator L so ˆ il, this would ma that all th zros of th fuctio would hav ral part ½, this is calld th Hilbrt-Polya [4 ] approach, i ordr to gt this Liar oprator Wu ad Sprug [ 9] cojcturd that if this oprator wr a Hamiltoia of th form H p V ( ), th th smooth part of this pottial would b V V V V ysm ( V ) V ( y) V V log V log V V V (.)
With V 9.743, th mthod by Wu ad Sprug is basd o smiclassical aalysis i physics or WKB [ 7] approach, th ida is that i th smiclassical approimatio dn / dv de E V ( E E) dp E p V ( ) (.) I this WKB approach,w rplac th sum ovr Ergis (imagiary part of th Rima zros) by a sum ovr th phas spac (p,), also w hav usd th kow ( u) proprtis of Dirac dlta fuctio ( f ( )) ad f '( u) f ( u) f ( ) ( a) f ( a) to itgrat ovr th variabl p (momtum). Equatio (.) is just a typ of Abl itgral quatio ( s [ ] ) V y( t) dt f ( ) t d f ( t) dt y( ) (.3) t If w approimat th smooth dsity of stats ( Numbr of imagiary parts of th E E E 7 Rima zros, who ar lss tha a giv quatity E ) by N( E) log 8 dn, th f ( E) solvig th last itgral o (.3) w obtai th smooth part of th de Wu-Sprug pottial dfid implicitly i (.) o Fractioal drivativ corrctio to Wu-Sprug pottial: Although Wu ad Sprug cosidrd oly th smooth part of th dsity of zros N(E) thr is a tra trm proportioal to arg is if w isrt this, isid th Abl itgral quatio (.3)., th th oscillatig trm cotributio to th Wu-Sprug pottial ca b dfid i trms of th Rima-Liouvill diffritgral [ ] d ' ysm ( V ) B iv dv B R (.4) This prssio (.4) ar th commo dfiitios for a fractioal drivativ/itgral q m d f ( ) d dtf ( t) ( q m) ( t) q m qm (drivativ) q d f ( ) q q dtf ( t)( t) ( q) (.5) (itgral) Eprssios (.5) ar also usful bcaus thy satisfy a smigroup compositio proprty for th fractioal drivativs/itgrals,amly D a. D b D ab this allows to writ
D. D D or D. D D, howvr th additio of pottial (.4) maks th implicit form of th pottial isid (.) hardr to solv.. A TRACE AND ANOTHER INTEGRAL EQUATION FOR THE RIEMANN HYPOTHESIS: I a prvious papr [ 6 ] w obtaid a Trac formula for Uitary oprator dfid for u >. U iuhˆ u u / u / u / d( ) iue ˆ ˆ iuh cos( ) 3u u du Tr U ue u > (.) Ad for u <.This trac (.) follows imdiatly from diffrtiatio with rspct to u ad sttig isid th plicit formula for th Chbyshv fuctio (.) () s log( ci ( s) ) ( ) ds () i ci ( s) s Usig th smiclassical approach isid (.) for u > ad th Eulr formula for th compl potial,w ca gt th itgral quatio with Awkb R a costat to b fid by mpirical or umrical obsrvatios u u / u / u / d( ) u A cos ( ) u wkb uv du u > (.3) 4 Hr, th drivativ of th Chbyshv fuctio ca b dscribd as a ifiit sum ovr d( ) prims p ad prim powrs ( p ), this itgral quatio log( ) p ca b mak smoothr by dfiig a pair of fuctios g() ad h() with th followig proprtis [ 6 ] Both g()=g(-) ad h()=h(-) ar v fuctios g( ) lim ists ad it is fiit Th fuctios h() ad g() ar rlatd by a Fourir Cosi trasorm i h( ) g( ) Th fuctio h() tds to as fastr tha iu g( ) / ists as a fuctio /, so th itgral 3
iu Applyig th Fourir itgral trasform h( u) g( ) fuctios g ad h (.3) bcoms for our tst i i log idhr idhr ( ) g(log ) ir ir u / du ug u ur ( )cos( ) A ( ) ( ) u wkb i h V r i h V r (.4) d With ( ) ( ) ( ) big th Magoldt fuctio ad D th halfdrivativ, this fractioal drivativ appars from th proprty of th Fourir trasform u u i d i f ( ) i f ( ) d. (.4) is ow th most gral o-liar itgral quatio that ca b obtaid usig th WKB approach for th pottial V() ad that is compatibl with th Tracial coditio (.) may authors forgt this fact ad do ot us (.) to costruct a mor gral pottial tha Wu-Sprug o (i fact this Wu-sprug pottial is icomplt sic it dos ot tak ito accout th oscillatig part of th Rima zros, which is vry importat i th sarch of a oprator ralizatio for th Rima Hypothsis), i ordr to covrt (.4) ito a liar quatio it is ough dfi th chag of variabl V ( y). A similar formla to (.4) ca b usd to comput sums ovr th imagiary part of th zros (trac formula) i i h r h r ( ) g(log ) ir ir u / cos( ru) g( u) ( ) ( ) u du A wkb h r h r (.5) o Trac formula for Rima Zros ad Wu-Sprug pottial: A mor gral formula tha (.5) valid oly i th distributioal ss prssig th sum ovr Rima zros without ivolvig th prim umbrs ' ' 4 (4 ) Awkb ( s ) Awkb ( s ) is is 4s s (.6) Itgratio from to E i th variabl s will giv th oscillatig trm arg is, th factor 4s is du to th pol of th Zta fuctio at s= ad th last sum coms from th o-trivial zros 4
Proof: if w st h( ) ( ) isid (.5) ad us th aalytic cotiuatio [ 5], [6] of th divrgt sris (is a rgularizatio mor tha a sum dfiitio) ( )cos( s log ) ' ' is is as as (.7) Togthr with th rsult,ad th zta rgularizatio ( s ) a for Dirichlt sris [ 6] ad [5] so w ca rgulariz th divrgt sris by ( ) is log ' is, th w ca giv a formal proof to (.6). A dirct applicatio of this formula ca b to valuat sums of th form h( ) ad to obtai a ( ) cos( ) s s a pasio for th smooth part of Wu-Sprug pottial CSM ( V ) i i i s i / i s i / ( s ia ) ( s ia ) With ( ) lim ( ) C applyig th half-drivativ oprator (.8) 3/ 3/ 3/ 3/ ad a this dfiitio is obtaid by just d ad th pasio isid th quatio (.6) 3. A POTENTIAL FUNCTION FOR THE PRIMES A fial qustio i our papr would b, could w obtai via smiclassical approimatios a Hamiltoia H p Q( ), so w could obtai th ivrs of pottial Q() usig th WKB approimatios for th dsity of stats or th compl potial sum of dsity of stats ( E E) dp E p Q( ) ue u( p Q( )) dp (3.) So E p is th -th prim umbr ad u > is a Ral umbr, for th scod cas w will d fudamtal proprtis of Laplac trasform [ ] t L f ( u) g( t u) du L f ( t) Lg( t) f g L f t ( ) st dtf ( t) (3.) Th first formula for th pottial Q() is imdiat, th dsity of prims is giv by d th drivativ of th Prim Numbr coutig fuctio [ ] ( p) suig p 5
(3.) (.) ad th solutio for th Abl itgral quatio (.3) togthr with th dfitio of th fractioal drivativ ad itgral w fid d dt d ( ) d ( ) ( ) p. p t Q A D D A (3.3) Th costat Ap ca b dtrmid by mpirical or umrical obsrvatios ad must b idpdt of th choic of pottial Q() ad its ivrs. Th scod mthod basd o th Laplac dirct ad ivrs trasform is th followig, usig agai th WKB approach to rplac th sum by a itgral ue u( p Q( )) uq( ) u dq (3.4) dp u u O th othr had thr is a act prssio for th dsity of prims/rgis dfid by ue u s ( ), so quatig (3.4) ad this prssio ad usig th uicity proprty for th Laplac trasform (if two fuctios hav th sam Laplac trasform, th ths fuctios ar qual f=g ) th (3.4) bcoms ue u u dq u (3.5) s ( ) s s Q ( ) Usig th covolutio ad th uicity proprtis of th Laplac trasform th w rach d ( ) to th sam coclussio as i (3.3) Q ( ) Ap, th problm hr is that still w do ot kow th valu of th Prim coutig fuctio ( ), o of th most commo approachs to this fuctio ar giv by th Prim Numbr thorm, or th Ramauja approimatio [ 3], th our pottial ow bcoms d Li( ) ( ) p or Q A / ( ) d lg (3.6) Q ( ) Ap dt With Li( ) big th logarithmic itgral, ad ( ) th Möbius fuctio. For l t a bttr dfiitio ad a good itroductio to Möbius ad othr Numr thortic fuctios Apostol s book is th bst rfrc [] CONCLUSIONS AND FINAL REMARKS: W hav ivstigatd th Wu-Sprug pottial ad its gralizatios (.4), w hav show rlatd to this problm, how if Trac formula for th Hamiltoia H that rproducs th imagiary part of th o-trivial zros is tru ( s (.--5) ) o ca gt 6
a smiclassical WKB approach to obtai th ivrs of th pottial V(), this improvs th approimat rsult by Wu ad Sprug, sic thy did ot tak car of this cssary coditio (.) i ordr to obtai a Hamiltoia oprator ralizatio of Rima Hypothsis, also i a similar mar w maagd to us fractioal Calculus to obtai th oscillatory part of th Wu-sprug pottial ad applid th sma i (3.6) to gt th ivrs of th pottial isid a Hamiltoia whos rgy lvls ar prcisly th prim umbrs E p APPENDIX A: A CONSTRUCTION OF THE INVERSE OF THE POTENTIAL V() FROM THE SUM iue W hav dvisd a Trac formula (.3) usig th smiclassical WKB plus th rlatio of th drivativ of th Chbyshv fuctio, th ida is ca w solv (.3) to gt a ral valud quatity?, first of all w will ot us our Trac formula, but a similar trac obtaid by Rima ad Wyl [] ( ) ' ir h( ) h( i / ) g() log g(log ) h( r) dr 4 (A.) Takig th ivrs fuctio for th pottial ( V ) th sum ovr th rgis plus itgratio by parts, ad usig th approimatio for iue iuv ( ) iuv iuv u dv iu dv( V ) dv (A.) So takig th Fourir ivrs trasform, ad takig ito accout that th trac is oly ozrop for u >, i ordr to gt (V) w should valuat th itgral iu iuv / 4 du (A.3) u I ordr to gt rid off th sum ovr th No-trivial zros, w could us (A.) with g( ) to gt th most gral pottial compatibl with (.3) ad (A.) A ( V ) BCos( cv / 4) D V i ( )cos( V log / 4) ir E Fp. v dr log 4 V r V r (A.4) Hr, A,B,C,D,E ad F ar REAL umbrs that dscrib th ivrs of th pottial V(), sic th Wu-Aprug pottial ad ours hav b obtaid by usig WKB mthods comparig th two pottials (at last th smooth parts) w could gt th valu of ths costats, th sum ivolvig ( ) is divrgt, ad ds to b rgularizd to 7
tract som maigful fiit iformatio, from th Dirichlt gratig fuctio, ad itgratig ovr s w gt via Zta rgulariztio ( )cos( V log / 4) d '( iv ) d '( iv ) i i log dv ( iv ) dv ( iv ) (A.5) (A.5) is gt by a simpl half-itgratio with rspct to s isd th zta rgularizd idtity ( ) is is. Aothr formula rquivalt to (A.4) ca b giv, if w cosidr i distributioal ss d H ( ) H ( ) ad f ( a) f ( a) H ( a), th th sum (up to a costat) could b viwd as th half-drivativ of th Dsity of d Nsmooth( ) d Nosc( ) zros isid th Critical li with N(T) giv by Arg it Arg i. log 4 8 8 (A.6) T T T 7 4 i T T T Th usig th proprty D. D D. D, ( V ) ca b rwritt (at first approimatio igorig possibl trms proportioal to O(/V) ) as A '( ) 7 R i d iv d V V V V osc log dv ( iv ) dv 6 3 3 (A.7) Th, th Wu-Sprug pottial is quivalt to our Trac formula (.) ad (.3), howvr Wu ad Sprug avoidd th oscillatig trm comig from th half-itgral of ' ( ) th prssio (i th ss of Zta rgularizatio) is, this trm is is d obtaid from th drivativ of th Chbyshv stp fuctio cotrollig how prim ad prim powrs ar distributd, Also th Trac (.3) for Tr ˆ iuh gral sic it ca b usd to giv maig to ay sum h( ) ot oly for dn ( ) as w hav provd i formula (.6) is mor 8
Rfrcs [] Abramowitz, M. ad Stgu, I. A. (Eds.). "Rima Zta Fuctio ad Othr Sums of Rciprocal Powrs." 3. i Hadbook of Mathmatical Fuctios. Nw York: Dovr, pp. 87-88, 97. [] Apostol Tom Itroductio to Aalytic Numbr thory ED: Sprigur-Vrlag, (976) [3] Brdt B. Ramauja's Notbooks: Part IV Sprigr; ditio (993) ISBN-: 3879496 [4] Cory, J. B. "Th Rima Hypothsis." Not. Amr. Math. Soc. 5, 34-353, 3. availabl at http://www.ams.org/otics/33/fa-cory-wb.pdf. [5] Elizald E. ; Zta-fuctio rgularizatio is wll-dfid, Joural of Physics A 7 (994), L99-34. [6] Garcia J.J Zta Rgularizatio applid torima hipótsis ad th calculatios of divrg itgralst Gral Scic Joural, -prit: http://wbabi.t/scic/morta3.pdf [7] Griffiths, David J. (4). Itroductio to Quatum Mchaics Prtic Hall. ISBN -3-89-7. [8] Hardy G.H Divrgt sris, Oford, Clardo Prss (949) [9] Hua Wu ad D. W. L. Sprug Rima zros ad a fractal pottial Phys. Rv. E 48, 595 598 (993) [] Kilbas, A. A.; Srivastava, H. M.; ad Trujiilo, J. J. Thory ad Applicatios of Fractioal Diffrtial Equatios. Amstrdam, Nthrlads: Elsvir, 6. [] Polyai D ad Mazhirov A.V Hadbook of Itgral Equatios. CRC Prss, Boca Rato, (998) ISBN -8493-876-4. 9