Proof of Analytic Extension Theorem for Zeta Function Using Abel Transformation and Euler Product

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1 Joual of athematic ad Statitic 6 (3): , 200 ISSN Sciece Publicatio Poof of Aalytic Eteio Theoem fo Zeta Fuctio Uig Abel Tafomatio ad Eule Poduct baitiga Zachaie Deatmet of edia Ifomatio Egieeig, Oiawa Natioal College of Techology, 905 Heoo, Nago, , Oiawa, Jaa Abtact: Poblem tatemet: I the ime umbe the Riema eta fuctio i uquetioable ad udiutable oe of the mot imotat quetio i mathematic whoe may eeache ae till tyig to fid awe to ome uolved oblem uch a Riema Hyothei. I thi tudy we ooed a ew method that ove the aalytic eteio theoem fo eta fuctio. Aoach: Abel tafomatio wa ued to ove that the eteio theoem i tue fo the eal at of the comle vaiable that i tictly geate tha oe ad coequetly ovide the equied aalytic eteio of the eta fuctio to the eal at geate tha eo ad Eule oduct wa ued to ove the eal at of the comle that ae le tha eo ad geate o equal to oe. Reult: Fom thi ooed tudy we oted that the eal value of the comle vaiable ae lyig betwee eo ad oe which may hel to udetad the elatio betwee eta fuctio ad it oetie ad coequetly ca ay the way to olve ome comle aithmetic oblem icludig the Riema Hyothei. Cocluio: The combiatio of Abel tafomatio ad Eule oduct i a oweful tool fo ovig theoem ad fuctio elated to Zeta fuctio icludig othe ubect uch a adio atmoheic occultatio. Key wod: Zeta fuctio, Abel tafomatio, Eule oduct INTRODUCTION Thee ae a umbe of mathematical fuctio called Zeta fuctio amed fo thei cutomay ymbol, the Gee lette ζ. Of them all, the mot famou i the Riema Zeta Fuctio, fo it ivolvemet i the Riema Hyothei, which i highly imotat i Pime Numbe Theoy (PNT). The Riema Zeta fuctio ζ() i the fuctio of a comle vaiable iitially defied by the followig ifiite eie: ξ() = () = Fo value of with eal at geate tha oe ad the aalytically cotiued to all comle with. comle vaiable i the egio { C: Re () >}of comle lae (Fig. ). A a eult, the Zeta fuctio become a holomohic i the egio { C: } of the comle lae ad ha a imle ole at with eidue =. The coectio betwee the Zeta fuctio ad ime umbe wa dicoveed by Leohad Eule who oved the idetity (Choudhuy, 995) whee, by defiitio, the left had ide i ζ() ad the ifiite oduct i the ight had ide eted ove all ime. Thu the eeio i called Eule oduct: = ( ) (2) = ime l = e, ice = Re Thi Diichlet eie covege fo all eal value of geate tha oe. Sice the 859 tudy of Behad Riema, (Catellao, 988; David, 998), it ha become tadad to eted the defiitio of ζ() to a comle value ; by howig that the eie i covege fo all comle whoe eal at Re() i geate tha oe ad defie a aalytic fuctio of the Fig. : Z ad it cougate Z i the comle lae 294

2 J. ath. & Stat., 6 (3): , 200 Hee age ove all oitive itege ( =, 2, 3, 4,.) ad age ove all ime ( = 2, 3, 5, 7,.). Thi fomula, which i ow ow a the Eule oduct eult fom eadig each of the facto o the ight ( ) 2 ( ) 3 ( ) = (3) Ad obevig that thei oduct i theefoe a um of tem of the fom: ( 2 3 ) 2 3,, = Ditict ime,, = Natual umbe (4) ad the uig the fudametal theoem of aithmetic, that i to ay evey itege ca be witte i eetially oly oe way a a oduct of ime to coclude that the um i imly: (5) Eule ued thi fomula icially a a fomal idetity ad icially fo itege value of. Diichlet (Ohtua,967) alo baed hi wo i thi field o the Eule oduct fomula ice Diichlet wa oe of Riema teache ad ice Riema efe to Diichlet wo i the fit aagah of hi tudy. It eem cetai that Riema ue of the Eule fomula wa iflueced by Diichlet. Diichlet, ulie Eule, ued fomula (2) with a a eal vaiable ad, alo ulie Eule he ooed igoouly that () i tue fo all eal >. Riema, a oe of the foude of the theoy of fuctio of a comle vaiable, would atually be eected to coide a a comle vaiable. It i eay to how that both ide of the Eule oduct fomula covege fo comle i the half lae Re()>, but Riema goe much futhe ad how that eve though both ide of (2) divege fo othe value of, the fuctio they defie i meaigful fo all value of eect fo the ole at =. ATERIALS AND ETHODS Re()>0. Thu ς ha a aalytic to {: Re ()>0, } ad ha a imle ole with eidue ad =. Poof: To ove thi eteio theoem we ued the Abel tafomatio geeally ow a ummatio by at: + + = = a b = a b a b b a (6) With: b = b b (7) + = Radiu of the comle vaiable = Simle ole Abel tafomatio i the dicete aalogue of the fomula fo itegatio by at of the fom u()dv() by a itegal of v()du(). Fo a defiite itegal the fomula of itegatio by at i: a b u() dv() = u(b) v(b)-u (a) v (a) v()du() (8) Thi itegatio by at i alied ude the aumtio that u, v ad thei deivatio u,v ae cotiuou o a b. The (8) i valid wheeve both u ad v ae abolutely cotiuou o the cloed iteval [a, b]. Abel tafomatio i ofte ued to ove may citeia of covegece of eie of umbe ad fuctio. Abel tafomatio of a eie ofte yield a eie with ad idetical um, but with a bette covegece. Subtitutig (7) ito (6) we obtai: a (b b ) = a b a b b a = = = a b a b + = = = a b a b b a + + = a b (9) Fo Re() > ad i ode to aly the Abel tafomatio we uoe that: a = ad: Eteio theoem fo eta: The fuctio ς() ha a aalytic eteio to the ight half lae b = 295

3 J. ath. & Stat., 6 (3): , 200 The ubtitutig them ito (6) we get the followig eeio: = ( + ) ( + ) = = Wee: = = - eectively Thu: (0) = 0 = 0 We obtai the followig itegal fo Re()>: ζ() = [ ] d + (5) Coide the cloely elated itegal: + + = = ( + ) = ( + ) + + = = ( + ) = ( ) + ( + ) + = = ( + ) = () d = d = + = 0 = = + (6) Combiig (5) ad (6), we ca wite the defiitio of the eteio theoem a follow: ζ() = + ( [ ] ) d (7) + Now, if we fi > ad coide the itegal: Whe eeig the ecod tem of the ight had ide of (), it eeet the imitive of the followig itegal: d d + + = [ ] (2) + + That i to ay: + = d ( + ) + + = [ ] d + (3) whee, [] i laget itege le tha o equal to. Ietig (3) i to the ecod tem of () we have: = + ( + ) = = [ ] + = = + d Now, lettig ad uig the covetio that: (4) [] d + ( ) We ca ee that thi itegal i a aalytic of Z ad if Re()>, the: ( ) + d = Re() Re() Re( + ) [] d Thi itegal imlie that the equece: + (8) () = ( [] ) d (9) Of a aalytic o eal at of the comle i geate tha eo that i to ay Re()>0 i uifomly bouded o comact ubet. Uig the Vitali theoem, let [b ] be a bouded equece i A (Ω). Ω = coected Suoed that [f ] covege fo oit wie o S Ω ad S ha a limit oit Ω. The [f ] i uifomly 296

4 J. ath. & Stat., 6 (3): , 200 Cauchy o comact ubet of Ω, hece uifomly covege i comact ubet of Ω eectively. Hece: + () = ( [] ) d (20) i aalytic ad thu the fuctio: ψ(0.4) = 3l2 + 2l3 + l5 + l7 Note that: m () l iff m ()l l iff m () (25) l Thu: + ( [ ] ) d + l m () = l (26) i aalytic o Re()>0. But thi fuctio agee with ζ() fo Re()> ad coequetly ovide the equied aalytic eteio of ξ to Re()>0. Hece the oof of the eteio theoem i comleted. We have ee that Eule oduct imlie that ξ ha o eo i the half lae Re ()>. But how about eo of the eteio of ξ i 0<Re(). The et theoem aet tell that ξ ha o eo o the lie Re() =. Theoem: Fo Re() >: ζ () ψ (t) = dt + ζ() (2) t ψ(t) = Λ() (22) With: l if = Λ () = 0 othewie Λ() = l m fo ome m (23) whee, i the geatet itege fuctio. The fuctio ψ will be ued to obtai the deied ζ itegal eeetatio. ζ Poof: Thoughout the tudy ad q age ove ime if Re() >. The Riema eta fuctio: ζ() = (27) = i give by the Eule oduct: ζ() = Π ( ) ζ() Π = ( ) = = = (28) = Iceaig equece of ime umbe ad the oduct covege uifomly o comact ubet of eal at i geate tha oe ξ = Aalytic o Re()> Futhemoe the oduct of ξ how that ξ ha o eo i Re()>. Hece: ζ () l l = ζ () = ζ() ζ() (- ) (- ) (29) if i a owe of the ime ad 0 if ot. A equivalet eeio of ψ i: ψ() = m()l (24) whee the um i ove the ime ad m () i the laget itege uch that m (). Fo eamle: 297 Subtitutig (26) ito (27) we have: l ζ () = Π ( ) (30) - 2 q (- ) Π = ζ() = ( ) (3)

5 J. ath. & Stat., 6 (3): , 200 Equatio 29 become: l - 2 (- ) ( ) ζ () = ζ()( ) l = ζ() ( ) - 2 (- ) 0 Uig the covetio that a 0 =, we have: l ζ () ζ () = ζ() = = l - 2 (- ) ζ () = (32) (33) A the iteated um of (33) i abolutely coveget fo Re()>, we ca eaage it a a double um a: ( ) l = l (34) ( > ) K ( + ) + t = = = + ψ()( ) ψ() dt + t t = ψ() dt = ψ(t) dt + + = (38) Becaue ψ i cotat i each iteval [, +]. Taig the limit of (38) if we have fially: ζ () t = ψ(t) dt fo Re > ( ) (39) ζ() + The oof i comleted. RESULTS I thi tudy we have leaed thee imotat thig: The fit thig i that the Riema eta fuctio whee, i fo um. Coequetly (33) become: ζ() = = ζ () = = ζ () Λ() (ψ()-ψ( -)) = = (35) By the defiitio of Λ ad ψ. But uig Abel tafomatio oce agai with: a = b + = ψ( ) b = ψ(0) ad = We have: = ψ()-ψ(-) = ψ()( + ) = + ψ() K ( + ) - Now fom the defiitio of: (36) ψ() = Λ() (37) We obtai ψ() I, ψ()( + ) o if Re()> we have 0 ad a, we ca wite: 298 Which i give by the followig oduct: Π ( = ) whee, P i equece of ime umbe covege uifomly o comact ubet of Re()>, hece ξ i aalytic o Re()>. Futhemoe we have oted alo that the Eule oduct eeetatio of ξ how that ξ ha o eo i Re()> ad coequetly it ca be eteded to a egio lage tha Re()> The ecod thig i the eta logaithmic deivative ξ ()/ξ() we have ued to ove that ξ ca have ome value i 0< Re(), which we ow i aalytic o Re()>. I fact the tue i that ξ ()/ξ() i aalytic oly o a eighbohood of {Z: Re() > ad Z }. Sice ξ ha a imle ole at = o doe ξ ()/ξ(), with eidue Re(ξ ()/ξ(z), ) = -. We et obtai a itegal eeetatio fo ξ ()/ξ() that i imila to eeetatio (5) of ξ The fuctio ψ defie i (2) ovide aothe coectio though (22) betwee the Riema eta fuctio ad the oetie of the ime umbe. The itegal that aea i (2) i called elli tafom of ψ

6 J. ath. & Stat., 6 (3): , 200 DISCUSSION The combiatio of Abel tafomatio (Camelia ad Guia, 2008) ad Eule oduct ha heled u to udetad the elatio betwee eta fuctio ad it oietie which may ay the way to olve ome comle theoem (Weima, 2007) ad aithmetic o geometic oblem icludig oblem elated to adio atmoheic occultatio. Fo eamle: Thomo (2007) ooed i hi eeach that adio occultatio meauemet made with a eceive iide the eath atmohee ca be iveted with a Abel tafom to ovide a etimate of atmoheic efactive ide ofile. Healy et al. (2002), alo ooed i thei tudy that the efomulatio of the ado tafom a ath itegal fo the cae of a adio ay efactig i a heical ymmetic atmohee ca be chage to Abel tafomatio a both ae equivalet i atmoheic adio occultatio ad I agee with thei ooed theoy. Some ucceful wo elated to Riema eta fuctio ad ca be foud i: Choudhuy (995); Coey (2003) ad Cviovic ad Kliowi (2002). CONCLUSION I thi tudy, we have ove the eteio theoem fo eta fuctio baed io Abel tafomatio ad Eule oduct whee the Abel tafomatio ha bee ued to ove the eal value of the comle vaiable that ae oly geate eo ad et the Eule oduct i ued to deal with the eal value le tha eo ad geate o equal to oe. The ooed tudy ha heled u ot oly to udetad the elatio betwee eta fuctio ad it oetie but alo how the fuctio ψ defie i (2) ovide aothe coectio though (22) betwee Riema eta fuctio ad oetie of the ime umbe. Catellao, D., 988. The ubiquitou Pi. ath. ag., 6: htt:// Choudhuy, B.K., 995. The Riema eta-fuctio ad it deivative. Poc. Roy. Soc., 450: DOI: 0.098/a Coey, J.B., The Riema hyothei. Notice Am. ath. Soc., 50: htt:// Cviovic, D. ad J. Kliowi, Itegal eeetatio of the iema eta fuctio fo odd-itege agumet. J. Comut. Alied ath., 42: David, R.W., 998. O the umbe of ime umbe le tha a give quatity. oatbeichte de Belie Aademie. htt:// ma/zeta/ezeta.df Healy, S.B., J. Haae ad O. Leme, Abel tafomatio iveio of adio occultatio meauemet made with a eceive iide the eat atmohee. J. Eu. Geohy. Soc., 20: htt:// Ohtua,., 967. Diichlet icile o Riema uface. J. D aal. ath., 9: DOI: 0.007/BF Thomo, F.S., Rado ad Abel tafom equivalece atmoheic adio occultatio. Radio Sci., 42: DOI: 0.029/2006RS Weima,., Cocavity, Abel tafom ad the Abel ivee theoem i mooth comlete toic vaietie; Joeh Fouie Uiveity. htt://aiv.og/ab/ REFERENCES Camelia, G. ad I. Guia, Solvig the itegal equatio of the ivee Abel tafom uig cubic lie iteolatio alicatio to lama ectocoy. Ecamig. htt:// 3.df 299