Zeta-reciprocal Eteded reciprocal zeta fuctio ad a alterate formulatio of the Riema hypothei By. Alam Chaudhry Departmet of athematical Sciece, Kig Fahd Uiverity of Petroleum ad ieral Dhahra 36, Saudi Arabia. E-mail: malam@kfupm.edu.a Abtract We defie a eteio of the reciprocal of the zeta fuctio. Some propertie of the fuctio are dicued. We prove a alterate criteria for the proof of the Riema hypothei ad the implicity of the zero of the zeta fuctio. athematic Subject Claificatio: 6, 6, 36, 99. Keyword ad phrae: Riema hypothei, zeta fuctio, critical trip, critical lie, o-trivial zero. öbiu fuctio, öbiu iverio formula, Hurwitz zeta fuctio Laplace traform, icomplete gamma fuctio, Ramauja formula.. Itroductio Riema proved that the zeta fuctio (ee [,, 3, 6, 7]), ζ (): = ( = σ + iτσ, > ), (.) ha a meromorphic cotiuatio to the comple plae. Thi atifie the fuctioal equatio (ee [6], p.3 (..)) ( ) π ζ( ) = ( π) co( ( )) Γ( ) ζ( ), (.) ad ha imple zero at =, 4, 6,... called the trivial zero. All the other zero, called the o-trivial zero, of the fuctio are ymmetric about the critical lie σ = i the critical trip σ. The multiplicity of thee o-trivial zero (i geeral) i ot kow. Riema cojectured that the o-trivial zero of the fuctio lie o the critical lieσ =. Thi cojecture i called the Riema hypothei. The zeta fuctio ha the itegral repreetatio ([6], p.8 (.4.)) t dt ζ () = Γ t () ( σ > ). (.3) e Writig the öbiu fuctio a µ ( ) = ( ) k if the p... p pk i a product of k ditict prime ad zero otherwie. It i kow that (ee [3], p. 6)
µ ( ) = ( σ ). (.4) ζ () The eteio of the idetity (.4) to the regio / < σ < would prove the Riema hypothei. There are everal eteio of the zeta fuctio. The zeta fuctio belog to a wider cla of L-fuctio (ee [, 3, 6]). Oe of the well kow eteio of the zeta fuctio i the Hurwitz zeta fuctio ([6], p. 36(.7)) ζ (, ): = ( = σ + iτσ, > ), (.5) ( ) + which ha the itegral repreetatio([6], p.37(.7.)) t t e dt ( ) t t e dt t t ζ (, ) = = Γ() e Γ() e It follow from (.) ad (.5) that (, σ > > ). (.6) ζ(,): = = = ζ() ( σ > ), (.7) ( ) + leadig to the fact that the Hurwitz zeta fuctio geeralize the Riema zeta fuctio i the mot atural way. It eem iroical that the fuctio / ζ (, ) doe ot eem to be a ueful eteio of the reciprocal / ζ ( ) of the zeta fuctio. We itroduced a fuctio that eem to be a atural geeralizatio of the reciprocal/ ζ ( ). Some propertie of the fuctio are dicued. We eploit the aymptotic repreetatio of our fuctio to give a alterate formulatio of the Riema hypothei ad implicity of the zero of the zeta fuctio. For the other eceary ad ufficiet coditio of the Riema hypothei we refer to (ee [6], ectio 4.3).. The öbiu iverio formula ad applicatio to the Hurwitz zeta fuctio The öbiu iverio formula ca be writte i the form ([3], p.7) g ( ) = f ( ) f( ) = µ ( g ) ( ) ( > ), (.) provided f ( ) ad g ( ) both coverge abolutely. The above formula ca alo be writte i the form g ( ) = f( ) f( ) = µ ( g ) ( ) ( > ). (.) Rewritig (.5) we fid that ζ (, + ): = ( = σ + iτσ, > ), (.3) ( + )
which lead to a ueful aalytic repreetatio ( ) µ ( ) ( + ) = ζ (, + ) ( = σ + iτσ, > ). (.4) The LHS i (.4) i a etire fuctio of for all. I particular for = i (.4) the claical idetity (.4) i recovered. Sice we have ( / + ) a, it follow from (.4) that µ ( ) ζ (, + ) ( ). (.5) Similarly we have ( ) + + a, it follow from (.4) that µ ( ) + ζ (, + ) ( ). (.6) 3. The Eteded reciprocal zeta fuctio A cloe look to the Hurwitz zeta fuctio how that it i baically the eteio of the erie repreetatio (.) of the zeta fuctio obtaied whe i replaced by + +. We follow the ame procedure i (.4) ad defie the our eteded reciprocal of the zeta fuctio by µ ( ) R (, ): = ( σ, ). (3.) ( ) + The fuctio R(, ) eted the reciprocal fuctio / ζ ( ) i the mot atural way a we have R(,) = / ζ ( ) ( σ ). (3.) The eteio of the idetity (3.) to the regio / < σ < hould prove the Riema hypothei (ee [3], p.6). A applicatio of the öbiu iverio formula (.) i (3.) lead to the relatio ( ) = R (, ) ( σ, ). (3.3) + The relatio (3.3) i importat i the ee that it how that R (, ) ( ). (3.4) ad + R(, ) ( ). (3.5) Theorem The fuctio R(, ha ) the itegral repreetatio t R(, ) = t e Θ() tdt Γ() ( σ, ), (3.6) where the fuctio Θ ( t) i defied by 3
t Θ (): t = µ ( ) e ( t > ). (3.7) t Proof ultiplyig both ide i (3.7) by e ad the takig the elli traform we fid that t µ ( ) [ e Θ ( t); ] =Γ ( ) =Γ( ) R(, ) ( σ > ). (3.8) ( + ) Dividig both ide i (3.8) by Γ ( ), a the gamma fuctio doe ot vaih i the comple plae, lead to (3.6). Corollary R (,) = ζ () ( σ > ) (3.9) Theorem The fuctio R(, ) ha the Taylor erie repreetatio at = give by Γ ( + )( ) ()( ) R (, ) = = ( σ, < ) (3.) Γ () ζ( + )! ζ( + )! Proof Replacig the epoetial fuctio i (3.5) by it erie epaio lead to the repreetatio ( ) + Γ ( + )( ) R (, ) = t Θ () tdt= Γ()! (3.) = Γ () ζ ( + )! which i eactly (3.). Remark Sice we have ζ ( ) =, the repreetatio (3.) may be viewed a the perturbatio of the geometric erie () ( + ) = ( ). (3.)! 4. Alterate formulatio of the Riema hypothei Takig the elli traform i of both ide of (3.7) ad uig Ramauja formula (ee [3], p.8) we fid that c+ i k ()( k ) Γ( z) Γ( z) z R(, ) = = dz ζ( + k) k! πi Γ( ) ζ( z) (4.) k = c i where the lie z = c pae through the regio of aalyticity of the itegrad. The pole z = ( =,,,... ) of the above itegrad are to the LHS of the lie of itegratio leadig to the repreetatio (3.). The itegrad i (4.) ha pole a well to the RHS of the lie of itegratio at z = + ( =,,,... ) ad that at z = where ' are the o-trivial zero of the zeta fuctio. Takig the um over the reidue lead to the aymptotic repreetatio for large value of to give R(, ) R + R ( ), (4.) 4
where, we defie + Γ( z) Γ( z) z R : = Re [ ; + ], (4.3) Γ() ζ ( z) Γ( z) Γ( z) z R (, ): = Re [ ; ]. (4.4) Γ() ζ ( z) It i to be oted that if the zero of the zeta fuctio i imple, R doe ot deped o ad i jut a fuctio of. oreover, the cotributio to the aymptotic due to the preece of everal zero of the zeta fuctio o the lie σ = σ will ot eceed C σ + ε ( ε > ). However, if the zero i of multiplicity N, the for each, R i a polyomial of degree N i log. The aymptotic repreetatio (4.) how, uder the aumptio of the implicity of the zero of the zeta fuctio, that R (, ) ( σ σ + R ε =Ο ) (, σ >, ε > ), (4.5) where σ : = up{re( ) : ζ ( ) = }.The Riema hypothei i true ad it zero are imple if σ = /. Hece it follow that the Riema hypothei i true ad it zero are imple if we have i particular for = 3/ +ε R(3/, ) = O( ) ( ε >, ), (4.6) which provide a alterate formulatio of the Riema hypothei ad the implicity of the zero of the zeta fuctio. We ue athematica to plot the fuctio i Figure I by takig the um of the firt te thouad term of the erie (3.) ad fid coitecy with (4.6). However a aalytic proof of (4.6) i eeded to prove the Riema hypothei..4.3.. 3 4 5 Figure I The graph of R(, ) for =.5 uig (3.) for large. 5
5. Cocludig Remark We have the erie repreetatio (ee [4], p.357 (54.6.)) () ζ ( + )( ) ζ (, + ) = ( < < ), (5.)! for the Hurwitz zeta fuctio. A compario of the repreetatio (3.) ad (5.) i ot without iteret. The eteded reciprocal zeta fuctio eted the reciprocal zeta fuctio i the mot atural way ad provide a alterate criteria for the proof of the Riema hypothei ad implicity of the zero of the zeta fuctio. A aalytic proof of (4.6) will reolve the claical problem of the Riema hypothei ad the implicity of the zero of the zeta fuctio. oreover, we have ot bee able to prove the fuctioal equatio for the eteded reciprocal zeta fuctio though it i epected that there hould be a fuctioal equatio for the fuctio imilar to the oe kow for Hurwitz zeta fuctio. Ackowledgemet The author i grateful to the Kig Fahd Uiverity of Petroleum ad ieral for ecellet reearch facilitie. Referece.. A. Chaudhry ad S. Zubair, O A Cla of Icomplete Gamma Fuctio With Applicatio, Chapma ad Hall/ CRC,.. H. Daveport, ultiplicative Number Theory,Secod editio, Spriger-Verlag, 98. 3. H.. Edward, Riema Zeta Fuctio, Academic Pre, 974. 4. E. R. Hae, A Table of Serie ad Product, Pritice-Hall, INC, 975. 5. R. B. Pari ad D. Kamiki, Aymptotic ad elli-bare Itegral, Cambridge Uiverity Pre,. 6. E. C. Titchmarh, The Theory of the Riema Zeta Fuctio, Oford Uiverity Pre, 95. 7. S. Wedeiwki, Zeta Grid-Computatio coected with the verificatio of the Riema Hypothei, Foudatio of Computatioal athematic coferece, ieota, USA, Augut. 8. D.V. Widder, The Laplace Tradform, Priceto Uiverity Pre, 97. 6