On the Real part of non-trivial zeros of Riemann s Zeta function.

Transcription

1 O th ral part of o-trval zro of Rma Zta Fucto O th Ral part of o-trval zro of Rma Zta fucto. A.M.S. Subjct Clafcato, 98: 3E, 45E5, 45M5, A7.

2 Rutlo Hrrra. To Th Salvaora Popl That Rvolt pt of Thr Mry a Igorac. Th Mathmatca thk that a problm I ay to olv Aftr t ha b olv

3 O th ral part of o-trval zro of Rma Zta Fucto 3 Prfac: Th followg ot ummarz fv yar of ffort. Th rarch proc ha ot b lar, uffrg momt of avac a momt of tagato, a racal chag th tactc u to olv th problm. I tr wth Dffrtal Equato orar a wth partal rvatv, Clacal Aaly a Harmoc Fucto, utl I fou a almot fortutou way th Bouary Valu Problm a th Sgular Itgral Equato Thory, whch I ralz t coul b of corabl u olvg th problm. Th rult ar urprg. Naturally, th ffrc of tapot ca b obrv a rgar th Rma zta fucto, btw Ttchmarh or Ivĉ Bblography a that prt hr. I th frt ca, a mathmatcal tool u that, although t ha vlop a avac gratly rpct to Rma Zta Fucto Thory, th pcfc ca of fg th ral part of o-trval zro, partcularly obcur. I blv that th ma rao th Bouary Valu Problm a Sgular Itgral Equato Thory hav ot b u for tryg to prov th Rma Hypoth th followg: th frt plac, th Bouary Valu Problm wa orgat Phyc problm, Quatum Mchac, Flu Mchac a th Mathmatcal Thory of Elatcty, a t wa hr that t wa vlop a trgth. O th othr ha, th problm a rgar th mto hypoth ha b tak up, gral, from th pot of vw of th rlatohp, btw th zta fucto a th trbuto of prm umbr ug Clacal Aaly: Itgral Traformato, bou, aymptotc appromato, tc. Ev mor, all ttbook a joural rv tll ow, th problm o trmg th ral part of o trval zro of th Rma Zta fucto ha b work arou th aymptotc formula for th umbr of zro th crtcal trp, a partcular wth th formula tablh by Rma a aftrwar prov by Vo Magolt: T T N T l OlT Whr NT th umbr of zro of th form βα wth < α T A kow, Hary wa th frt provg that th ζ fucto ha fty umbr of zro o th l R z a aftrwar Slbrg prov that th umbr of zro N T o th l atf th qualty N T > AT lt for om A >.

4 4 Rutlo Hrrra. Th appromatly th tat of th gam th rarch of th zro of ζ fucto th crtcal trp, that ca truly b corr, a part of aalytc umbr thory. Th ma a w propo to chag th pot of vw of th problm, ug th thory of Cauchy typ gular tgral, th bouary valu problm a th gular tgral quato thory, for th horzotal tuy of th problm. If th ot ta th tt of aaly a opo of pcalt, two mportat ubjct wll b hghlght Aalytc Numbr Thory: a- Sgal of uty of th Bouary Valu Problm a Aalytc Numbr Thory b- Proof of Rma Hypoth. A th rar wll apprcat t ha b uffct to am th problm wth th w tapot ta of th uual o. Th a th vlopmt of th bouary valu problm a gular tgral quato hav ma th ot pobl. I wh to t my grattu to Wllam Lambrt Mart R.I.P a Héctor Fguroa for th rgor, th thuam a th curoty for Mathmatc thy oc taught m. I partcularly wh to thak Profor Ew Catro Fráz who cla th thr mto ubjct wr ythz. Wthout th avc of Profor Catro th ot woul t t. H wa th frt ra th maucrpt a ma th rpctv obrvato a ot. Fally, I wh to thak Profor Maul Barahoa Drogutt, for havg ough htorcal prpctv to tmulat m to urtak th ffort, whch th publhg of a papr lk th volv. Th author. School of Mathmatc. Uvrty of Cota Rca. Sa Joé, Cota Rca, 987.

5 O th ral part of o-trval zro of Rma Zta Fucto 5 Abtract: Th Rma zta fucto ha b ubjct of amout of rarch c t troucto Mathmatc, a wll a th Rma HypothRH. Tll ow, th bac rcto o rarch rpct to RH ha b clmbg th crtcal l R, a wll a th rgo am crtcal trp : < R <. W propo to chag th pot of vw orr to approach th zta fucto th crtcal trp but horzotal orr to attmpt to prov th RH: If a zro of th ζ fucto o th crtcal trp th R. Th wll b o by ma of th thory of Sgular Itgral Equato, maly tryg to prov that o- trval zro of ζ ar aocat to th aymptotc bhavor of crta k of oluto of th gular tgral quato: ϕ, t L For om ϕ Hölr fucto. All tgral that L t appar houl b urtoo th of VP Cauchy. Ky wor: zta fucto, gular tgral quato. A.M.S. Subjct Clafcato 98: 45E5, 45M5, A7, 3E.

6 Rutlo Hrrra. 6 Th followg quc of propoto ha a a ma objctv th Proof of Rma Hypoth: If ζ o th crtcal trp, th R Propoto : For R > : l ζ Proof: Lt pa th krl of th tgral:, That covrg uformly for > I coquc: l l l * A th trchag of th um a th tgral jutf by Lv omat covrgc thorm. If ow w put th tgral ur th um g : l l a by puttg: th lat tgral of th rght ha w wll hav: t t t t. Puttg th rult * wll follow: l ζ, that th Propoto

7 O th ral part of o-trval zro of Rma Zta Fucto 7 Th Formula abl u to t th ζ fucto to th whol pla a th followg propoto. Propoto : For C, : l ζ L Whr L ta to b th cotour pct blow: l l A l ; th logarthm of l trm uch a way that t ral for gatv valu of l. Proof: Puttg: z l w gt to: ζ C z z z 3 Whr C ta to b th cotour of blow: Th fuctoal rlatohp 3 tally th tartg pot of Rma ow aalytcal to of ζ fucto : Ewar, H.M.: Th Rma zta fucto. Acamc Pr, N.Y.974 th tralato of Rma papr Ubr Azahl r Prmzahl. It mportat to ot, at th pot, that [ 5 ], Ttchmarh, E.C, that ha b mayb th mot mportat htorcal rfrc to Rma ζ

8 8 Rutlo Hrrra. fucto, th aalytcal to wa wrog, roppg o t th mu g of th umrator ur th tgral g 3.: pag8 of 5], Chap,.4 From 3, mot of Mathmatca tt th cotour to th puctur pla C{ } aftr avog th gulart: z kι, for k Z, uch a mar that from that pot of vw, t ca b uc aly th fuctoal quato: ζ ζ that charactrz plty th ζ fucto. W wll avo th pot of vw, a ucat rathr th followg: Propoto 3 For C, : ζ l 4 Proof: From Rma pot of vw, w ca wrt: Π ζ ζ for R > Rma ot Π ; w wll kp th otato a alo that: C Hr C:

9 O th ral part of o-trval zro of Rma Zta Fucto 9 But f w put : w ca wrt: l l * L C Whr L th cotour pct blow: Rma hmlf wrot that qualty: C ζ rma val for all a art alo that th tgral qual *, thrfor w ca wrt: l ζ or, what th am: l ζ for But c:, wll follow: l ζ, a multplyg a vg by, w wll hav: l ζ, that prov th Propoto

10 Rutlo Hrrra. From th lat propoto, aly follow that ζ, by th mply valuato of th rprtato, c co qual zro for, N, a th tgral bou for tho valu of. Th zro ar am trval zro of th ζ fucto. It to ot th ffrc btw th tapot a th o th tratoal ucto of th trval zro from th fuctoal quato of ζ, mto bfor. From hr w gt th fuamtal thortcal tur th oluto of th problm of fg th ral part of o-trval zro: th mpropr gular tgral that appar th fuctoal quato 4 houl b urtoo a a lmt pot of th prcpal valu of a tgral of Cauchy typ ovr a cotour that t to fty: l lm t l t I orr to cofrm th lat qualty, w gv th followg Propoto 4: Lt ϕ b a arbtrary Hölr fucto o ], [, th: lm t ϕ t ϕ, a th lmt uform. Proof: Th a taar rult of th thory of Sgular Itgral,: Mukhlhvl, N,I. Sgular Itgral Equato. Woltr-Noorhoof Publhg Co. Grog, Th Nthrla, 97, pag 38 a ff.

11 O th ral part of o-trval zro of Rma Zta Fucto Dfto5: Φ W f a fucto: o th t ], [ by ma of o gular tgral of Cauchy typ ovr a cotour that tt to fty: Φ t ϕ t ], [ 5 t l 5* wth ty fucto: ϕ ], [ Whr y, a ], [, th ral part of, f but ukow a y a arbtrary gv ral umbr. Propoto 6: Th tgral 5 wth ty 5* a Cauchy typ tgral ovr a cotour that t to fty. Proof: It ough to vrfy that th two followg coto ar atf: a- Th ty fucto ϕ a Lpchtz fucto o ], [. A b- For ough gratr valu of t thr hol: ϕ t ϕ µ t for om A >, µ > a- I a ay mattr : ϕ ffrtabl rpct to t a th rvatv ar ft o ], [. Th mattr that ϕ o t mak ffrc rgar to th cocluo. b- Lt frt prov that lm ϕ : I: ϕ l. Sc ], [, <, w gt thu:

12 Rutlo Hrrra. ϕ f, that : ϕ 6 l B: ϕ f l Th thr t A > a µ, uch that th propoto rma val Th fact ϕ 6 guarat th covrgc of th tgral. Gakhov, F.D. Bouary Valu Problm, Prgamo Pr, Ofor,966, 4, Scc 4.6. If w cor ow ϕ a a ukow fucto, c w o t kow th valu of R wh th fucto 5 wth ty 5* qual to zro wh t, th w hav th rght to vokg th followg propoto: Propoto 7: Th oluto of th gular tgral quato: ϕ 7 t th cla of Hölr fucto ubou at tal trm of th cotour of tgrato : a ϕ t, 7* t for om compl cotat a a a that o t p o t. Proof: Th a taar rult of th thory of Sgular Itgral Equato. S, for tac : Gakhov, F.D. op. ct. 4.3.

13 O th ral part of o-trval zro of Rma Zta Fucto 3 If w ar ow lookg for th oluto of th tgral quato: l, for t t 8 th cla of Hölr fucto ubou at th tal trm of th cotour of tgrato, th th oluto of 8 mut bhav th am way tha 7* a rght ghborhoo of, that both t mut b aymptotcally qual wh t. W hav thu th followg Propoto 8: If: y lt a a t t for t th : Proof: W hav: lm t lt t a a y t th thr mut t ral fucto mar that: v a t, wth lm v t t a a uch a lt t a a y t w gt to: v yl l t l But c : l a t. By takg logarthm both of th qualty [ a a t] l v t l t a [ a a t] l a a t arg a a t k

14 4 Rutlo Hrrra. W gt to th par of quato: l a a t l va t l t 9 llt llt llt [ arg a a t ] k y, for k Z l l t By approachg t from th frt of th two quato w wll hav: lm t l a a l l t t, lm t l va t l l t, lm t l t l l t a coquc :. Th th Rma Hypoth. It to ot that th mtho u hr for approach th Rma Hypoth o ot gv formato about th magary part of o-trval zro, c from follow: y ± f k ± a t Rutlo Hrrra. E mal: luca54@latmal.com Pho Ecula Matmátca Uvra Cota Rca Acama Matmátca MAC. 999-

15 O th ral part of o-trval zro of Rma Zta Fucto 5 REFERENCES:. Apotol, T.M..: Itrouccó a la Toría Aalítca Númro. E. Rvrté, Mar, 98.. Btaz. A.V. Equato of Mathmatcal Phyc. Mr. Publhr, Mocow, Buak, B.M.,Fom, S.V.:Multpl Itgral, Fl Thory a Sr. Mr Publhr, Mocow, Charakara,K.: Arthmtcal Fucto. Sprgr Vrlag, Brl Coway, Joh.: Fucto of O Compl Varabl. Sprgr Vrlag, N.Y Sco Eto. 6. D Bruj, N.G.: Aymptotc Mtho Aaly. Dovr Publcato Ic. N.Y. rpublcato of thr E. 7. Drrck, W.: Itrouctory Compl Aaly. Acamc Pr, N.Y Ewar, Harol: Rma zta Fucto. Acamc Pr, N.Y., Gakhov, F.D. Bouary Valu Problm, Prgamo Pr, Ofor,966. Ivĉ, A.: Th Rma Zta Fucto. Holt, Rhart a Wto, N.Y Markuhvch, A.: Toría la Fuco Aalítca. E. Mr, Mocú, trauccó l ruo Mukhlhvl, N.I. Sgular Itgral Equato. Woltr Noorhoo Publhg, Grog, Th Nthrla. Rprt Nvala, R. Paatro,V.: Itroucto to Compl Aaly. Chla Publhg Co., N.Y. 98. Sco Eto 4. Ramachr, R.: Topc Aalytc Numbr Thory. Sprgr Vrlag, Brl, Svhkov, A., Tkhoov, A.: Th Thory of Fucto of a Compl Varabl. Mr Publhr, Mocow, Ttchmarh, E.C.: Th Thory of Rma Zta Fucto. Ofor Uvrty Pr, Loo, Trcom, F.: Itgral Equato. Itrcc Publhr Ic. N.Y. Pur a Appl Mathmatc, vol. 5 Et by R. Courat, L.Br, a J.J.Stokr. Loo Uvrty Pr. Loo Frt E. 957.