rather basic, using double extensions of analytic and harmonic functions.

Transcription

1 REFUTATIO OF THE RIEMA HYPOTHESIS By Hr Brocch Absrac W rov ha h Ra hyohss for h Raς fco s fas W rov v ha h ra ar of o rva zros of ς ads ad for accao os Th roof s rahr basc, sg dob sos of aayc ad haroc fcos Irodco b h o rva zros of h Ra ς fco of osv agary ar Or roof s basd o h sdy of h fco F où R > For Prr CarrIHES Frac h sdy of hs fco char o o sv s ow b o bshd For Rao RobrCRS Fracohr sr roof s rhas ossb Th rs s ha F ca b dd o h doa { IR }ad s roorhc o hs doa wh s o a h o ± wh Λ wh rsd Λ Is ra ar o h osv agary haf as s I 5 I I Th sv frs aragrah ar dvod o sabsh hs rs Th d s dvod o sdy h fco : R F ad R > Λ whr F df for > W s hod of haroc so ad hooorhc so, o oba a coradco f w sos R < δ, δ >

2 Ths wor s a Egsh vrso of a boo bshd a Edos Ecooca Pars wh h agr of h bshr Ja Pavvs Rca ha h Ra fco ς s dfd for R s > by ς s s W sha s h Ra fco [] s s s s ς s Γ fco of Ra: s whch s a gra fco of ordr o[] ad scay h whch s v ad ra o h agary as R b a ah h co a coosd of wo haf s: R I a < a <, R I a ad h haf crc R a W sha a h drao of ra ogarh o h a who gav ras ad w or h drc ss -cog ' d K R >, I < W s h fora of rodcs of h fco []: ² C whr C osv ra ² I P v w v, w ras w <

3 3 W sha co d I ' wh > 4 R I ad ² ² C h ² ² ' W ca ay h rsds hor for cog h gra by cog ar of o a ah by a b of crc of cr ad wh rads R aos hgh W oba : I ow for cog K w ca rac by a crv ad by a crc of cr ad of rads < cod by haf s sord by h osv ras ad ad dow Wh grows o fy ' covrg o ' fory o h crc Th ² 4 ² ² ² ² ² ² v w w v hs w hav a ajord covrgc wh grows o fy as by h bsg hor: K ' d

4 4 W ca d by hooorhy wh > R by sg h rory of h zros of a gra fco of ordr [] W ca aso d by hooorhy o h bad < I - Cog ' ζ ζ d > R W wr: ζ ζ ' ' ' Γ Γ W hav fro[3], ' Γ Γ s s s s γ whr γ s h Er cosa O h osv ras, h fco: M s ajord by a grab fco o ddg o M h rs ad-γ gv a gra, h cacs W hav o co d by h rsds hor, ad h aragrah o, hs qay s

5 5 W hav hc h fora : ' ζ ζ d I I I W hav for > R ' ' d Th scod br s aayc o a doa D o coag h o ad coag h haf-, δ ad Hc: ' ζ ζ d I I I Ths fco s aayc for > R ad aara à D

6 ζ ' 3-Aayc c coao of Γ d ζ W ow [] ha ζ ' s ζ s Λ s s Rs > wh Λ f wh rr >, f o b h co a wh W hav: I < f R ad a I f R > ζ ' ζ Λ f R Hc : Λ a d C, C> f, f a IR C W ddc ha h gra ζ ' d ζ ss, ha s co, by h bsg hor, ha s hooorhc for R > ad gra by h Cachy-Morra hor W cosdr oy aaycy o varab, h ohr bg fd W do' s aacy, R For > R D Z R w hav hs rory for R > cf aragrah3, whr w hav a c rsso of h fco 6

7 b C bg a coos ah of C cass by cs bgg, o coag h o ad bg h haf v R > for < < I cf, whr h fco s c b w Cw, w bg a arar of C wh RC w > for w > ad IC w < W sha co: < for ], w ] w IR wh C w, > ad w I, ϕ d 3 R! 4 C W ca s frs ha hs gra ss: For > w hav : I Λ d, Γ Λ Γ d Ths fora s asy vrfd for a crv bogg o a ghborhood of f R < I rss ha I, has h ordr of a dcrasg oa,hc h gra ca b co for a b ow wh R < Cog J I, ϕ d! C w hav for 3 R 4 : J I, ϕ d! I d, C C I, 7

8 ow J s hooorhc wh h ra ar of s srcy bw grs Z For R gr, w cosdr h wo dfos of ϕ By grao by ars, w s ha h wo dfos gv h sa J, sg h rao I, I, Z ad h fcoa rory of Γ for R Th by h Cachy Morra hor J s aayc coao of I ζ ' d Γ, R 3, o Z, R 5 ζ, W hav gv h aayc coao rqs 4-Asyoc sao of h aayc so of I o R < Z b wh R <, ϕ, I W hav I, ϕ d! C C I!, ϕ d I!, ϕ d Whr C s a ar of C bw ad W hav ow I, ϕ d! C C ϕ ϕ! Γ ϕ d Th, ϕ ϕ 8

9 I!, ϕ d Ad! wh Λ d A Γ ϕ ϕ W hav: Λ ϕ d A Λ ϕ Th fco d β s gra β If β s gav ra w hav: d s β d β β β sβ d β β s β Γ β β s β Γ β β Γ β Hc d ϕ Fay ϕ Γ ϕ A Λ ϕ Th w hav o co 9

10 Λ! d ϕ ϕ d Λ! By sccssv graos by ars w oba:! d!! b > h r! s a for wh h va! By h Srg fora hs r s ajord by K K Ta ow sch ha K < - wh ϕ for Th ϕ d Λ! Λ Co ow h s sc By h Tayor fora w hav:,!! < < θ θ Hc: ϕ d Λ! R Λ < ϕ wh R 3 < * W rca ha ϕ

11 W hav of cors < Λ ϕ < Λ * s r o h doas ha w d R P I, P >, R < Fay h ar corrsodg o h gra o h ah C s asy ajord by 5-Cog Γ d Whr >, D v, v IR D s K ' Rar ha h gra Γ d s covrg by sg h D s co Srg fora [3] for Γ ad h fac ha h rs hav a arg ar Γ b > w frs co d D By h Phrag-dof hod [4], Γ d w b sar ad sar o cs of crc wh cr ad rads R I W ca ay h rsd hor ad so oba d by h D ad coag -IR Γ D d! ow for h w rar ha

12 Γ d s aayc v o a doa coag wh ra ad v v D s Th Γ d - D s Fay sg h hor of covrgc of h gra of srs absoy ajord w oba Γ d D s Rar ha h fco s ra for ra srcy osv Char 6 Mroorhc so of d R > o I IR I, W cosdr h fco f d for osv ra D s Ths fcos w df by f ss ory covrg by gras by h rsso of I of o, W ca rbod I by K K K > by sg h Srg co, I I fora As w hav dvdd s h gra s w dfd > b W cosdr H I, Λ s s dfd a o 4 By h co Srg fora ad h Prag dof hod for a scor, w hav ha: H d s qa o D H d whr s h o of wo cojgad haf-

13 s ϑ v v, θ δ > ad ϑ v,v h va or s bodd by R So: H d s a aayc fco IR, < A whr A Γ wh W ca cosdr d sad of s Γ d by sg aga h Phag dhof hod o a D s scor [4] Ths ha aars ha h gra w co as for scod drva : Γ Γ d d s D s Γ d whch s dfd for R > W o ha h D s scod drva of H o s bodd by h Cachy sas by [ ] Ra 6 cosdrg h dscs of cr a of rads Th w hav o co : 4 4 Λ d for ré > sch as > w hav : D s ² d D s ² ² by h résd hor o haf crc of cr o h sd of osv ras dby D ² For <, w oba ² ² W hav h résds a h os ±, Λ ² W wr ² ² Λ So h résd s Γ Γ W hav o sbsrac d d s, for D s a h fco, R > aars whch ads a oorhc so wh h 3

14 s os a h os ±, Λ o os To co - C - IR wh h résd Λ ad o ohr o h agary as, w s h fac ha h fco s ra o h osv ra as Hc, h ra ar s h haf s of vas v, v > ad v W cosdr frs R v D Γ s d 5 I I I Γ Ths s oba by sfcao by s ad h coao of d cf o5 D W hav o co: H v d H v d wh v> B v v cos v s v W fay hav o co Γ v Γ v d d s wh v s By h Phrag-dof hod ad h cos Srg fora, w hav o co h 'égra o E whch s h dg of h haf bad I δ R Th w oba by h rsd hor by bodg scssvy by h sgs v w w δ ha h égra has for va v! Fay R > has a roorhc so à ± Λ wh h résds I 5 I C R, as s os Λ Is ra ars o h agary as f : I Th h sdy of h fco R > s cod Th ohr os ar dvod:o rf h Ra Hyohss 4

15 7- Cog a c aayc coao R > IR, I > O h osv agary as h ra ar of F R >, Λ s I 5 I I Th fco F s so coos, R,, Λ, I > W ca h ay h Schwarz rfco rc o h fco 5 M F W wr: F F Th aayc coao a h o, I > s 5 R F 5 - I F I s sffc ow o y by o oba h roorhc coao wshd : w b hooorhc wh w f ad roorhc wh w f W hav aso a haroc coao of R >, C, I > ' I fac w s h aayc coao of R >, C, I' < 8-Coao o δ < < < of h fco Λ >, fd, R, I >, f s I < δ <, ad of h fco Λ > b ϕ h s of h wo fcos, whch s haroc 5

16 6 For, > R C h srs fd #, > Λ s covrg W sar r fro h, b ] [, W hav sg grao by ar for Sjs asr d > Hr s h br of r grs Sos ow ha h zros of sasfd s < < I δ ; h casscay Marc Hdry, ora cocao: δ O v dv Hc w hav d d v dv d O δ Th d v dv v dv - d

17 d f R > 4 δ Th rs coag O hav a asy aayc coao ow, f w cosdr h, sg h grao by ars, hy gv a hooorhc fco for ra # I > R > δ W s ow fd a coao of : v d W a h varab chag ad oba: v dv v W wr ow v dv v v dv v Th as r s a gra fco, hooorhc for I > ad covrgg o zro wh I ra fd W ow a a grao by arsw oba: v dv v v dv v * W cosdr h qadra d by h ra osv haf as ad h osv agary haf as W cos h ass by a qarr of crc wh cr O ad rads R growg h fy W ca ay h rsd hor ad oba: f R < ad I > hr s a o v 7

18 Hc: * w dw w If R >, I > hr s o o w * dw w f w f, w ca hav a ao aayc coao wh Fay, f w hav h coao by coy for R, R > of Λ > δ I < < <, fd I rsrvs h aaycy R as w ca s by h Cachy-Morra hor [4] W hav ow o cosdr Λ R > W oba forr h va of aayc fco h ao rob bg o fd a coao of v dv v By h hor of rsd w hav o fd coao by h sa way ha ar Th o s for v, hc h rsd aars for R >, I > ad gvs: 8

19 w dw w for R < ad w dw w 9-So rors of cofora rasforaos ad h Posso r W cosdr a rcag h co a wh ss A B C T DT T > Th [] h haf a Rz s cofory qva o a sar rcag by h cofor z rasforao Z K v v v dv wh sch ha v v v dv T v v v dv To acy h haf a o ABCD w s a K a wh a sch ha a v v v dv Wh T a a cosa Th h cofor rasforao o h rcag covrgs fory o vry coac o h cofor rasforao of h haf a A ad B fd, hs haf bad s { z Iz, Rz } W cosdr ow a ago A'B'C'D'E' wh: A'-, B',C'T,E'-, gv by h rory ha h ag E' D'C' s ] [ θ ad D' o h agary as θ,, fd b D'v T Cosdr h fco 9

20 R v η T <η < whch s haroc ad gv by s Posso r cf [] Ths fco s osv o h ago Cosdr h rsrco of h fco o h sg E' C', h s bodd by Ths fco s osv o h rcag A'B'C'E'ad gv by s Posso Kr R T η ' Th corbo h Posso r of h va of h fco o h sg E'C' covrgs o wh T - Cog h fco Λ Λ G R > R cod a δ < < I R f S I < δ O h haf bad I R Th fco rodcd s haroc vrywhr for R > ad dfd W hav ha h fco Λ Λ ca b cod for R ad δ < <,rodcg, by a ao way Th fco s roorhc o h haf bad ad s haroc for # Λ

21 O h haf bad,h fco Λ Λ φ G, U δ, by s ajord f [ [ η > bcas: -for R > h fco s bodd o h haf bad -for δ < < h bad s η Cf aragrah 8 W hav o rov ow ha G s gv by h Posso r o h haf bad b v> sch ha h oygoa whch s h rsco of h scor of s v ad agar a h s < θ < ad R < w hav ha 4 Λ Λ v K for R > ad for v gog o v fy Sch v ca oba by drvg o absrdy Rca ha h br of zros : vv O v hc o h oygoa w hav: wh < v s gv by v A v ow h fco R η <η < w b haroc o h ago d by h v dg of h haf bad ad h dg of h scor wh h c wrg of h cofor

22 rasforao of hs doa o o h dsccf[] Th corbo of h dg of h scor covrgs o wh v by h aragrah 9 R η v o So G s bodd o h dg of h haf bad ad gv by s Posso r I s haroc wh s fd b Q T, h Posso r o h haf bad rasord fro h dsc T s o h dg, dt s h bsg sr o h dg W hav : G Q T, G T dt f R > Ths qay dd wh s fd aaycay a δ < < wh s h haf bad, by a ao way B wh, h wo brs covrg f R so : G Q T, G T dt, δ < < Fay Q T, G T dt s haroc sd h haf bad, so G s haroc sd h haf bad Is rsso for R s gv by aragrahs7 ad 8 Th fco G, δ < < s gv by K for R < whr K s haroc ad covrgs o wh I, < R <, R fd ad gv by J f < R <, J wh I R fd -Ed of h roof

23 3 Foowg os 8 ad, G s dd o fd R < < δ f s < < I δ h sa ad q way By h ar 8, w hav ha h fco ϕ ca b wr for y > I > R : ² ² w w dw w dw w d y o δ d y o _ δ rr rr _ K wh K wh R I fdossby zro W sbsrac ad w oba : ] [ R R d y o δ R d y o _ δ M R wh M bodd wh I R fd h haf bad I R, w b wh [ ] [ ] φ,, r C b h haf crc of cr ad or rads ocad o h sd of ra gav ars Th δ s aayc o h dsc D of cr ad rads Th : { d y o > R R δ } d y o _ δ { d o C R δ

24 δ o C { R [ y ] [, ] [ y ] [, ] δ o, δ o, d } d d Ths rssos ar B R for w Λ has h foowg va W hav o sbsrac 5 w w w obad by coy o h agary as B for w sffcy ar w w >, w hav a va w } 5 w fco s o bodd o h haf bad ad w g a coradco 4-Cocso W hav rovd h : C R >, I, ar So h Thor: Th Ra hyohss s fas Thr s a fy of o rva zros s of ζ s sch ha hy ra ar ads for accao os ad Rfrcs [] Varo, Thor ds focos, Masso [] Davor, Mcav br horycr advacd ahacs [3] Ddoé, cac fésahra [4]Rd, ra ad cos aayss Mac Graw H Paasa, 4/3/6 4