A Bessel polynomial framework to prove the RH
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1 A Bl poloil frwork o prov h RH Dr lu Bru Friburg i Br wwwri-hpohid Jur Abrc Th Gu-Wirr di fucio f : bl rprio of Ri duli quio i h for f d f d Th odifid Bl-Hkl fucio : rc Y / J J Y co d dd ih coh : coh r likd b h rlio d d wih log igulri zro [7] G N Wo For ppropril dfid / d R i hold d / d A which i igrl of qur of rl-vlud fucio h rprio i h for P 3 P : d dfi diribuio fucio for R h corrpodig di fucio d igh lvrg iig chiqu or cojcur o d prov h RH g Ruj Mr Thor Pol rgu bou h zro of cri ir fucio or h Brr cojcur wh rplcig h Gu- Wirr di b h Bl di fucio kig dvg of ppropri choic of h prr
2 Iroducio d Prliiri Our riolog follow ho of [8] HM Edwrd d [7] G N Wo Throughou hi ppr w do wih log h url logrih i log log l Th Gu-Wirr di fucio f : fulfill h followig k propri i d f covrg dfiig h G fucio g i h for / d : d d ii f f i dcrig fucio for iii h Fourir rfor rlio For h ir fucio 3 : / d wih : Ri duli quio vlid for ll copl vlu 5 : c b wri i h for 6 f d f d pplig Jcobi rlio [8] HM Edwrd 3 7 G : f : G A quivl forul o 7 i giv i [] H Hburgr b 8 ico i
3 Th hprbolic c fucio 9 g : coh fulfill iilr propri h g for i Fourir rfor i hold : g coh gˆ d [7] B E Pr chpr 3 coh k k k k k Fro [5] N Nil chpr IV 3 5 w rcll / i / i coh i ih Fro [] IS Grdh IM Rzhik [7] GN Wo 7-5 w o 3 B! co B! E coh! co E! wih Broulli ubrb d Eulr ubr E Th corrpodig rlio of 9 o 7 i giv b whrb G : Z coh Z coh Z G coh 5 d coh d d ih i forll d ih wih 6 k : d k E! Th rld forul of h Z fucio r : d k! B 3
4 Th corrpodig proof of 5 for ld o rprio i h for 8 d d Our Bl poloil pproch u lrivl o h Bl fucio of d d kid 9 : J Y i H co ih coh coh dd ih coh d ppdi d [6] GN Wo 3-75 Wih h oio fro [6] GN Wo 7-35 Q : 3 5 P d Y rc J w rcll fro [6] GN Wo h followig lriv rprio Th Mlli rfor of giv i [6] GN Wo 3- w i L : For R R i hold i d ii for N d iii L i ud o how k d! k 35!
5 L : For d R h followig poic pio hold ru P wih P :! /! / W rcll h proof of l fro [6] GN Wo i h ppdi o giv o iigh io h li chiqu So forul o b ud i hi proof w rcll i L 3 For N i hold! /! d Sirlig forul i h for / i li! li li!!!! / Proof of l 3 i giv i h ppdi Subiuig h vribl i l 3 ld o Corollr For : k N d R i hold i k d k d k k k k for k N ii k k d k d k Bk k! Fro [6] GN Wo 5-6 w rcll h idi i J Y d o h pcil rlio / 5
6 W u h bbrviio 3 : 3 p / : / d g fro [6] GN Wo 7- d l 3 u/ u / v v dv wih h coour igrl u / u giv i [6] GN Wo 7- d 5 p! p!!! p!! p! Th rlio o h poloil i w uri i L 5: For h poloil i i hold i P 3 p : ii p 3 3 P :! ii P p! iv p p v F : P P F Proof of l 5 i giv i h ppdi 6
7 Th objciv of hi ppr r: w propo h di fucio d d d d P l lriv o h Gu-Wirr di fucio o ovrco curr iu i h Z hor g [8] HM Edwrd 3 w giv h Mlli rfor of 9 which i igrl of qur of rl-vlud fucio blig g h li chiqu fro [] G Gpr o prov h cri Mlli rfor of u of qur hv ol rl zro 3 w kch fw opio o prov h RH bd o ppropri choic of h fr prr uig pcific propri of h Bl poloil d h fucio P giv i l g h i dfi diribuio fucio for I ordr o k fir lik o Pol rgu w rcll L A Pol: If i poloil which h ll i roo o h igir i or if i ir fucio which c b wri i uibl w lii of uch poloil h If du u F u h ll i zro o h criicl li o do u du u u F u log u I hi co w lo rcll fro [6] D Bup l Rrk B A opror which k v fucio q d rplc i b q q h h propr of ovig h zro of fucio clor o h igir i d o igfucio of hi opror hould hv i zro o h igir i 7
8 Mi rul Thi cio giv h Mlli rfor of h Bl fucio of d d kid i : J Y W uri h k propri of i l Th proof rp h rfrc o h lirur r giv i h ppdi W u h followig bbrviio [7] GN Wo 3-7 : h h for d : h h ih for L I hold i co ih coh co ih coh coh dd coh ih dd for R R ii d J Y d d wih Y : rc J iii for d h fucio i icrig wih du o d d for rp ih h coh d iv for h fucio i icrig wih du o d d for rp ih h coh d 8
9 9 W cobi l d l i Corollr I hold 3 P d d I ordr o forul h Mlli rfor of w u h oio fro Dfiiio 3 For R w pu : b d d d : 5 co : 3 / L bl h clculio of h Mlli rfor of which w giv i Propoiio For R i hold i / d d d ii B d iii / / / B d whrb 6 / / Proof of Propoiio i giv i h ppdi
10 Applig h rgu for 3 ld o Propoiio 5 For R d R i hold : d d d Proof of Propoiio 5 i giv i h ppdi Applig h Müz forul ppdi o 7 : giv 8 d d d for R W u 5 d 8 o kch fw opio o prov h RH
11 Opio If for R : hr i ppropri ig of : fulfillig rprio i h for l o h criicl li h hr i rprio of h Z fucio rfor of lf-djoi igrl opror which i poiiv dfii h i Thrfor hr i udrlig igfucio/igvlu dfiig corrpodig Hilbr pc which giv h doi of h opror Thi i h Brr cojcur Th fou G idi for R z g z g z wih g z : z i z igh b courpr of : I proof i uig h Hr ur propr o h ulipliciv group of poiiv rl ubr R lo [8] HM Edwrd for h corrpodig Fourir li chiqu d hdicp i h co of lf-djoi opror d i rfor of o b pplid o d d : wih d c d c d z z : d I hold b rplcig wih d d z z z z z d d d d Echgig h ordr of igrio d rplcig wih d d z z z z d d d d z d d z z z d
12 Opio G Polá ] obrvd h H : 9 / 3 5 / i poicll iilr o d provd h h ol rl zro H 9 coh H : 8 coh iz z H d H co z d iz 9 iz 9 Pol udrlig rgu uig h ifii Wirr produc rprio provd b uig diffrc quio i z for h odifid Bl fucio of h hird kid i L O: If c d G z i ir fucio of gu or h u rl vlu for rl z h ol rl zro d h l o rl zro h h fucio G z ic G z ic lo h ol rl zro Epcill i hold h h fucio F c z : i zic i zic h ol rl zro for c d Uforul z i o pproiio o z [6] E C Tichrh Thi didvg igh b ovrco b lriv fucio z proprl dfid uig i : J Y H grig fucio I [] G Gpr h rli of h zro of z i prov lzig igrl of qur of cri rl-vlud pcil fucio
13 Opio 3 I [] G Polá obid h followig grl hor bou zro of h Fourir rfor of rl fucio: L O3: L b d l g b ricl poiiv coiuou fucio o b d diffribl hr cp poibl fiil poi Suppo h g g vr poi of b whr g i diffribl Suppo furhr h h igrl G : d g I covrg for R Th ll zro of G i hi rip if R Th prr igh b cho ppropril o chiv for g 7 i g : g g W o ppdi h pplig Pol' l o Müz forul 8 ild o iforio bou h locio of o-rivil zro of h Z fucio Opio L O i r-foruld i [6] D Bup l h opror which k v fucio q d rplc i b q q h h propr of ovig h zro of fucio clor o h igir i d o igfucio of hi opror hould hv i zro o h igir i: If o rric o / R : d pu Auig h : : R i hold d d : d h l o zro igh ld o cordicio kig io ccou h i hold d d ih coh d co coh ddd 3
14 Opio 5 Lik h Bl poloil ppropril wih Ruj Mr Thor [] B C Brd Th fir qurrl rpor Thor I i k k F d for F k! i h ighborhood of kig dvg of i propri giv i l 5 Moivio Wih k : k h Ri Hpohi r h Hrd/Lilwood rp h Riz quivlc criri of 5 RH hold if d ol if k k F O / k! k 5 RH hold if d ol if k / O k! k Ruj oivd hi forul wih h followig wordig [] B C Brd chpr Er 8: S: If wo fucio of b qul h grl hor c b ford b ipl wriig id of i h origil hor 3 Soluio: Pu d ulipl i b f h chg o d ulipl f f f rpcivl d dd up ll h rul Th id of w hv f for!! 3! poiiv wll for giv vlu of Chgig f o w c g h rul Epl: rc rc Ruj buildig proc: f rc rc f f f rc rc!! f f rc rc!! Rplc rc z b i Mcluri ri i z whr z i igrl powr of Now dd ll h qulii bov O h lf id o obi wo doubl ri Ivr h ordr of uio i ch doubl ri o fid h f f f Rplc f b o coclud h Of cour hi forl procdur i frugh wih urou difficuli bu h hor w fill corrcl provd b GH Hrd
15 Opio 6 Thr i odifid Z fucio rprio ~ : which c b rlizd ihr i covoluio / i G df G z iu df u or ii Fourir igrl z u z du F u log u u whrb g z : u z F u du u h ll i zro o h criicl li 5
16 6 Appdi L : For d R h followig poic pio hold ru P Proof of L : Fro h pio A ih coh ih!! coh coh giv i [6] GN Wo w g b ubiuig h vribl fir b u ih d h b u h idii ih coh ih coh ih d d u du u u u du u u u d d Fro l w g d rulig io Uig h forul A / /!! i i follow /!! i!!!!
17 W o h pcil c 3 P : for : b! P / : 3 3! for : / Proof of l 3 For N i hold Sirlig forul! c whrb : k k k k c li c li c! li!! Puig z : rp z wih i io h wo forul z d z z i z ld o d i 3 which copl h proof of l 3 Proof of l 5: i P 3 p : follow fro Sirlig forul ii p 3 3 P :! ii P p! follow fro l 3 iv p 3 3 p 3 3 d hrfor v vid p p 7
18 Proof of L i fro [] IS Grdh IM Rzhik 658 w rcll co J Y ih d for R R d fro [7] G N Wo 6-3 h forul ih coh ih coh d which giv Nu-Nicholo igrl i h for co ih coh coh dd Wih i pcill hold coh d / ih coh ih d dd d / / ih d ih d ih ih d ih i ih / d ih ii d J Y Thi rlio o Hkl fucio d d wih Y : rc J H : J iy R i wih R : J Y H Y : rc J d Y J Y J J Y J i giv b [6] GN Wo / Y
19 d J Y d iii iv r giv [7] GN Wo 3-7 d To prov propoiio 3 w will u L A I hold i ih d ih d B for R ii for R coh d B coh iii d B coh for R ih iv d B for R R coh v co i d coh / for / vi i co d B for R R Proof of l A Fro [] IS Grdh IM Rzhik 35 & w g i For R d R i hold ih d B coh ii For R d b i hold coh b d coh b b B Puig d rul io R d ih which giv L A i d co B 9
20 Puig d giv L A ii Puig b : : : giv R R d l iii iv i giv i [] B C Brd qurrl rpor 33 v Fro [] IS Grdh IM Rzhik 36 w g / i co d B R R Proof of Propoiio i i giv i [7] G N Wo 3-7 ii Applig h vribl ubiuio ih coh i ih coh i follow co ih coh coh dd d co ih coh coh dd d d d co ih coh coh dd co ih d coh coh d d co coh coh d co coh co coh d B d coh coh Wi : : : d R l iii i hrfor hold co B B ii wih ih / d i follow ih ih d / d / / d ih ih ih d / ih d d / d / ih ih d d / ih / ih d d
21 / / / ih ih d d d d Wih l A i follow for R 3 / 3 / B d B d Applig h forul / / h giv iii Proof of Propoiio 5 coh h ih d d d d d coh h ih d d / d h h coh h ih / 6 d ih coh ih ih coh coh / 6 d ih coh ih coh ih coh / 6 d ih coh ih ih coh ih coh / 6 d ih ih ih ih coh / 6 d 3 coh ih ih coh ih / 6 d d 3 3 coh ih d coh ih coh ih ih coh ih ih 3 3
22 ih 3 ih ih 3 3 d coh d coh L Müz forul For coiuou d boudd i fii irvl wih o d o for d i hold d d d for R Proof: i bcu o i hold i coiuou d boudd i fii irvl wih d i for i h ivrio ldig o h lf hd id of 3 i juifid ii d d O d O d O Th fir ud i juifid bcu h cod ud i juifid bcu o i i hold / / i coiuou d boudd i fii irvl c O wih c d Hc d c d d c for cp Alo c c d for d hrfor 3 for R
23 L Pol cririo A rl lf-djoi opror of h for which h h propr h f / z d z G i G rl d G G G i o-dcrig o h li zro of i rfor ll li o h li R z Pol' cririo i giv i [8] HM Edwrd 5 h h propr h h Rrk Th l bov i vlid for igrl ovr fii irvl d o d i o ifii irvl i d cri codiio I ordr o ppl Pol' cririo dircl o Müz forul d d d for R for coiuou d boudd i fii irvl wih o d o for d o d o how h h fucio P d i poiiv d icrig for Howvr P co b poiiv d icrig i h whol rg for bcu ohrwi i vlu ifii would b poiiv d o i h c bcu Muz' forul rquir o vih ifii o ordr wih Hc h fucio P vih o l ordr d i priculr i h h vlu ifii Thrfor h prio i h for co b boh poiiv d icrig r ifii i Pol' cririo i for of l vr ppli o forul of Muz' p L Thor of Frulli L f b coiuou igrbl fucio ovr irvl A B Th for b f f b f F d log b whr f li f for d f li f for W io grlizio of hi l du o Hrd Qur J Mh 33 9 p 3- i h for b p d log 3
24 Rfrc [] B C Brd Ruj Nobook Sprigr-Vrlg Nw York Brli Hidlbrg Toko 985 [] Brr M V & ig J P H p d h Ri zro i Suprr d Trc Forul: Cho d Diordr Ed IV Lrr JP ig DEhliki luvr Nw York 999 pp [3] Ph Bi J Pi M Yor Probbili lw rld o h Jcobi Th d Ri Z fucio d Browi curio Bull Ar Mh Soc 38 p [] E Bobiri Rrk o Wil qudric fuciol i h hor of pri ubr I Rd M Acc Licri [5] R P Br A poic pio ipird b Ruj Aur Mh Soc Gz [6] D Bup - Choi P urlbrg J Vlr A Locl Ri Hpohi I Mh Zi 33 pp -9 [7] D A Crdo Covoluio Opror d zro of ir fucio o b pprd i Proc Ar Mh Soc rcivd b h dior Jul 999 [8] G Doch D Eulr Prizip Rdwrprobl dr Wärliughori ud phiklich Duug dr Igrlglichug dr Thfukio Ali dll Scuol Norl Suprior di Pi Cl di Sciz ri o o pp 35-3 [9] H M Edwrd Ri Z Fucio Dovr Publicio Ic Miol Nw York 97 [] G Gpr Uig u of qur o prov h cri ir fucio hv ol rl zro i Fourir Ali: Alic d Goric Apc W O Br P S Milojvic d C V Sojvic d Mrcl Dkkr Ic [] IS Grdh IM Rzhik Tbl of Igrl Sri d Produc Fourh Ediio Acdic Pr Nw York S Frcico Lodo 965 [] H Hburgr Übr iig Bzihug di i dr Fukiolglichug dr Rich -Fukio äquivl id Mh A 85 9 pp 9- [3] Li X-J Th Poiivi of Squc of Nubr d h Ri Hpohi J Nubr Th pp [] H Mlli Abri ir ihilich Thori dr G- ud dr hprgorich Fukio Mhich Al 68 9 pp [5] N Nil Hdbuch dr Thori dr Gfukio Lipzig BG Tubr Vrlg 96 [6] R Pro Th Rod o Rli Alfrd A opf Nw York 5 [7] B E Pr Iroducio o h Fourir Trfor & Pudo-diffril Opror Pi Publihig Liid Boo Lodo Mlbour
25 [8] G Polá Brkug übr di Igrldrllug dr Rich Fukio Ac Mhic 8 96 pp [9] G Pol Ubr rigoorich Igrl i ur rll Nullll Jourl für di ri ud gwd Mhik pp 6-8 [] G Pol Ubr di Nullll gwir gzr Fukio Mh Zi 98 pp lo Collcd Ppr Vol II [] B Ri Ubr di Azhl dr Prizhl ur ir ggb Grö Mobrich dr Brlir Akdi 859 pp [] M Riz Sur l'hpohè d Ri Ac Mhic 96 pp85-9 [3] P Srk Spcr of hprbolic urfc Bull Ar Mh Soc 3 pp - 78 [] M du Suo Wh uolvd Probl i Mhicl Mr Fourh E Lodo [5] E Schrodigr Spc-Ti Srucur Pr Sdic of Uivri of Cbridg Cbridg-Nw York Mlbour rprid [6] E C Tichrh Th Thor of h Ri Z-fucio Oford Uivri Pr Ic Nw York fir publihd 95 Scod Ediio 986 [7] G N Wo A Tri o h Thor of Bl Fucio Cbridg Uivri Pr Cbridg Scod Ediio fir publihd 9 rprid
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