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1 UD 5 The Geeralized Riema' hypohei SV aya Khmelyy, Uraie Summary: The aricle pree he proo o he validiy o he geeralized Riema' hypohei o he bai o adjume ad correcio o he proo o he Riema' hypohei i he wor [], obaied by a iie expoeial ucioal erie ad iie expoeial ucioal progreio Keyword: Riema' hypohei,aural erie, ucio o öbiu, ere ucio, iie expoeial ucioal erie, iie expoeial ucioal progreio Iroducio I hi paper we give he proo o he geeralized Riema' hypoheio he bai o adjume ad correcio o he proo o he Riema' hypohei o he zea ucio, which wa uderae i [], a well a value o peciied i o he coicie The paper alo provide a reuaio o he hypohei o ere The ormulaio o he problem (Geeralized Riema' hypohei) All o-rivial zero o Dirichle ucio L, have a real par ha i equal o )For hi we ir i will prove he Riema' hypoheior he zea ucio The oluio For he coirmaio o he Riema' hypohei we will give he deiiio ad prove he ollowig heorem DeiiioThe expreio [ ] ( ) u( ) ( ) ( ) [ ] ( ) () called iie expoeial ucioal erie wih repec o he variable expoe,,,, where DeiiioThe progreio o he ype ( ) ( ) ( ) ( ) ( ) ( ) ( ) ()

2 i called iie expoeial ucioal progreio, i heir ir member a i a ucio o x or i equal o ad he deomiaor q i a ucio o he variable x x Theorem I he e o aural umber,,,,,,,, 5, 6 i he uio o ube, ad hee ube are dijoi ad have appropriaely or m, r,,, v, u he eleme, he umber o eleme o he e o 5 equal m r v u ProoThe heorem i proved imilarly o he heorem 7, [,p50] Theorem o a erie Proo Le he umber,,,, The poiive ieger equal, 5 be he quaiy o eleme o he e o aural umber coi: o ; prime, he quaiy o which i deoed m by П П() ; o aural umber, which are divided o p where quoie diere rom wih m, he umber o which i deoed by К к к К ; o aurel umber which are decompoed o produc o dual accumulaio acor, le i deoe he umber o hee cogruou umber hrough T T umber o prime, will be deoed by e o aural umber, ad he amou o umber, ha ca be covered io produc o upaired T T, accordig o he heorem, equal The umber o he aural erie expoeial ucio erie The he amou o aural umber, o he к П Т Т K К () approximaely ca be expreed a a iie () We will wrie he umber o aural umber ucio erie coiig o he ir he K, approximaely a iie expoeial member o he erie (), we deoe i a, (5)

3 I he um o he erie () each erm o he erie i ae, a he amou o aural umber For example, ; 00 ;;;;5;6;7;8;9;0 ; 00 ;;; A erie poiive ieger ad o o i greaer ha he erie o he aural umber o Deiiio The aural umber ha overlap, are called he iie expoeial ucio erie o (5), i which hey occur more ha oce Le ha ad K, becaue i he ucio excep he umber K which overlap Deiiio Two iiiely grea are called equivale i ad The ucio are icluded he umber, which are o equal o oe aoher whe 0 We aume ha he e o aural umber,,,,, wih algebra A,,,,0, i vecor pace P I hi pace we e he adard he e o umber a vecor pace P wih adard,,, wih algebra x max, ad A,,,,0, will be aumed x max The, he deomiaor o a expoeial ucio o iie progreio, which operae i he pace P will ae a q, ad he deomiaor o a expoeial ucio o iie progreio, which operae i he pace P, a q The iie expoeial ucioal erie () i approximable by he um o he iie expoeial ucioal progreio (6)

4 Propoiio The iie expoeial ucioal erie () ad he um o he iie expoeial ucioal progreio (6) are equivale ProoTo prove he equivalece o he iie expoeial ucioal erie wih he iie ucioal progreio he um o he ucioal erie i wrie i he orm o, The le u wrie ha Thereore, i accordace wih he deiiio, he ucioal erie ad ucioal progreio will be equivale Propoiio i proved Propoiio The iie expoeial ucioal erie() ad he um o he iie expoeial ucioal progreio (6) S are equivale ProoThe um ohe iie expoeial ucioal progreio ca be calculaed by he ormula S (7) ad he he i will be equal o:

5 5, Where ad 0 [5,p67] Thereore, i accordace wih he deiiio, he ucioal erie () ad he iie um S o he ucioal progreio (6) a Propoiio i proved, whe,will be equivale From he expreio () ad (7) oe ca ee ha i wihi i o S S The um o he iie expoeial ucioal progreio (6) wih q Whe, S We will compare he ucio We id ucio S Thereore, S equal wih he Lemma The umber o aural umber ha overlap i le ha,5

6 6 Proo To prove hi propoiio le u deoe hrough K i he umber ha occur more ha oce i he iie expoeial ucioal erie () whe, ad ue he expoeial ucioal erie The erie (5) i ae i hi orm o be becaue i iclude all umber ha overlap Thi ollow rom he expreio, l l Two i ae becaue i i he malle prime umber ha ca o be decompoed io prime acor The iie expoeial ucioal erie (5) will be replace by he um o iie expoeial ucioal progreio (8) Propoiio The iie expoeial ucioal erie (5) ad he iie expoeial ucioal progreio(8) are equivale ProoThe ucioal erie () ca be wrie a, ad he ucioal progreio (6) i a Dicard he ir erm o he erie ad progreio, we id ha, or We how ha ad

7 7 are equivale: I ollow ha he iie expoeial ucioal erie ad he iie expoeial ucioal progreio are equivale Propoiio i proved To prove he heorem, we iroduce he ucio erie 6, (9) where ={ [ ] } ad ucioal progreio 8 (0) I we exprehe erie (5), a a erie, ha he erie (8) i ae 6 i uch orm o ha each eleme o he erie (8) overlap he each eleme o he erie (5) wih upaired expoe o he roo Ad he we ca wrie ha 5 6 () Hece he amou o umber ha cover more umber ha overlap Propoiio The iie expoeial ucioal erie (9) ad he iie expoeial progreio (0) are equivale

8 8 ProoTo prove he equivalece o he iie expoeial ucioal erie wih he iie expoeial ucioal progreio i he orm o he relaio Thereore, i accordace wih he deiiio, he ucioal erie (9) ad a ucioal progreio (0) are equivale Propoiio i proved Propoiio The iie expoeial ucioal erie (9) ad he iie um S o he expoeial ucio progreio (0) are equivale whe ProoThe um o ucioal erie i more ha, ad he um o ucioal progreio i coidered, a he um o he ucioal progreio wih q The we id ha S I order o calculae he ucio al, le u e or, ad he we obai Sice he 8 Ad he we will have 8 S, () Uig he deiiio we will have

9 Thereore,a ucio o erie (9) ad he um S o ucioal progreio (0) are equivale Propoiio i proved From he expreio (0) ad () i i clear ha 9 i wihi S The we compare ucio wih he ucio whe ad we obai 8 8 Hece, we have ha, or 0 erie Thereore, Le u ae io accou he value o he iie expoeial ucioal, ad wrie ha K Lemma i proved The we ca wrie ha Uig he iequaliy K I we ubiue value o he ucio Uig Lemma, we obai 0,, () we will wrie ha K K () () io (), we obai K K 0

10 0 K 0 Hece; we id ha K The value rom () i ubiued iead, we obai,whe к,5 K П Т Т К K к П Т Т К,5 П Т Т К к, or (5) The we ca wrie ha appropriaely o he properie o he ucio o obiu- ;,where i he amou o prime acor o he umber p p p ad 0 We wrie ha whe i muliple m p or m, П Т Т П Т T,5 (6) T П Т The he expreio(6) ae he orm Thereore The heorem i proved, 5 0,5 (7) ) For he ere ucio you ca id a more precie eimae equal Lemma The accurae aeme i a erie will be

11 ,5,5 ProoI order o id a more accurae eimae ha 0,5 he um o he iie expoeial ucioal erie (5) For ha we ue he ucioal progreio (6), le u id, he we obai ha,5, 5 We id rom he expreio (0) ha he quaiy o umber ha overlap i le ha becaue 0,5 5 ad he expreio Uig he mehod give deermie 5,5,5 we obai ha Hece, we id ha he upper i o he value ucio will be he value up 0,5, ad he lower i i i 0,5 Thereore, he evaluaio 0,5,5 eimae ha 0,5 Lemma i proved whe i a more accurae

12 ) The heorem prove ha he upper i value o he ucio ad he lower i i up i Propoiio 0,5 0,5 0,5 whe,, equal O ProoAccordig o he heorem 5 [7p] we have ha O 0,5 i compared wih O, we will wrie ha The value l 0,5 0 0,5 whe Hece we id ha 0 whe l, 0 0 Thereore,we ca aume ha 0, where i a radom mall umberad here we id ha 0,5 Propoiio i proved Theorem The erie 0,5 whe coverge i where i a radom mall umber orollary o Theorem (he Riema' hypohei) All o-rivial zero o he zea ucio have a real par equal o Proo A eceary ad uicie codiio or he validiy o he Riema' hypohei i he covergece o he erie he erie, whe whe O 0,5 ad ad [7,p] We id he covergece o

13 0,5 0,5 he erie diverge Ad whe we have 0,5 0,5 0,5 0,5 d he erie coverge,where ε i a arbirary mall umber Thereore, he erie coverge uiormly or,ad ice i i a ucio i, or he heorem o aalyic coiuaio, i i alo a i Thereore, he Riema' hypohei i rue The heorem i proved ) A deermiaio he value o coeicie The we ca wrie ha accordig o he properie o obiu ucio -, he ;,where he umber o prime acor o he umber p p p ad 0, whe i he muliple o m p or m ha 0,5 Т П Т Т П T п, The

14 0,5 ; 0,5 From he expreio К 0 be wrie ha,5 Ad rom he expreio 0 we id ha,uig he properie o öbiu ucio, i ca Thi coicide wih he reul [] The we ca id he exe o which he coeicie i locaed From he double iequaliy 0,5, 5 Ad here we id ha,5 Uig a more precie value М, we id ha 0,5,5, 5, we id ha Ad here we id ha 5 ad rom he double iequaliy we id ha he coeicie с will be i he rage,5, 5) Theorem All o-rivial zero o Dirichle ucio L, i equal o Proo Le' coider he Dirichle' erie where i he characer o modulu m have a real par ha L,, i, (8) There i m o uch erie where i he Euler' ucio Sice, he erie (8) coverge whe, a ca be ee rom a compario o hi erie wih he erie

15 5 We deoe i by he um hrough erie L, diere ucio L, For variou characer, we obai They are called L i he Dirichle' ucio I udyig he properie o hee ucio i i coveie o diiguih he cae where i he mai characer ad whe a) I ha he erie (8) coverge i he emiplae 0 Le u how rom he begiig, ha he parial um x ox io clae o deducio by are iedwe divide he ieger umber rom mod m ad wriex mq r, 0 r m The we have x x m m mq mq r m m mq Becaue o he orhogoaliy relaio m, ha 0, ha mod m x mqr mq, hece x mqr mq r m erie Sice a 0 decreae moooically ad ed o zero whe / coverge or real 0 0 whe I, however, 0, he he, ad, coequely,or all i he emiplae, he hi he erie obviouly diverge I abcia coverge 0 0 ad he abcia o abolue covergece By he heorem, "The Dirichle' erie a i he emiplae o he covergece i a regular aalyic ucio rom

16 6, he ucceive derivaive o which are obaied by he erm diereiaio o hi he erie [8, p5],he ucio, b) I we ue L, i a regular aalyic ucio rom whe 0 L,, Re (9) mod m, he From he heorem i ollow ha L, 0 whe I i he mai characer by Uig he codiio L,, he a, m a 0, he a, m he ucio (9) ca be wrie a, whe = ad Uig he reul o he heorem, i ca be argued ha he geeralized Riema' hypohei i rue, ad accordigly o i: "All o-rivial zero o he Dirichle' ucio have a real par equal o " The heorem i proved Appedix The proo o he Riema' hypohei ad he cojecure o Birch ad Suieroa-Daeyr wihou coeicie С A beer proo o he Riema' hypohei ad he proo o he cojecure o Birch ad Suieroa-Daeyr ad a a reuaio o he ere hypohei, ca be oud o he bai o hi wor ad whe we ae he erie o

17 7 ad i ca be wrie i he orm,5,5,5,5, where The we ca wrie,5, 5,5,5 0 or,5,5 (0) The value rom he expreio i ubiued iead o, we obai П Т From he expreio (0) we obai п Т К к К п п,5,5 п к П Т Т К К () The properie o obiu ucio [, p ] will be applied o he expreio () ad we obai ha or,5,5,5,5 I will be he malle value o he ucio o ere ucio o ere From he expreio 6 ad he bigge value or he

18 8 we id ha 0, () whe ad We wrie he expreio i he orm п п к П Т Т К К () Le u apply he properie o obiu ucio o he expreio ad we obai The we ca ae ha he ucio o ere i wihi,5,5 Ad i rejec he hypohei o ere Ad he Riema' hypohei repecively by he heorem i he rue Ad he coeicie i he paper [, p ] will be equal, 67 Wha clariie he proo o he cojecure o Birch ad Suieroa-Dyer Jule, 05 Reerece [] SVaya Proo o he Riema' hypohei // arxiv:0587 [mahg], Apr 0 [] SVayaThe proo o he correce o he Birch ad Swiero Diyer cojecure//arxiv:0670 [mahg],6 Ju 0 []A Odlyzo ad Herma e Riele Diproo o he ere ojeure Joral ur die reie ud agewade ahemai 57 (985) pp 8-60

19 9 [] YS Lyapi, AY Yeveyev Algebra ad heory o umber, p umber Educaioal boo or ude o he aculie o phyic ad mahemaic o eachig college : " Provehcheiye", 97, 8 p [5] G Fihegolc Diereial ad iegral calculu coure V :"aua", 969, 607 p [6] LY Kuliov Algebra ad heory o umber Wor-boo or eachig college :Vyhaya hola, p [7]Y K Tichmarh Riema zeaucio : IL, 97, 5 p [8]K hadraehara Iroducio oaalyic umber heory : ir,97, 87p ma@yadexua