The Geometric Proof of the Hecke Conjecture

Transcription

1 The Geometc Poof of the Hecke Cojectue Kada Sh Depatmet of Mathematc Zhejag Ocea Uvety Zhouha Cty 6 Zhejag Povce Cha Atact Begg fom the eoluto of Dchlet fucto ug the e poduct fomula of two fte-dmeoal vecto the complex pace the autho poved the afflg polem---- Hecke cojectue Keywod: Dchlet fucto Hecke cojectue o-tval zeoe old of otato axco ecto ay-cetc coodate e poduct fte-dmeoal vecto MR: M Itoducto Fo vetgatg the dtuto polem of pme ume wth the athmetc ee ft Dchlet mpoted the Dchlet fucto Although t popety ad effect mla wth Rema Zeta fucto ζ ut the dffeet matte the eeach aout the dtuto of the eal zeo of the Dchlet fucto whch coepod eal popety a vey dffcult thg But ecaue of th t ha vey mpotat meag Fo olvg aove-metoed polem Hecke put fowad a famou popoto a follow: Hecke Cojectue: Suppoe that a eal popety the Now let pove the popoto < < The eoluto of Dchlet fucto Codeg the geometc meag of the Dchlet fucto the followg fgue:

2 w C A B A B A B O D D D z Fg I th fgue the aea of the ectagle ADOC ADDB ADDB ae epectvely: theefoe the geometc meag of the Dchlet fucto the um of the aea of a ee of ectagle wth the complex pace Ug the e poduct fomula etwee two fte-dmeoal vecto the Dchlet fucto euato t ca e eolved a t t t t o t t t t Fom the expeo ad we ota:

3 t co N N ad co N N t 5 But the complex pace f the e poduct etwee two vecto eual to zeo the thee two vecto ae pepedcula amely ad π N N π N N theefoe co N N co N N The elatohp etwee the volume of the otato old ad the aea of t ax-co ecto wth the complex pace Fom the expeo we ca kow that whe Re the ee hamoc ee wth the ft adcal expeo of left de dveget Becaue co N N theefoe the tuato of the complex ee wth the ecod adcal expeo of left de wll chage uale to eeach Hece we mut tafom the Rema Zeta fuctoζ euato Fo th am let deve the elatohp etwee the volume of the otato old ad the aea of t ax-co ecto wth the complex pace We call the co ecto whch pa though the ax z ad teect the otato old a ax-co ecto Accodg to the aycetc fomula of the complex plae lama:

4 ξ η a a a z[ f z g z] dz [ f z g z] dz a [ f z g z] dz [ f z g z] dz πη π a a [ ] f z g z dz [ f z g z] dz 6 The umeato of the facto of ght de of the fomula ** the volume of the otato old ad the deomato the aea of the ax-co ecto Theη of left de of the fomula 6 the logtudal coodate of the aycete of ax-co ecto It geometc explaato : takg the ax-co ecto aoud the ax z to otate the agle of π we wll ota the volume of the otato old The poof of the Hecke cojectue ADD Becaue the volume fomula of the otato old fomed y otatg the ectagle B aoud the ax z: π π π V dz z theefoe the um of the volume of a ee of the cylde fomed y otatg a ee of the ectagle ADOC ADDB ADDBaoud the ax z : V V π But the um of the aea of a ee of the ectagle ADOC ADDB ADDB:

5 By the fomula 6 π πη Suttutg the Dchlet fucto euato to we ca ota the tafomed euato: Th ca e explaed geometcally a: f the aea of the ax-co ecto eual to zeo the t coepod to the volume of the otato old alo eual to zeo Becaue theefoe aove expeo ecome Accodg to the e poduct fomula etwee two fte-dmeoal vecto aove expeo ca e wtte a: co N N t t t t I the expeo whe t although the ee ut ecaue the ee l 5

6 6 l co theefoe the complex ee ] l [co l Becaue fte multpled y zeo doe t eual to zeo theefoe O the othe had fom the expeo : co N N t t t t A kow to all amog the ee: l l l l we ca kow Accodg to the chaacte of the popety alo theefoe the eal popete ae fte Hece we ca ota:

7 co log log theefoe So poved that whe the eal popety the < < Refeece Pa C D Pa C B Bac of Aalytc Nume Theoy Bejg: Scece Pe pp- pp-6 ou S T Wu D H Rema hypothe Sheyag: aog Educato Pe pp5-5 Che J R O the epeetato of a lage eve tege a the um of a pme ad the poduct of at mot two pme Scece Cha 6 No pp - Pa C D Pa C B Goldach hypothe Bejg: Scece Pe pp- pp- 5 Hua G A Gude to the Nume Theoy Bejg: Scece Pe 5 pp-6 6 Che J R Shao P C Goldach hypothe Sheyag: aog Educato Pe pp- pp-5 Che J R The Selected Pape of Che Jgu Nachag: Jagx Educato Pe pp5- ehma R S O the dffeece πx-lx Acta Ath 66 pp~ Hady G H ttlewood J E Some polem of patto umeoum III: O the expeo of a ume a a um of pme Acta Math pp- Hady G H Ramauja S Aymptotc fomula comatoy aaly Poc odo Math Soc pp 5-5 Rema B Uee de Azahl de Pmzahle ute ee gegeee Goße Ge Math Weke ud Wechaftlche Nachlaß Aufl 5 pp 5-55

8 E C Ttchmah The Theoy of the Rema Zeta Fucto Oxfod Uvety Pe New Yok 5 A Seleg The Zeta ad the Rema Hypothe Skadavke Mathematke Koge 6 Appedx The e poduct fomula etwee two vecto eal pace ca e exteded fomally to complex pace Dea M Refeee Thak you fo you evew pleae oeve followg example Example A{ } B{ } C{ } { } { } { } 6 Accodg to Coe theoem co O the othe had 6 co

9 Accodg to the Coe theoem co 6 O the othe had co 5 θ S A 5 θ S B Example A{ } B{ } C{ } { } { } { } Accodg to the Coe theoem co O the othe had co Accodg to the Coe theoem

10 co 6 O the othe had 6 co 6 S A θ 6 6 S B θ Example } { } 5 {6 } { C B A } 6 { } { } 5 { Accodg to the Coe theoem co O the othe had

11 co Accodg to the Coe theoem co O the othe had we had 6 co S A θ S B θ