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MomentsoftheRiemannZetaFuntionandEisensteinSeriesIJenniferBeinekeandDanielBumpJune19,2003AbstratItisshownthatiftheparametersofanEisensteinseriesonGL(2k)arehosensothatits(integrated)L-funtionisthe2k-thmomentoftheRiemannzetafuntion,thenthe 2kk termsinitsonstanttermagreewith 2kk fatorsappearinginaonjeturalformulaforthe2k-thmomentofzetabyConrey,Farmer,Keating,RubinsteinandSnaith.Whenk=1,anexplanationforthisphenomenonisfoundbydedu-ingOppenheim’sgeneralizationoftheVorono summationformulafromtheEisensteinseriesandrepresentationtheoretionsiderations.Thepossibilityofeliminatingtheproblematial\arithmetifatorisdisussed.AMSSubjetClassi ation:Primary11M06,Seondary11F55and11F12.Thereisreasontoexpetthatthe2k-thmomentoftheRiemannzetafuntionanberelatedtothespetraltheoryofGL(k)orGL(2k).TheworkofMotohashi[27℄supportstheideaofseekingsuhanapproah,by ndinganexpliitformulaforthefourthmomentof involvingspeialvaluesofL-funtionsofMaassuspformsforSL(2;Z).Stillanautomorphiattakonthehighermomentsofthezetafuntionhasprovedanelusivegoal.ReentlyConrey,Farmer,Keating,RubinsteinandSnaith[9℄gaveonje-turalasymptotisforthehighermoments.TheseonjeturesaresupportedbyheuristisfromRandomMatrixTheoryandAnalytiNumberTheoryandbynumerialomputation.Theyarealsoimpliedbyanindependentonje-tureofDiaonu,GoldfeldandHo stein[11℄.Wewillarguethatthesereent1onjeturesprovideluesastohowsuhanautomorphiattakmightbeformulated.Infat,wewillargueforaloseonnetionbetweenthe2k-thmomentofzetaandanEisensteinseriesonGL(2k).Oneitisunderstoodthatsuhaonnetionmayexist,evenfortheseondmoment,itisnotimmediatelylearhowthelassialresultsanberelatedtotheEisensteinseriesonGL(2).Thepurposeofthispaperistopresenttheevideneforalinkbetweenthe2k-thmomentandtheEisensteinseriesonGL(2k),andtoestablishasolidbasisforthisonnetionwhenk=1.Theseondandfourthmomentsof arewellunderstood.Beyondthefourthmoment,therearereentonjetures,beginningwiththatofConreyandGhosh[10℄.AlthoughthemomentofgreatestinterestisZT0 12+it 2kdt;(1)reentauthors,inludingMotohashi[27℄andConrey,Farmer,Keating,Ru-binsteinandSnaith[9℄haveemphasizedthatitisbettertoonsideranintegralsuhasZT0 ( 1+it) ( k+it) ( k+1 it) ( 2k it)dt;(2)sinetheasymptotisofsuhamomentrevealastruturenotapparentin(1).Iftheasymptotisof(2)areknown,thentheasymptotisof(1)anbededuedasalimitingase.Theauthorsof[9℄foundthatthedominanttermsin(2)are 2kk innum-ber,andeahinvolvesaprodutofk2zetafuntions.WewillshowthatthisidentialstrutureisexhibitedintheonstanttermofaertainEisensteinseriesonGL(2k).Beginningwiththeseondmoment,Ingham[16℄provedthatif0 1and 6=12thenZT0j ( +it)j2dt= (2 )T+(2 )2 12 2 (2 2 )T2 2 +O(T1 log(T)):(3)WemayomparethiswiththeonstanttermofthelassialEisensteinseriesonSL(2;Z),E (z)=12 (2 )X(;d)=1 yjz+dj2 ;z=x+iy;y0:2Theseriesisonvergentifre( )1buthasmeromorphiontinuationtoall .ThisEisensteinseriesisrelevantto(3)beauseitsL-funtionisL(s;E )= s+ 12 s +12 ;soL 12+it;E = ( +it) (1 +it)= (1 +it)j ( +it)j2;where (s)= s 1=2 1 s2 s2 1.Ontheotherhand,theonstanttermZ10E (x+iy)dx= (2 )y + 2 1 (1 ) ( ) (2 2 )y1 :(4)We ndthatiftheEisensteinseriesisseletedsothatitsL-funtionmathestheintegrandontheleftsidein(3),thenthezetafuntionsinthetwoomponentsofitsonstanttermmaththetwotermsontherightsideof(3).Assumingtheonjeturalasymptotisin[9℄,wewillshowinSetion1thatthisphenomenonextendstothe2k-thmoment.Forexampleinthefourthmomentof thelargesttermsaresixinnumber,eahaprodutoffourzetafuntions.ThesemaybeseenintheanalysisinSetion1.7of[9℄oftheresultsofMotohashi[27℄.WewillshowthatthereexistsanEisensteinseriesonGL(4)whoseL-funtionmathesthefourthmoment,andwhoseonstanttermZ10Z10Z10Z10E0BB0BB10xy01zw001000011CCA;s1CCAdxdydzdwonsistsofsixterms,eahinvolvingaprodutoffourzetafuntions,whihmaththesixtermsontheright-handsideof(1.7.6)in[9℄.AndwewillhekthatthissamepreiseorrespondeneworksforallkbyexhibitinganEisensteinseriesonGL(2k)whoseL-funtionandonstantterm,asumof 2kk produtsofk2zetafuntions,bothmathperfetlythe2k-thmomentanditsonjeturedasymptotis.Thereisoneaspettothisorrespondenewhihremainsproblematial.Thisisthearithmetifatorwhihoursintheonjeturalasymptotisof[9℄.WewilldisussthearithmetifatorbelowinSetion2.3SofartheonnetionthatwehavedesribedbetweenmomentsandEisen-steinseriesappearsasasimpleoinidenebetweendataassoatedwiththeEisensteinseriesanddataassoiatedwiththemoments.TheomplexityofthisdataissuÆientthatwedonotbelieveitpossiblethatitisoiniden-tal.HoweverourasewillbestrengthenedbyexhibitingadiretonnetionbetweentheseondmomentandtheEisensteinseriesE .Thisonnetionomesaboutthroughageneralization,duetoOppen-heim[28℄,ofthefamousVorono [31℄summationformula.LetusstateOp-penheim’sformulainasmoothedversion.Ifa2Clet a(n)bethelassialdivisorfuntion,andlet a(n)=Xdjn dn=d a= 2a(n)n abethesymmetrialdivisorfuntion,so a= a.Let beaontinuousfun-tionwithompatsupportin(0;1).IntermsofstandardBesselfuntions(Watson[32℄)lets(y)=Z10 (x)[ 2 os(s )J1 2s(4 pyx) 2 sin(s )Y1 2s(4 pyx)+4sin(s )K1 2s(4 pyx)℄dx:(5)WewillshowinProposition7thats(y) !0rapidlyasy !1,andwewillprovethefollowingtheorem.Theorem1If ha