Moments of the Riemann zeta function and Eisenstei

MomentsoftheRiemannZetaFuntionandEisensteinSeriesIJenniferBeinekeandDanielBumpJune19,2003AbstratItisshownthatiftheparametersofanEisensteinseriesonGL(2k)arehosensothatits(integrated)L-funtionisthe2k-thmomentoftheRiemannzetafuntion,thenthe2kktermsinitsonstanttermagreewith2kkfatorsappearinginaonjeturalformulaforthe2k-thmomentofzetabyConrey,Farmer,Keating,RubinsteinandSnaith.Whenk=1,anexplanationforthisphenomenonisfoundbydedu-ingOppenheim’sgeneralizationoftheVoronosummationformulafromtheEisensteinseriesandrepresentationtheoretionsiderations.Thepossibilityofeliminatingtheproblematial\arithmetifatorisdisussed.AMSSubjetClassiation:Primary11M06,Seondary11F55and11F12.Thereisreasontoexpetthatthe2k-thmomentoftheRiemannzetafuntionanberelatedtothespetraltheoryofGL(k)orGL(2k).TheworkofMotohashi[27℄supportstheideaofseekingsuhanapproah,byndinganexpliitformulaforthefourthmomentofinvolvingspeialvaluesofL-funtionsofMaassuspformsforSL(2;Z).Stillanautomorphiattakonthehighermomentsofthezetafuntionhasprovedanelusivegoal.ReentlyConrey,Farmer,Keating,RubinsteinandSnaith[9℄gaveonje-turalasymptotisforthehighermoments.TheseonjeturesaresupportedbyheuristisfromRandomMatrixTheoryandAnalytiNumberTheoryandbynumerialomputation.Theyarealsoimpliedbyanindependentonje-tureofDiaonu,GoldfeldandHostein[11℄.Wewillarguethatthesereent1onjeturesprovideluesastohowsuhanautomorphiattakmightbeformulated.Infat,wewillargueforaloseonnetionbetweenthe2k-thmomentofzetaandanEisensteinseriesonGL(2k).Oneitisunderstoodthatsuhaonnetionmayexist,evenfortheseondmoment,itisnotimmediatelylearhowthelassialresultsanberelatedtotheEisensteinseriesonGL(2).Thepurposeofthispaperistopresenttheevideneforalinkbetweenthe2k-thmomentandtheEisensteinseriesonGL(2k),andtoestablishasolidbasisforthisonnetionwhenk=1.Theseondandfourthmomentsofarewellunderstood.Beyondthefourthmoment,therearereentonjetures,beginningwiththatofConreyandGhosh[10℄.AlthoughthemomentofgreatestinterestisZT012+it2kdt;(1)reentauthors,inludingMotohashi[27℄andConrey,Farmer,Keating,Ru-binsteinandSnaith[9℄haveemphasizedthatitisbettertoonsideranintegralsuhasZT0(1+it)(k+it)(k+1it)(2kit)dt;(2)sinetheasymptotisofsuhamomentrevealastruturenotapparentin(1).Iftheasymptotisof(2)areknown,thentheasymptotisof(1)anbededuedasalimitingase.Theauthorsof[9℄foundthatthedominanttermsin(2)are2kkinnum-ber,andeahinvolvesaprodutofk2zetafuntions.WewillshowthatthisidentialstrutureisexhibitedintheonstanttermofaertainEisensteinseriesonGL(2k).Beginningwiththeseondmoment,Ingham[16℄provedthatif01and6=12thenZT0j(+it)j2dt=(2)T+(2)2122(22)T22+O(T1log(T)):(3)WemayomparethiswiththeonstanttermofthelassialEisensteinseriesonSL(2;Z),E(z)=12(2)X(;d)=1yjz+dj2;z=x+iy;y0:2Theseriesisonvergentifre()1buthasmeromorphiontinuationtoall.ThisEisensteinseriesisrelevantto(3)beauseitsL-funtionisL(s;E)=s+12s+12;soL12+it;E=(+it)(1+it)=(1+it)j(+it)j2;where(s)=s1=21s2s21.Ontheotherhand,theonstanttermZ10E(x+iy)dx=(2)y+21(1)()(22)y1:(4)WendthatiftheEisensteinseriesisseletedsothatitsL-funtionmathestheintegrandontheleftsidein(3),thenthezetafuntionsinthetwoomponentsofitsonstanttermmaththetwotermsontherightsideof(3).Assumingtheonjeturalasymptotisin[9℄,wewillshowinSetion1thatthisphenomenonextendstothe2k-thmoment.Forexampleinthefourthmomentofthelargesttermsaresixinnumber,eahaprodutoffourzetafuntions.ThesemaybeseenintheanalysisinSetion1.7of[9℄oftheresultsofMotohashi[27℄.WewillshowthatthereexistsanEisensteinseriesonGL(4)whoseL-funtionmathesthefourthmoment,andwhoseonstanttermZ10Z10Z10Z10E0BB0BB10xy01zw001000011CCA;s1CCAdxdydzdwonsistsofsixterms,eahinvolvingaprodutoffourzetafuntions,whihmaththesixtermsontheright-handsideof(1.7.6)in[9℄.AndwewillhekthatthissamepreiseorrespondeneworksforallkbyexhibitinganEisensteinseriesonGL(2k)whoseL-funtionandonstantterm,asumof2kkprodutsofk2zetafuntions,bothmathperfetlythe2k-thmomentanditsonjeturedasymptotis.Thereisoneaspettothisorrespondenewhihremainsproblematial.Thisisthearithmetifatorwhihoursintheonjeturalasymptotisof[9℄.WewilldisussthearithmetifatorbelowinSetion2.3SofartheonnetionthatwehavedesribedbetweenmomentsandEisen-steinseriesappearsasasimpleoinidenebetweendataassoatedwiththeEisensteinseriesanddataassoiatedwiththemoments.TheomplexityofthisdataissuÆientthatwedonotbelieveitpossiblethatitisoiniden-tal.HoweverourasewillbestrengthenedbyexhibitingadiretonnetionbetweentheseondmomentandtheEisensteinseriesE.Thisonnetionomesaboutthroughageneralization,duetoOppen-heim[28℄,ofthefamousVorono[31℄summationformula.LetusstateOp-penheim’sformulainasmoothedversion.Ifa2Cleta(n)bethelassialdivisorfuntion,andleta(n)=Xdjndn=da=2a(n)nabethesymmetrialdivisorfuntion,soa=a.Letbeaontinuousfun-tionwithompatsupportin(0;1).IntermsofstandardBesselfuntions(Watson[32℄)lets(y)=Z10(x)[2os(s)J12s(4pyx)2sin(s)Y12s(4pyx)+4sin(s)K12s(4pyx)℄dx:(5)WewillshowinProposition7thats(y)!0rapidlyasy!1,andwewillprovethefollowingtheorem.Theorem1Ifha