a note on the fourth power moment of the riemann zeta-function

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1、ANOTEONTHEFOURTHPOWERMOMENTOFTHERIEMANNZETA-FUNCTIONJ.B.ConreyAbstract.WegiveexplicitformulaeforallofthetermsintheasymptoticexpansionofthemeanfourthpoweroftheRiemannzeta-functiononthecriticalline.Heath-Brownhasevaluatedthe4thpowermomentoftheRiemannzeta-functiononthecriticallineviatheformulaZTT

2、47=8+I(T)=j(1=2+it)jdt=TPlog+O(T)2420wherePisa4thdegreepolynomial.HerewegiveanexplicitformulaforP.In44particular,weshowthatthecoecientsofPareintheeld4000Q(;;;;(2);(2);(2))0123wherethe'sarethecoecientsintheLaurentexpansion1(s)=++(s1)+:::01s1of(s)ats=1.Theorem.WehaveP(x)=g(x)+g(x)401w

3、heres42x(s+1)g(x)=Res0s=0s(s+1)(2s+2)and2112s20(xe)((s+1)(2s+1)(2s+2))d2sg(x)=:1ds(s+1)(s+2)s=0IvichasalsowrittendownformulaeforthecoecientsofP.4Proof.WerefertoHeath-Brown'spaper.ThereheshowsthatPisnaturally4expressedasasumofg+g.Thersttermgarisesfromthediagonalterms010Researchsupport

4、edinpartbyagrantfromtheNSF.TypesetbyAS-TXME12J.B.CONREYandisexactlyasabove.Theotherpiecearisesfromtheodiagonaltermsanditistheexplicitcalculationofthispartthatisthepurposeofthispaper.FollowingHeath-Brown,wewriteXd(n)d(n+r)=m(x;r)+E(x;r)nxwherem(x;r)=xisapolynomialofdegree2inxwithcoecientsdepen

5、dingonr.Infact,sxm(x;r)=ResD(s;r)s=1swith1Xd(n)d(n+r)D(r;s)=:snn=1ThenZ1T=(2)XT0g=2m(x;r)sin(Tr=x)dx:120r=1Lemma.Withtheabovenotation,1XX2(s)xm(x;r)=log+21+10222sdss=1djrand1XX2(s)x0m(x;r)=log+20222sdss=1djrProof.AccordingtoHeath-Brown,m(x;r)maybecalculatedbyconsideringthemaintermsi

6、nXX1=22R(x;q;r)R(qx;q;r)1=21=2qxqxwhere2XXxxR(x;q;r)=d(q=d)log+21:022qqdj(q;r)jq=dWritingq=sdwehaveXXXX1(s)x1R(x;q;r)=xlog+21:0222dsds1=21=21=2djrqxsx=dx=sdUsingX1=logy++O(1=y)0yANOTEONTHEFOURTHPOWERMOMENTOFTHERIEMANNZETA-FUNCTION3(andwenoteXlog12=logy+O(logy=y);12yf

7、orfuturereference)wendthattheabovesumisXXx1(s)xx=log+21log+200222222dsdsds1=2djrsx=d01XX11=2@A+Ox:s1=2djrsxExtendingthesumoversto1theaboveis1XX(s)xx1=2=log+21log+2+O(xlogxd(r)):0022222sdsdss=1djrSimilarly,usin