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1SomeseriesandintegralsinvolvingtheRiemannzetafunction,binomialcoefficientsandtheharmonicnumbersVolumeVDonalF.Connon18February2008AbstractInthisseriesofsevenpapers,predominantlybymeansofelementaryanalysis,weestablishanumberofidentitiesrelatedtotheRiemannzetafunction,includingthefollowing:()121212102(21)(1)()cot()(21)!nnnnBxxdxnπςπ++++=−+∫01()()cos2bbnaapxdxpxnxdxα∞==∑∫∫01()cot(/2)()sin2bbnaapxxdxpxnxdxαα∞==∑∫∫()820111cotlog221222,221168648xxdxGππςπ⎡⎤⎛⎞⎡⎤⎡⎤=−+−+−+⎜⎟⎢⎥⎣⎦⎣⎦⎝⎠⎣⎦∫41()55(4)log(4)32nCinnπςγπς∞=⎡⎤′=+−−⎢⎥⎣⎦∑20(1)(2)240(1)(2)(3)2nnnnnnnπς∞=−=+++∑(1)(2)231111791log2(3)log226203nnnnHHπς∞==−−∑2122log8=+Gπ⎟⎟⎠⎞⎜⎜⎝⎛+∑∞=nnnnn2)12(21023where()pxisasuitablybehavedcontinuouslydifferentiablefunctionand()Cixisthecosineintegral.Whilstthepaperismainlyexpository,someoftheformulaereportedinitarebelievedtobenew,andthepapermayalsobeofinterestspecificallyduetothefactthatmostofthevariousidentitieshavebeenderivedbyelementarymethods.2CONTENTSOFVOLUMESITOVI:Volume/pageSECTION:1.IntroductionI/112.AnintegralinvolvingcotxI/17TheRiemann-LebesguelemmaI/213.ResultsobtainedfromthebasicidentitiesI/24SomestuffonStirlingnumbersofthefirstkindI/69Euler,LandenandSpencepolylogarithmidentitiesI/86AnapplicationofthebinomialtheoremI/117SummaryofharmonicnumberseriesidentitiesI/1544.ElementaryproofsoftheFlajoletandSedgewickidentitiesII(a)/5SomeidentitiesderivedfromtheHasse/SondowequationsII(a)/13Aconnectionwiththegamma,betaandpsifunctionsII(a)/26AnapplicationofthedigammafunctiontothederivationofEulersumsII(a)/31GausshypergeometricsummationII(a)/48StirlingnumbersrevisitedII(a)/51LogarithmicseriesforthedigammaandgammafunctionsII(a)/64AnalternativeproofofAlexeiewsky’stheoremII(a)/74AdifferentviewofthecrimesceneII(a)/91AneasyproofofLerch’sidentityII(a)/109Stirling’sapproximationforlog()uΓII(a)/114TheGosper/VardifunctionalequationII(a)/125AlogarithmicseriesforlogAII(a)/137Asymptoticformulaforlog(1)Gu+II(a)/139Gosper’sintegralII(a)/1443ThevanishingintegralII(a)/147AnothertriptothelandofGII(a)/165Evaluationof0cotxnuuduππ∫II(a)/169AnobservationbyGlasserII(a)/188AnintroductiontoStieltjesconstantsII(b)/5ApossibleconnectionwiththeFresnelintegralII(b)/16EvaluationofsomeStieltjesconstantsII(b)/21AconnectionwithlogarithmicintegralsII(b)/67Ahitchhiker’sguidetotheRiemannhypothesisII(b)/84AmultitudeofidentitiesandtheoremsII(b)/101VariousidentitiesinvolvingpolylogarithmsIII/5Sondow’sformulaforγIII/42EvaluationofvariouslogarithmicintegralsIII/61SomeintegralsinvolvingpolylogarithmsIII/66Alittlebitoflog(/2)πIII/88AlternativederivationsoftheGlaisher-KinkelinconstantsIII/108SomeidentitiesinvolvingharmonicnumbersIV/5Someintegralsinvolvinglog()xΓIV/77Anotherdeterminationoflog(1/2)GIV/88AnapplicationofKummer’sFourierseriesforlog(1)xΓ+IV/92SomeFourierseriesconnectionsIV/109FurtherappearancesoftheRiemannfunctionalequationIV/124MoreidentitiesinvolvingharmonicnumbersandpolylogarithmsIV/1335.AnapplicationoftheBernoullipolynomialsV/546.Trigonometricintegralidentitiesinvolving:-RiemannzetafunctionV/7-BarnesdoublegammafunctionV/31-SineandcosineintegralsV/45-YetanotherderivationofGosper’sintegralV/1087.SomeapplicationsoftheRiemann-LebesguelemmaV/1418.SomemiscellaneousresultsV/145APPENDICES(VolumeVI):A.SomepropertiesoftheBernoullinumbersandtheBernoullipolynomialsB.Awell-knownintegralC.Euler’sreflectionformulaforthegammafunctionandrelatedmattersD.Averyelementaryproofof22018(21)nnπ∞==+∑E.SomeaspectsofEuler’sconstantγandthegammafunctionF.ElementaryaspectsofRiemann’sfunctionalequationforthezetafunctionACKNOWLEDGEMENTSREFERENCES55.ANAPPLICATIONOFTHEBERNOULLIPOLYNOMIALSIn(2.24)ofVolumeIweshowedthatforsuitablefunctions111()sin2()cot2bbnnnkaanpxkxdxpxxdxkαα∞==⎛⎞=⎜⎟⎝⎠∑∑∫∫Itwouldbepossibletoreplace)(xpbyNxin(2.24),forexample,andcarryoutthedetailedcalculationsasbefore,butconsiderablesimplificationsareachievedbyemployingtheBernoullipolynomialswhoseimportantpropertiesareconsideredinAppendixAofVolumeVI.Therefore,inthispart,welet)()(12xBxpN+=,theselectedrangeofintegrationis[0,1/2]andweusethegeneralisedidentity(2.24)withα=π.Since(5.1)0)2/1()0(1212==++NNBBtheRiemann-Lebesguelemmaconditionswillbesatisfiedatbothendpointsof[0,1/2].Wethereforehavefrom(2.24)(5.2)1122212111001()sin2()cot2nNNnnknBxkxdxBxxdxkππ∞++==⎛⎞=⎜⎟⎝⎠∑∑∫∫anditwillbenotedthattherighthandsideappearsin(1.13),anintegralinvolving(21)Nς+.Integrationbypartsgives(5.3)12210()sin2kNIBxkxdxπ+=∫(5.4)112221200cos221()()cos222NNkxNBxBxkxdxkkππππ++=−+∫Theintegratedpartin(5.4)vanishesinviewoftherelationscontainedin(5.1).Afurtherintegrationbypartsgives(5.5)()112222120021()sin2(21)2()sin222(2)NNNBxkxNNBxkxdxkkkπππππ−++=−∫6()122120212()sin2(2)NNNBxkxdxkππ−+=−∫Integratingbypartsatotalof12−Ntimes,weobtain(5.6)1112222200(1)(2)!()cos2cos22(2)NNNNNBxkxdxBkxdxkπππ−−=+∫∫andthelastintegraliszero.Thereforeweget(5.7)121(1)(21)!2(2)NkNNIkπ−+−+=Substituting(5.7)in(5.2)gives(5.8)112212121110(1)(21)!11()cot2(2)2NnNNnNnknNBxxdxkkππ−∞+++==⎧⎫⎛⎞−+=⎨⎬⎜⎟⎝⎠⎩⎭∑∑∫WehavealreadyshowninLemma(3.4)thattheseriesinparenthesesisequalto)12(2+Nςandtherefore,uponrearranging(5.8)weh