Multiple Gamma Function and Its Application to Com

arXiv:math/0308074v1[math.CA]7Aug2003MultipleGammaFunctionandItsApplicationtoComputationofSeriesV.S.AdamchikCarnegieMellonUniversity,Pittsburgh,USAAbstract.ThemultiplegammafunctionΓn,definedbyarecurrence-functionalequationasageneralizationoftheEulergammafunction,wasoriginallyintroducedbyKinkelin,Glaisher,andBarnesaround1900.Today,duetothepioneerworkofConrey,KatzandSarnak,interestinthemultiplegammafunctionhasbeenrevived.ThispaperdiscussessometheoreticalaspectsoftheΓnfunctionandtheirapplicationstosummationofseriesandinfiniteproducts.Keywords:multiplegammafunction,Barnesfunction,gammafunction,Riemannzetafunction,Hurwitzzetafunction,Stirlingnumbers,Stieltjesconstants,Catalan’sconstant,harmonicnumbers,Glaisher’sconstant2000MathematicsClassification:Primary33E20,33F99,11M35,11B731.IntroductionTheHurwitzzetafunction,oneofthefundamentaltranscendentalfunctions,istraditionallydefined(see[11])bytheseriesζ(s,z)=∞Xk=01(k+z)s,ℜ(s)0.(1)Itadmitsananalyticcontinuationtotheentirecomplexplaneexceptforthesimplepoles=1.TheRiemannzetafunctionζ(s)isaspecialcaseofζ(s,z)ζ(s,1)=ζ(s).TheHurwitzfunctionhasquiteafewseriesandintegralrepresentations(see[11,8]).ThemostfamousistheHermiteintegral:ζ(s,z)=z−s2+z1−ss−1+2Z∞0sin(sarctan(xz))(x2+z2)s/2(e2πx−1)dx,s6=1,ℜ(z)0(2)c2008KluwerAcademicPublishers.PrintedintheNetherlands.rama.tex;1/02/2008;14:05;p.12V.AdamchikfromwhichonecandeducemanyfundamentalpropertiesoftheHur-witzfunction,includingtheasymptoticexpansionatinfinity:ζ(s,z)=z1−ss−1+z−s2+m−1Xj=1B2jΓ(2j+s−1)(2j)!Γ(s)z−2j−s+1+O(1z2m+s+1)TheHurwitzfunctioniscloselyrelatedtothemultiplegammafunctionΓn(z)definedasageneralizationoftheclassicalEulergammafunctionΓ(z),bythefollowingrecurrence-functionalequation(forreferencesandashorthistoricalsurveysee[6,19]):Γn+1(z+1)=Γn+1(z)Γn(z),z∈C|,n∈IN,Γ1(z)=Γ(z),(3)Γn(1)=1.Themultiplegammafunction,originallyintroducedover100yearsago,hassignificantapplicationsintheconnectionwiththeRiemannHypothesis.Montgomery[14]andSarnak[17]haveconjecturedthatthelimitingdistributionofthenon-trivialzerosoftheRiemannzetafunctionisthesameasthatoftheeigenphasesofmatricesintheCUE(thecircularunitaryensemble).IthasbeenshowninworksbyMehta,Sarnak,Conrey,Keating,andSnaiththataclosedrepresentationforstatisticalaveragesoverCUEofN×Nunitarymatrices,whenN→∞canbeexpressedintermsoftheBarnesfunctionG(z)=1/Γ2(z),definedbyG(z+1)=G(z)Γ(z),z∈C|,(4)G(1)=1.KeatingandSnaith[9,10]conjecturedthefollowingrelationshipbe-tweenthemomentsof|ζ12+it|,averagedovert,andcharacteristicpolynomials,averagedovertheCUE:log1a(λ)limT→∞1logλ2(T)TZT0|ζ(12+it)|2λdt=λ2(γ+1)−2λ∞Xk=2(−λ)k(2k−1)ζ(k)k+1,(5)wherea(λ)isaknownfunctionofλ∈IN.Theseriesontherighthandsideof(5)isunderstoodinasenseofanalyticcontinuation(see[1]rama.tex;1/02/2008;14:05;p.2MultipleGammaFunction3forthemethodofevaluationofsuchsums),providedbytheBarnesfunction2∞Xk=2(−z)kζ(k)k+1=2zlogG(z+1)+z(γ+1)−log(2π)+1,(6)|z|1,whereγdenotestheEuler-Mascheroniconstant.Conversely,wefindthatthemomentsof|ζ12+it|in(5)arejustaratiooftwoBarnesfunctions:log1a(λ)limT→∞1logλ2(T)TZT0|ζ(12+it)|2λdt=G(λ+1)2G(2λ+1)TheevidenceinsupportofKeatingandSnaith’sconjectureisconfirmedbyafewparticularcasesλ=1,2,3,4,andthenumericalexperiment,conductedbyOdlyzko[15],forTupto1022thzerooftheRiemannzetafunction.Theidentity(6)canbefurthergeneralizedto∞Xk=2(−z)kζ(k)k+r−1,r∈Q|,|z|1andthenevaluatedintermsofther-tiplegammafunctions.Forin-stance,withr=3wefind∞Xk=2(−z)kζ(k)k+2=2z2logΓ3(z+1)+1z2logG(z+1)+(7)6z2+3z−112z+γz3−log(2π)2−2ζ′(−1)z,|z|1.Inthispaperweaimatdevelopingamathematicalfoundationforsymboliccomputationofspecialclassesofinfiniteseriesandproducts.Generallyspeaking,allseries(subjecttoconvergence)oftheform:∞Xj=1R(j)logmP(j),m∈IN,whereR(z)andP(z)arepolynomials,canbeexpressedinaclosedformbymeansofthemultiplegammafunction,whichmayfurthersimplifytoelementaryfunctions.Thisalgorithmalsocomplementstherama.tex;1/02/2008;14:05;p.34V.AdamchikworkpreviouslystartedbyAdamchikandSrivastava[1]forsymbolicsummationofseriesinvolvingtheRiemannzetafunction.Thealgo-rithmwillexpandthecapabilitiesofexistingsoftwarepackagesfornon-hypergeometricsummation.2.Asymptoticsofζ′(−λ,z)Inthissection,basedontheintegral(2),wederivetheasymptoticexpansionofζ′(t,z)=ddtζ(t,z),whent=−λ,λ∈IN0andz→∞.Differentiatingbothsidesof(2)withrespecttos,weobtainζ′(−λ,z)=zλ+1λ+1logz−zλ2logz−zλ+1(λ+1)2+2Z∞0tan−1(xz)cos(λtan−1(xz))(e2πx−1)(x2+z2)−λ/2dx(8)+Z∞0log(x2+z2)sin(λtan−1(xz))(e2πx−1)(x2+z2)−λ/2dxNext,weexpandtheintegrandsintotheTaylorserieswithrespecttoz.Takingintoaccountthat(x2+z2)λ/2cos(λtan−1(xz))=λ/2Xk=0(−1)kλ2k!zλ−2kx2kand(x2+z2)λ/2sin(λtan−1(xz))=λ/2Xk=0(−1)kλ2k+1!zλ−2k−1x2k+1wecomputeZ∞0tan−1(xz)cos(λtan−1(xz))(e2πx−1)(x2+z2)−λ/2dx=λ/2Xk=0zλ−2k(−1)kλ2k!Z∞0x2ktan−1(xz)e2πx−1dx(9)andrama.tex;1/02/2008;14:05;p.4MultipleGammaFunction5Z∞0log(x2+z2)sin(λtan−1(xz))(e2πx−1)(x2+z2)−λ/2dx=−λ/2Xk=0z−2k+λ−1(−1)kλ2k+1!Z∞0x2k+1log(x2+z2)e2πx−1dx.(10)Inthenextstep,wefindasymptoticexpansionsofintegralsontherighthandsideof(9)and(10)whenz→∞.Expandingtan−1(xz)andlog(x2+z2)intotheTaylorserieswithrespecttoxandp