Lecture Notes in Mathematics 2185 Bernard Candelpergher Ramanujan Summation of Divergent Series Lecture Notes in Mathematics 2185 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryBoard: MichelBrion,Grenoble CamilloDeLellis,Zurich AlessioFigalli,Zurich DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GáborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,NewYork AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Bernard Candelpergher Ramanujan Summation of Divergent Series 123 BernardCandelpergher LaboratoireJ.A.Dieudonné.CNRS UniversitédeNice Côted’Azur,Nice,France ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-319-63629-0 ISBN978-3-319-63630-6 (eBook) DOI10.1007/978-3-319-63630-6 LibraryofCongressControlNumber:2017948388 Mathematics Subject Classification (2010): 40D05, 40G05, 40G10, 40G99, 30B40, 30B50, 11M35, 11M06 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Introduction: The Summation of Series Thestrangesums CX1 nD0C1C2C3C4C5C6C7C8C9C::: n(cid:2)0 and CX1 n3 D0C13C23C33C43C53C63C73::: n(cid:2)0 appearinphysicsaboutthestudyoftheCasimireffectwhichistheexistenceofan attractiveforcebetweentwoparallelconductingplatesinthevacuum. Theseseriesareexamplesofdivergentseriesincontrasttoconvergentseries;the notionofconvergenceforaserieswasintroducedbyCauchyinhisCoursd’Analyse inordertoPavoidfrequentmistakesinworkingwithseries.Givenaseriesofcomplex numbers a ,Cauchyconsidersthesequenceofthepartialsums n(cid:2)0 n s0 D 0 s1 D a0 s2 D a0Ca1 ::: sn D a0C:::Can(cid:3)1 v vi Introduction:TheSummationofSeries P andsaysthattheseries a isconvergentifandonlyifthesequence.s /hasa n(cid:2)0 n n finitelimitwhenngoestoinfinity.Inthiscasethesumoftheseriesisdefinedby CX1 a D lim s n n n!C1 nD0 P 1 TheclassicalRiemannseries isconvergentforeverycomplexnumbers n(cid:2)1 ns suchthatPRe.s/>1anddefinestheRiemannzetafunctiondefinedforRe.s/>1by (cid:2) Ws7! C1 1. nD1 ns Non-convergentseries are divergent series. For Re.s/ (cid:2) 1 the Riemann series is a divergent series and does not give a finite value for the sums that appear in the Casimir effect. A possible strategy to assign a finite value to these sums is to performananalyticcontinuationofthezetafunctionthishasbeendonebyRiemann (Edwards2001)whofoundanintegralformulafor(cid:2).s/whichisvalidnotonlyfor PRe.s/ > 1 but also for s 2 Cnf1g. By this method we can assign to the series nk withk>(cid:3)1thevalue(cid:2).(cid:3)k/;weget,forexample, n(cid:2)1 X 1 n0 D 1C1C1C1C1C1C:::7!(cid:2).0/D(cid:3) 2 n(cid:2)1 X 1 n1 D 1C2C3C4C5C6C:::7!(cid:2).(cid:3)1/D(cid:3) 12 n(cid:2)1 X n2 D 1C22C32C42C52C:::7!(cid:2).(cid:3)2/D0 n(cid:2)1 X 1 n3 D 1C23C33C43C53C:::7!(cid:2).(cid:3)3/D 120 n(cid:2)1 ::: Fork D(cid:3)1wehavethe“harmonicseries” X1 1 1 1 1 1 1 1 1 D1C C C C C C C C C::: n 2 3 4 5 6 7 8 9 n(cid:2)1 whichiseasilyprovedtobeadivergentseriessincethepartialsumss verify n Z Z Z 1 1 2 1 3 1 nC1 1 s D1C C:::C (cid:4) dxC dxC:::C dxDLog.nC1/ n 2 n 1 x 2 x n x Butthestrategyofanalyticcontinuationofthezetafunctiondoesnotworkinthis casesince(cid:2) hasapoleatsD1thatislims!1(cid:2).s/D1. Introduction:TheSummationofSeries vii Divergent series appear elsewhere in aPnalysis and are difficult to handle; for example,byusingtheprecedingvaluesof nk,itseemsthat n(cid:2)1 1 1 (cid:3) (cid:3) D1C2C3C4C5C6C:::C.1C1C1C1C1C1C:::/ 12 2 D2C3C4C5C6C7C::: D.1C2C3C4C5C6C7C:::/(cid:3)1 1 D(cid:3) (cid:3)1 12 Thisabsurdityshowsthatwithdivergentserieswecannotusetheclassicalrulesof calculation,andforagivenclassofseries,weneedtodefinepreciselysomemethod ofsummationanditsrulesofcalculation. Before and after Cauchy, some methods of summation of series have been introduced by several mathematicians such as Cesaro, Euler, Abel, BoPrel, and others.Thesemethodsofsummationassigntoaseriesofcomplexnumbers a n(cid:2)0 n a number obtained by taking the limit of some mePans of the partial sums sn. For example,theCesarosummationassignstoaseries n a thenumber n(cid:2)0 n XC s1C:::Csn an Dlimn!C1 (whenthislimitisfinite) n n(cid:2)0 FortheAbelsummation,wetake XA CX1 an D lim.1(cid:3)t/ snC1tn (whenthislimitisfinite) t!1(cid:3) n(cid:2)0 nD0 P wheretheseries n(cid:2)0snC1tnissupposedtobeconvergentforeveryt2Œ0;1(cid:3).Note thatthisexpressioncanbesimplifiedsince CX1 CX1 CX1 .1(cid:3)t/ snC1tn Ds1C .snC1(cid:3)sn/tn D antn; nD0 nD1 nD0 andwehave XA CX1 a D lim a tn (whenthislimitisfinite)I n n t!1(cid:3) n(cid:2)0 nD0 thisgives,forexample, XA CX1 1 1 .(cid:3)1/n D lim .(cid:3)1/ntn D lim D t!1(cid:3) t!1(cid:3)1Ct 2 n(cid:2)0 nD0 viii Introduction:TheSummationofSeries Theclassicalmethodsofsummationusethesetypesofmeansofpartialsumand canbebrieflypresentedinthefollowingform. Let T be a topological space of parameters and l some “limit point” of the compactification of T (if T D N, then l D C1; if T D Œ0;1(cid:3), then l D 1). Let .pn.t//n2N be aPfamily of complex sequences indexed by t 2 T such that for all t2T theseries p .t/isconvergent;thenweset n(cid:2)0 n P XT C1p .t/s a Dlim PnD0 n n n t!l C1p .t/ n(cid:2)0 nD0 n P when this limit is finite, and in this case, we say that the series a is T- n(cid:2)0 n summable. AtheoremofToeplitz(Hardy1949)givesnecessaryandsufficientconditionson thefamily.pn.t//n2Ntoensurethatincaseofconvergencethissummationcoincides withtheusualCauchysummation. Thesesummationmethodsverifythelinearityconditions XT XT XT .a Cb /D a C b n n n n n(cid:2)0 n(cid:2)0 n(cid:2)0 XT XT Ca DC a foreveryconstantC2C n n n(cid:2)0 n(cid:2)0 andtheusualtranslationproperty XT XT an Da0C anC1 n(cid:2)0 n(cid:2)0 ThislaPstpropertywhichseemsnaturalisin factveryrestrictive.Forexample,the series 1can’tbeT-summablesincethetranslationpropertygivesanabsurd n(cid:2)1 relation XT XT 1D1C 1 n(cid:2)0 n(cid:2)0 P Thus, if we need a method of summation such that the sums T nk are n(cid:2)0 well defined for any integer k, then we must abandon the translation property requirement and find a way to define summation procedures other than the way ofthe“limitofmeansofpartialsums.” Thiscanbedonebyusingasortofgeneratingfunctionforthetermsoftheseries. Itisbasedonthefollowingalgebraicframework(Candelpergher1995).LetEbea C-vectorspace(ingeneralaspaceoffunctions)equippedwithalinearoperatorD Introduction:TheSummationofSeries ix andalinearmapv0 W E ! C.Givenasequenceofcomplexnumbers.an/n(cid:2)0,we callanelementf 2Eageneratorofthissequenceif an Dv0.Dnf/ Wecanwriteformally X X X an D v0.Dnf/Dv0. Dnf/Dv0..I(cid:3)D/(cid:3)1f/I n(cid:2)0 n(cid:2)0 n(cid:2)0 thusifRverifiestheequation .I(cid:3)D/RDf; (1) thenweget X an Dv0.R/ n(cid:2)0 Of course, such an algebraic definition of summation needs some hypotheses, especially to assure uniqueness of the solution of Equation (1); this is presented inChap.5. It is easy to see that the Cauchy summation is a special case of this algebraic formalism: we take E as the vector space of convergent complex sequences uD.un/n(cid:2)0and D W .un/7!.unC1/ v0 W .un/7!u0 Inthiscasethegeneratorofasequenceofcomplexnumbers.an/n(cid:2)0ispreciselythis sequencef D.an/n(cid:2)0since.Dnf/k DakCn:IfwesetRD.rn/n(cid:2)0,Eq.(1)becomes thedifferenceequation rn(cid:3)rnC1 Dan The solution of this equation is defined up to an arbitrary constant; thus to get a uniquesolution,weneedtoimposeaconditionon.rn/n(cid:2)0.Since r0(cid:3)rn Da0C:::Can(cid:3)1; weseethatifweaddthecondition lim r D0; n n!C1