Table Of Content

The Mathematical Legacy of Srinivasa Ramanujan M. Ram Murty (cid:2) V. Kumar Murty The Mathematical Legacy of Srinivasa Ramanujan M.RamMurty V.KumarMurty DepartmentofMathematicsandStatistics DepartmentofMathematics Queen’sUniversity UniversityofToronto Kingston,Ontario Toronto,Ontario Canada Canada ISBN978-81-322-0769-6 ISBN978-81-322-0770-2(eBook) DOI10.1007/978-81-322-0770-2 SpringerNewDelhiHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012949894 ©SpringerIndia2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Onaheighthestoodthatlookedtowards greaterheights. Ourearlyapproaches totheInfinite Aresunrisesplendoursonamarvellousverge Whilelingersyetunseen theglorioussun. What nowweseeisashadow ofwhatmust come. SriAurobindo, Savitri1.4 HowIwishI couldshowyoutheworld through myeyes. Vivekananda Preface 22December2012marksthe125thbirthanniversaryoftheIndianmathematician SrinivasaRamanujan.Beinglargelyself-taught,heemergedfromextremepoverty to become one of 20th century’s most influential mathematicians. His story is a phenomenal “rags to mathematical riches” story. In his short life, he had a wealth ofideasthathavetransformedandreshaped20thcenturymathematics.Theseideas continuetoshapemathematicsofthe21stcentury. This book is meant to be a panoramic view of his essential mathematical con- tributions. It is not an encyclopedicaccount of Ramanujan’s work. Rather, it is an informalaccountofsomeofthemajordevelopmentsthatemanatedfromhiswork inthe20thand21stcenturies.Thetwelveessaysfocusonasubsetofhissignificant papersandshowhowthesepapersshapedthecourseofmodernmathematics. These essays are based on lectures given by the authors over the years at the Chennai Mathematical Institute, Harish-Chandra Research Institute, IISER (Kolkata), IISER (Bhopal), IIT (Powai), IIT (Chennai), Institute for Mathematical Sciences(Chennai),andtheTataInstituteforFundamentalResearch(Mumbai)as wellasQueen’sUniversity,theFieldsInstitute,andtheUniversityofToronto.The lectures were given so that the material is accessible to undergraduates and grad- uate students. We have striven to not be too technical. At the same time, we tried toconveysomedepthofthemathematicaltheoriesemergingfromtheworkofRa- manujan.Surely,itisimpossibletobecomprehensiveinsuchamammothtask.Still, wehopethatthereaderwillseehowthevastlandscapeofRamanujan’sgardenhas blossomedoverthepastcentury. Toronto,Canada M.RamMurty V.KumarMurty vii Contents 1 TheLegacyofSrinivasaRamanujan . . . . . . . . . . . . . . . . . . 1 2 TheRamanujanτ-Function . . . . . . . . . . . . . . . . . . . . . . . 11 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Theτ-FunctionandPartitions . . . . . . . . . . . . . . . . . . . . 12 3 RelatedGeneratingFunctions . . . . . . . . . . . . . . . . . . . . 13 4 Valuesoftheτ-Function . . . . . . . . . . . . . . . . . . . . . . . 14 5 Parityoftheτ-Function . . . . . . . . . . . . . . . . . . . . . . . 17 6 CongruencesSatisfiedbytheτ-Function . . . . . . . . . . . . . . 17 7 Vanishingoftheτ-Function . . . . . . . . . . . . . . . . . . . . . 18 8 Divisibilityofτ(p)byp. . . . . . . . . . . . . . . . . . . . . . . 20 9 Lehmer’sConjectureandHarmonicWeakMaassForms . . . . . . 21 3 Ramanujan’sConjectureand(cid:3)-AdicRepresentations . . . . . . . . 25 1 TheWeilConjectures . . . . . . . . . . . . . . . . . . . . . . . . 26 2 TheCaseofEllipticCurves . . . . . . . . . . . . . . . . . . . . . 28 3 (cid:3)-AdicRepresentations . . . . . . . . . . . . . . . . . . . . . . . 30 4 EllipticCurvesandModularForms . . . . . . . . . . . . . . . . . 31 5 GeometricRealizationofModularFormsofHigherWeight . . . . 35 4 TheRamanujanConjecturefromGL(2)toGL(n) . . . . . . . . . . . 39 1 TheRamanujanConjectures . . . . . . . . . . . . . . . . . . . . . 39 2 MaassFormsofWeightZero . . . . . . . . . . . . . . . . . . . . 45 3 UpperBoundforFourierCoefficientsandEigenvalueEstimates . . 47 4 EisensteinSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5 EisensteinSeriesandNon-vanishingofζ(s)on(cid:2)(s)=1 . . . . . 51 6 TheRankin–SelbergL-Function . . . . . . . . . . . . . . . . . . 54 7 PoincaréSeriesforSL (Z) . . . . . . . . . . . . . . . . . . . . . 57 2 8 FourierCoefficientsandKloostermanSums . . . . . . . . . . . . 60 9 TheKloosterman–SelbergZetaFunction . . . . . . . . . . . . . . 64 10 Rankin–SelbergL-FunctionsforGL . . . . . . . . . . . . . . . . 65 n ix x Contents 5 TheCircleMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2 ThePartitionFunction . . . . . . . . . . . . . . . . . . . . . . . . 69 3 Waring’sProblem . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.1 SchnirelmannDensity . . . . . . . . . . . . . . . . . . . . 75 3.2 SchnirelmannDensityandWaring’sProblem . . . . . . . . 78 3.3 ProofofLinnik’sTheorem . . . . . . . . . . . . . . . . . 80 4 Goldbach’sConjecture . . . . . . . . . . . . . . . . . . . . . . . . 87 4.1 BasicLemmas . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2 MajorArcs . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 ApplicationofPartialSummation . . . . . . . . . . . . . . 91 4.4 PrimesinArithmeticProgressions . . . . . . . . . . . . . 91 4.5 TheSingularSeries . . . . . . . . . . . . . . . . . . . . . 92 4.6 TheMinorArcsEstimateUsingGRH. . . . . . . . . . . . 95 6 RamanujanandTranscendence . . . . . . . . . . . . . . . . . . . . . 97 1 Nesterenko’sTheorems . . . . . . . . . . . . . . . . . . . . . . . 97 2 SpecialValuesoftheΓ-FunctionatCMPoints . . . . . . . . . . . 100 3 SpecialValuesofJacobi’sThetaSeries . . . . . . . . . . . . . . . 101 4 TheRogers–RamanujanContinuedFraction . . . . . . . . . . . . 102 5 Nesterenko’sConjectures . . . . . . . . . . . . . . . . . . . . . . 103 6 SpecialValuesoftheRiemannZetaFunctionandq-Analogues . . 104 7 ArithmeticofthePartitionFunction . . . . . . . . . . . . . . . . . . 109 1 Ramanujan’sCongruences . . . . . . . . . . . . . . . . . . . . . . 109 2 HigherCongruences . . . . . . . . . . . . . . . . . . . . . . . . . 113 3 Dyson’sRanksandCranks . . . . . . . . . . . . . . . . . . . . . 114 4 ParityQuestions . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8 SomeNonlinearIdentitiesforDivisorFunctions . . . . . . . . . . . . 119 1 AQuadraticRelationAmongstDivisorFunctions . . . . . . . . . 119 2 QuadraticRelationsAmongstEisensteinSeries . . . . . . . . . . . 120 3 AFormulafortheτ-Function . . . . . . . . . . . . . . . . . . . . 121 4 DerivativesofModularForms . . . . . . . . . . . . . . . . . . . . 122 5 DifferentialOperatorsandNonlinearIdentities . . . . . . . . . . . 124 6 Quasi-modularForms . . . . . . . . . . . . . . . . . . . . . . . . 125 7 Non-linearCongruencesandTheirInterpretation . . . . . . . . . . 127 9 MockThetaFunctionsandMockModularForms . . . . . . . . . . 129 1 HistoricalIntroduction . . . . . . . . . . . . . . . . . . . . . . . . 129 2 Ramanujan’sExamples . . . . . . . . . . . . . . . . . . . . . . . 130 3 TheWorkofZwegers . . . . . . . . . . . . . . . . . . . . . . . . 130 4 TheSpaceofMockModularForms . . . . . . . . . . . . . . . . . 132 5 SomeApplications . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10 PrimeNumbersandHighlyCompositeNumbers . . . . . . . . . . . 135 1 TheDivisorFunctions . . . . . . . . . . . . . . . . . . . . . . . . 135 Contents xi 2 RamanujanandthePrimeNumberTheorem . . . . . . . . . . . . 138 3 HighlyCompositeNumbers . . . . . . . . . . . . . . . . . . . . . 141 4 RelationtotheSixExponentialConjecture . . . . . . . . . . . . . 143 5 CountingHighlyCompositeNumbers . . . . . . . . . . . . . . . 144 6 MaximalOrderofDivisorFunctionsandOtherArithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 MaximalOrdersofFourierCoefficientsofCuspForms . . . . . . 147 11 ProbabilisticNumberTheory . . . . . . . . . . . . . . . . . . . . . . 149 1 TheNormalOrderMethod . . . . . . . . . . . . . . . . . . . . . 149 2 TheErdös–KacTheorem . . . . . . . . . . . . . . . . . . . . . . 150 3 TheHardy–Ramanujan-TypeTheoremfortheτ-Function . . . . . 151 4 Non-abelianGeneralizationsoftheHardy–RamanujanTheorem . . 153 12 TheSato–TateConjecturefortheRamanujanτ-Function . . . . . . 155 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2 Weyl’sCriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 3 Wiener–IkeharaTauberianTheorem. . . . . . . . . . . . . . . . . 161 4 Weyl’sTheoremforCompactGroups . . . . . . . . . . . . . . . . 163 5 SymmetricPowerL-SeriesofEllipticCurves . . . . . . . . . . . 164 6 AnOutlineoftheProofoftheSato–TateConjecture . . . . . . . . 166 7 AChebotarev–Sato–TateTheoremandGeneralizations . . . . . . 168 8 ConcludingRemarks. . . . . . . . . . . . . . . . . . . . . . . . . 170 Erratumto:TheRamanujanτ-Function . . . . . . . . . . . . . . . . . . E1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Chapter 1 The Legacy of Srinivasa Ramanujan Mathematicsenjoysthefreedomofartandtheprecisionofscience.Thereisfree- domofcombinationofideasandconcepts,butthereisalsotheprecisionoflogicand theringoftruth.Itislikeamastersymphony.TheSovietmathematician,I.R.Sha- farevich[186]onceremarkedthat“asuperficialglanceatmathematicsmaygivean impressionthatitisaresultofseparateindividualeffortsofmanyscientistsscattered aboutincontinentsandinages.However,theinnerlogicofitsdevelopmentreminds one much more of the work of a single intellect, developing its thought systemat- icallyandconsistentlyusingthevarietyofhumanindividualitiesonlyasameans. Itresemblesanorchestraperformingasymphonycomposedbysomeone.Atheme passesfromoneinstrumenttoanother,itistakenupbyanotherandperformedwith irreproachableprecision.” This is no doubt true and yet, the music reaches a crescendo in the hands of certainluminaries.OnesuchluminarywasSrinivasaRamanujan.Whatisfascinat- ingaboutRamanujanisthathewaslargelyself-taughtandemergedfromextreme povertytobecomeoneofthe20thcentury’sinfluentialmathematicians.Hisstoryis a“ragstomathematicalriches”story.Inthecosmicsymphonyofmathematics,he playedamajorrole. The music of Ramanujan emanates both from his life and his work. Born on 22December1887inhumbleandpoorsurroundingsinthetownofErodesituated in present day Tamil Nadu, India, Ramanujan cultivated his love for mathematics singlehandedlyandintotalisolation.Asachild,hewasquietandoftentohimself. Thosethatknewhimwereimpressedbyhisshininglargeeyeswhichwerehismost prominentfeatures.Hehadaprodigiousmemory,andatschool,hewouldentertain hisfriendsbyrecitingthevariousdeclensionsofSanskritrootsandbyrepeatingthe valueoftheconstantπ toanynumberofdecimalplaces. Attheageof12,heborrowedabookontrigonometryfromanolderstudentand completelymastereditscontents.ThisbookwasLoney’sPlaneTrigonometrypub- lishedbyCambridgein1894andcontainsagreatdealofinformationonsummation of series, logarithms of complex numbers, calculation of π and Gregory’s series. Thiscertainlygoesfarbeyondanymoderncurriculumoftrigonometrytaughtinour highschoolstoday.ButthebookthatinfluencedhimthemostwasCarr’sAsynopsis M.R.Murty,V.K.Murty,TheMathematicalLegacyofSrinivasaRamanujan, 1 DOI10.1007/978-81-322-0770-2_1,©SpringerIndia2013