On Advanced Analytic Number Theory

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1、OnAdvancedAnalyticNumberTheoryByC.L.SiegelTataInstituteofFundamentalResearchMumbai,IndiaiFOREWORD“AdvancedAnalyticNumberTheory”wasfirstpublishedbytheTataInsti-tuteofFundamentalResearchintheirLectureNotesseriesin1961.Itisnowbeingmadeavailableinbookformwithanappendix–anEnglishtranslati

2、onofSiegel'spaper“BerechnungvonZetafunktionenanganzzahligenStellen”whichappearedintheNachrichtenderAkademiederWissenschafteninG¨ottingen,Math-Phy.Klasse.(1969),pp.87-102.WearethankfultoProfessorSiegelandtotheGottingenAcademyfor¨accordinguspermissiontotranslateandpublishthisimportant

3、paper.WealsothankProfessorS.Raghavan,whooriginallywrotethenotesofProfes-sorSiegel'slectures,formakingavailableatranslationofSiegel'spaper.K.G.RamanathaniiPREFACEDuringthewintersemester1959/60,IdeliveredattheTataInstituteofFundamentalResearchaseriesoflecturesonAnalyticNumberTheory.It

4、wasmayaimtointroducemyhearerstosomeoftheimportantandbeautifulideaswhichweredevelopedbyL.KroneckerandE.Hecke.Mr.S.Raghavanwasverycarefulintakingthenotesoftheselecturesandinpreparingthemanuscript.Ithankhimforhishelp.CarlLudwigSiegelContents1Kronecker'sLimitFormulas11Thefirstlimitformul

5、a......................12TheDedekindη-function....................133ThesecondlimitformulaofKronecker.............204Theelliptictheta-functionϑ1(w,z)...............295TheEpsteinZeta-function...................412ApplicationsofKronecker'sLimitFormulastoAlgebraicNumberTheory521Kroneck

6、er'ssolutionofPell'sequation.............√522ClassnumberoftheabsoluteclassfieldofQ(d)(d<0)...693TheKroneckerLimitFormulaforrealquadraticfieldsanditsapplications...........................√754RayclassfieldsoverQ(√d),d<0...............895RayclassfieldsoverQ(D),D>0...............1026SomeEx

7、amples.........................1333ModularFunctionsandAlgebraicNumberTheory1491Abelianfunctionsandcomplexmultiplications........1492FundamentaldomainfortheHilbertmodulargroup......1653Hilbertmodularfunctions....................1841EllipticModularForms.....................2202Modular

8、FormsofHecke.......