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1、OnAdvancedAnalyticNumberTheoryByC.L.SiegelTataInstituteofFundamentalResearchMumbai,IndiaiFOREWORD“AdvancedAnalyticNumberTheory”was?rstpublishedbytheTataInsti-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'sLimitFormulas11The?rstlimitformul
5、a......................12TheDedekindη-function....................133ThesecondlimitformulaofKronecker.............204Theelliptictheta-function?1(w,z)...............295TheEpsteinZeta-function...................412ApplicationsofKronecker'sLimitFormulastoAlgebraicNumberTheory521Kroneck
6、er'ssolutionofPell'sequation.............√522Classnumberoftheabsoluteclass?eldofQ(d)(d<0)...693TheKroneckerLimitFormulaforrealquadratic?eldsanditsapplications...........................√754Rayclass?eldsoverQ(√d),d<0...............895Rayclass?eldsoverQ(D),D>0...............1026SomeEx
7、amples.........................1333ModularFunctionsandAlgebraicNumberTheory1491Abelianfunctionsandcomplexmultiplications........1492FundamentaldomainfortheHilbertmodulargroup......1653Hilbertmodularfunctions....................1841EllipticModularForms.....................2202Modular
8、FormsofHecke.......
