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1、VIIAnalyticNumberTheoryAnalyticnumbertheorycontainsoneofthemostbeautifulapplicationsofcomplexanalysis.Inthefollowingsectionswewilltreatsomeofthedistinguishedpearlsofthisfascinatingsubject.WehavealreadyseeninSect.VI.4,thatquadraticformsorthecorrespondinglatticescanservetoconstructmo
2、dularforms.TheFouriercoefficientsofthethetaseriesassociatedtoquadraticformsortolatticeshavenumbertheoreticalimportance.Theyappearasrepresentationnumbersforquadraticforms,andrespectivelyasnumberoflatticepoints.Duetothegeneralstructuretheoremsformodularforms,onecanbringtogetherthetaser
3、iesandEisensteinseries.WewillcomputetheFouriercoefficientsoftheEisensteinseries,obtainingthusnumbertheoreticalapplications.Inparticular,inSect.VII.1wewillfindthenumberofrepresentationsofanaturalnumberasasumoffouroreightsquares(ofintegers)bypurelyfunctiontheoreticalmeans.Startingwithth
4、esecondsection,wewillbeconcernedwithDirichletseries,in-cludingtheRiemannζ-function,averyimportantinstanceofthem.ThereisastrongconnectionbetweenmodularformsandDirichletseries(Sect.VII.3).WeproveHecke’sTheorem,claimingaone-to-onecorrespondencebetweenDirichletserieswithafunctionalequa
5、tionofspecialtypeandFourierserieswithspecialtransformationpropertyunderthesubstitutionz→−1/z,withcertainasymptoticgrowthconditions.ThiscorrespondencewillbeobtainedbymeansoftheMellintransformoftheΓ-function.Asanapplication,weobtaininparticulartheanalyticcontinuationoftheζ-functioni
6、ntheplane,andalsoitsfunctionalequation.SectionsVII.4,VII.5,VII.6containaproofofthePrimeNumberTheoremwithaweakformfortheerrorterm.WeweretryingtoprovethePrimeNumberTheoremusingasfewinstrumentsaspossible.Forthisreason,wehaveoncemorecoveredsomefactsabouttheζ-functionwithafferentrathersi
7、mpleproofs,incasewewillneedthem.Indeed,lessthenthecontentofSect.VII.3isenoughtoproceed.Forinstance,forthepurposeofprovingthePrimeNumberTheoremthereisnoneedoftheanalyticcontinuationoftheζ-functionintothewholeplane,orofthefunctionalequation.Itissufficienttohavethecontinuationslightlybe
8、yondtheverticallineRes=1,a
