關于最小原根的兩個結果.pdf

1、河南大學碩士學位論文關于最小原根的兩個結果姓名：龔克申請學位級別：碩士專業(yè)：應用數學指導教師：王天澤20050501AbstractTheestimateofthemagnitudeoftheleastpositiveprimitiveroot90)moduloaprimePisimportantinnumbertheoryAtabout1960，YWangandDABurgessobtainedindependentlythatg(p

2、)=O(p1／4‘)forarbitraryf0，usingthedeepworkofAWeilontheRiemannHypothesisinthealgebraicfunctionfieldsoverafinitefieldUndertheGeneralizedRiemannHypothesisinrationalnumberfield，YWangalsoobtainedthat9(p)=O(m6l092P)，where7ndeno

3、testhenumberofdistinctprimedivisorsofP一1Thisresultwasimprovedtog(p)=O(m4(109m1)4l092P)byVShoupin1992withtheuseof1waniec’SshiftedsieveinsteadofthatofBrunIn2004，YHWangandCBauergeneralizedYWang’sconditionalresulttoalgebraic

4、numberfieldsOntheotherhand，Ch一NHsugaveanestimateontheleastprimitiverootmoduloanirreduciblepolynomialinfunctionfieldsin1998Inthisthesis，wewilldevoteourinvestigationstotheleastprimitiverootunderbothalgebraicnumberfieldcase

5、andfunctionfieldcaseTheexistingresultsofWangBauerandHsuaxesharpenedbytheuseof1waniec’SshiftedsieveinsteadofBrun’sasthatWasdonebyVShoupThereaxethreechaptersinthisthesisInchapter1，weintroducethedefinitionofprimitiverootand

6、thehistoricalrecordsontheestimateoftheprimitiveroot，thensketchthemethodsusedinthisthesisInchapter2，wefirstintroduceGrSssencharactersandHeckezetafunctions，thenweproveaPerronformulainalgebraicnumberfieldsFurthermore，usings

7、omeestimatesonHeckezetafunctions，weachieveakeycharactersumestimateinalgebraicnumberfieMsConsequentlybyapplying1waniec’Sshiftedsieve，wegetthemainresult：undertheassumptionoftheGRHinnumberfields，thereisatotallypositiveprimi

8、tiverootvmoduloaprimeidealnsatisfyingNv《m4(109m1)4(109Np)2，wherem=u(≯(Ⅳp))，andu(n)denotesthenumberofdistinctprimedivisorsof札Inchapter3，wegiveanimprovementofCh，NHsu’sresultbyusing1waniec’Sshiftedsieveagainweprove：let尸bean