padic_measures

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1、p-AdicMeasuresandBernoulliNumbersAdamBowers1IntroductionTheconstantsBkintheTaylorseriesexpansiontX1tk=Bket1k!k=0areknownastheBernoullinumbers.Therstfeware11111;;;0;;0;;


:263042Thesenumbersplayanimportantroleinnumbertheory.Forexample,theReimannzeta-functionessentiallyequalsaBernoullinu

2、mberatnegativein-tegers:Bk(1k)=kfork2.AfewoftheinterestingclassicalresultsaboutBernoullinumbersaregiveninthefollowingtheorem.Theorem.Letk2Z+andpbeaprime.Then1.Ifa2Z,thenka(a1)Bk2Z:2.Ifa2Z,thenkkBka(a1)2Z:k1ThispaperwaswrittenasanassignmentinacoursetaughtbyKeithConradduringFallSemester

3、2004attheUniversityofConnecticut.AnythingcorrectisonlysoduetoProfessorConrad'sdiligentrevision.23.Ifk60modp1,thenBk2Zp:k4.Fork1andk2even,ifk1k260modp1,thenBk1Bk2modp:k1k25.Forkevenork=1,X1Bk+2Zpp1jk.Thispaperwilldeveloptheideasneededtoprovetheseclaimsbtp-adicmethodsandintheprocesssho

4、wtheintimaterelationshipbetweenBernoullinumbersandp-adicmeasures.p-AdicdistributionsLetXbeanycompact-opensubsetofQ,suchasZorZ.Ap-adicpppdistributiononXisdenedtobeanadditivemapfromthecollectionofcompact-opensetsinXtoQp.Additivitymeans[nXnUk=(Uk);k=1k=1wheren1andfU1;:::;Ungisanycollec

5、tionofpair-wisedisjointcompact-opensetsinX.Justfromthedenition,itisnotclearhowtoconstructap-adicdistri-bution.Fortunately,Qphassomeforgivingtopologicalproperties.ThesetQhasatopologicalbasisofcompact-opensetsoftheforma+pnZ,whereppa2Qpandn2Z(thesearetheclosedballsinQp).Consequently,ifUisanyc

6、ompact-opensubsetofQp,itcanbewrittenasanitedisjointunionofsets[kNU=aj+pZpj=13forsomeN2Zanda1;:::;ak2Qp.Therearemanysuchrepresenta-tions.Infact,eachp-adicball"a+pnZcanberepresentedastheunionofpsmaller"balls,p[1nnn+1a+pZp=a+bp+pZp:b=0Thisfollowsfromthefactthattheseballsareclosedunderi

7、ntersection,sothattheyintersectifandonlyifoneiscontainedintheother.Theabilitytodecomposecompact-opensubsetsofQpintosetsoftheforma+pNZallowsonetoreformulatetheadditivityconditionintoamoreppracticalcondition.Proposition.Everymapfromthecollectionofcompact-