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GeneralizedDedekindSumsArisingfromEisensteinSeriesTristieStucker&AmyVennosAdvisor:Dr.
MatthewYoungDepartmentofMathematics,TexasA&MUniversityNSFDMS–1757872July16,2018MobiusTransformationsSL2(Z)=abcda,b,c,d∈Z,adbc=1.
MobiusTransformationsSL2(Z)=abcda,b,c,d∈Z,adbc=1.
Givenγ∈SL2(Z),theMobiustransformationassociatedtoγisthecomplexmapdenedbyz→az+bcz+d,wherez∈H={x+iy|x,y∈R,y>0}.
MobiusTransformationsSL2(Z)=abcda,b,c,d∈Z,adbc=1.
Givenγ∈SL2(Z),theMobiustransformationassociatedtoγisthecomplexmapdenedbyz→az+bcz+d,wherez∈H={x+iy|x,y∈R,y>0}.
Wewriteγz=az+bcz+d.
AutomorphicFormsAfunctionf:H→CisanautomorphicformifAutomorphicFormsAfunctionf:H→Cisanautomorphicformif1.
fobeyssometransformationproperty.
e.
g.
faz+bcz+d=(cz+d)kf(z)AutomorphicFormsAfunctionf:H→Cisanautomorphicformif1.
fobeyssometransformationproperty.
e.
g.
faz+bcz+d=(cz+d)kf(z)2.
fsatisesacertaindierentialequation(complexanalytic,harmonicfunctions,AutomorphicFormsAfunctionf:H→Cisanautomorphicformif1.
fobeyssometransformationproperty.
e.
g.
faz+bcz+d=(cz+d)kf(z)2.
fsatisesacertaindierentialequation(complexanalytic,harmonicfunctions,3.
fexhibitssomeboundarybehavior.
(polynomialgrowth,boundednessasfunctionapproachesi∞EisensteinSeriesFork≥4andkeven,theweight-kEisensteinSeriesisEk(z)=12gcd(c,d)=11(cz+d)k.
EisensteinSeriesFork≥4andkeven,theweight-kEisensteinSeriesisEk(z)=12gcd(c,d)=11(cz+d)k.
Forallγ=abcd∈SL2(Z),Ek(γz)=(cz+d)kEk(z).
DirichletCharactersADirichletcharacterχ(modq)isafunctionχ:Z→Cwiththefollowingproperties:DirichletCharactersADirichletcharacterχ(modq)isafunctionχ:Z→Cwiththefollowingproperties:1.
χ(n+ql)=χ(n)n,l∈ZDirichletCharactersADirichletcharacterχ(modq)isafunctionχ:Z→Cwiththefollowingproperties:1.
χ(n+ql)=χ(n)n,l∈Z2.
χ(n)=0igcd(n,q)=1DirichletCharactersADirichletcharacterχ(modq)isafunctionχ:Z→Cwiththefollowingproperties:1.
χ(n+ql)=χ(n)n,l∈Z2.
χ(n)=0igcd(n,q)=13.
χ(mn)=χ(m)χ(n)m,n∈ZDirichletCharactersADirichletcharacterχ(modq)isafunctionχ:Z→Cwiththefollowingproperties:1.
χ(n+ql)=χ(n)n,l∈Z2.
χ(n)=0igcd(n,q)=13.
χ(mn)=χ(m)χ(n)m,n∈ZExample:Jacobi/LegendreSymbolsEisensteinSerieswithDirichletCharactersEχ1,χ2(z,s)=12gcd(c,d)=1(q2y)sχ1(c)χ2(d)|cq2z+d|2s|cq2z+d|cq2z+dkwhereχ1andχ2areDirichletcharactersmoduloq1,q2,respectively.
EisensteinSerieswithDirichletCharactersEχ1,χ2(z,s)=12gcd(c,d)=1(q2y)sχ1(c)χ2(d)|cq2z+d|2s|cq2z+d|cq2z+dkwhereχ1andχ2areDirichletcharactersmoduloq1,q2,respectively.
Eχ1,χ2(γz,s)=ψ(γ)Eχ1,χ2(z,s),whereψ(γ)=χ1(d)χ2(d),forallγ=abcd∈Γ0(q1q2).
EisensteinSerieswithDirichletCharactersEχ1,χ2(z,s)=12gcd(c,d)=1(q2y)sχ1(c)χ2(d)|cq2z+d|2s|cq2z+d|cq2z+dkwhereχ1andχ2areDirichletcharactersmoduloq1,q2,respectively.
Eχ1,χ2(γz,s)=ψ(γ)Eχ1,χ2(z,s),whereψ(γ)=χ1(d)χ2(d),forallγ=abcd∈Γ0(q1q2).
Γ0(N)=abcd∈SL2(Z)c≡0(modN)PeriodicityofEχ1,χ2LetT=1101∈Γ0(q1q2).
ThenTz=1z+10z+1=z+1,soEχ1,χ2(z+1,s)=(0z+1)kχ1(1)χ2(1)Eχ1,χ2(z,s)=Eχ1,χ2(z,s).
PeriodicityofEχ1,χ2LetT=1101∈Γ0(q1q2).
ThenTz=1z+10z+1=z+1,soEχ1,χ2(z+1,s)=(0z+1)kχ1(1)χ2(1)Eχ1,χ2(z,s)=Eχ1,χ2(z,s).
Thus,Eχ1,χ2isperiodic.
FourierExpansionfortheCompletedEisensteinSeriesDenethecompletedEisensteinseriesasEχ1,χ2(z,s):=(q2/π)sikτ(χ2)Γ(s+k2)L(2s,χ1χ2)Eχ1,χ2(z,s)FourierExpansionfortheCompletedEisensteinSeriesDenethecompletedEisensteinseriesasEχ1,χ2(z,s):=(q2/π)sikτ(χ2)Γ(s+k2)L(2s,χ1χ2)Eχ1,χ2(z,s)TheFourierexpansionforthecompletedEisensteinseriesisEχ1,χ2(z,s)=eχ1,χ2(y,s)+n=0λχ1,χ2(n,s)|n|e2πinx·Γ(s+k2)Γ(s+k2sgn(n))Wk2sgn(n),s12(4π|n|y).
EvaluatingEχ1,χ2(z,s)atk=0ands=1Eχ1,χ2(z,1)=n>0e2πinz√nab=nχ1(a)χ2(b)ba12+χ2(1)n>0e2πinz√nab=nχ1(a)χ2(b)ba12Theη-functionandDedekindSumsη(z)=eπiz/12∞n=1(1e2πinz)Theη-functionandDedekindSumsη(z)=eπiz/12∞n=1(1e2πinz)logηaz+bcz+d=logη(z)+πia+d12c+s(d,c)+12log(i(cz+d))Theη-functionandDedekindSumsη(z)=eπiz/12∞n=1(1e2πinz)logηaz+bcz+d=logη(z)+πia+d12c+s(d,c)+12log(i(cz+d))s(h,k)=k1r=1rkhrkhrk12EvaluatingEχ1,χ2(z,s)atk=0ands=1Eχ1,χ2(z,1)=n>0e2πinz√nab=nχ1(a)χ2(b)ba12fχ1,χ2(z)+χ2(1)n>0e2πinz√nab=nχ1(a)χ2(b)ba12fχ1,χ2(z)EvaluatingEχ1,χ2(z,s)atk=0ands=1Eχ1,χ2(z,1)=n>0e2πinz√nab=nχ1(a)χ2(b)ba12fχ1,χ2(z)+χ2(1)n>0e2πinz√nab=nχ1(a)χ2(b)ba12fχ1,χ2(z)Wehavebeeninvestigatingthefunctionfχ1,χ2.
TransformationPropertiesoffχ1,χ2(z)Deneφχ1,χ2(γ,z):=fχ1,χ2(γz)ψ(γ)fχ1,χ2(z).
TransformationPropertiesoffχ1,χ2(z)Deneφχ1,χ2(γ,z):=fχ1,χ2(γz)ψ(γ)fχ1,χ2(z).
MainGoal.
Findanitesumformulaforφχ1,χ2.
Propertiesofφχ1,χ2Lemma1.
Thefunctionφχ1,χ2isindependentofz.
Propertiesofφχ1,χ2Lemma1.
Thefunctionφχ1,χ2isindependentofz.
Proof.
SinceEχ1,χ2(γz,1)=ψ(γ)Eχ1,χ2(z,1)andEχ1,χ2(z,1)=fχ1,χ2(z)+χ2(1)fχ1,χ2(z),φχ1,χ2(γ,z)=χ2(1)φχ1,χ2(γ,z).
Sinceφχ1,χ2isaholomorphicfunctionandφχ1,χ2isanantiholomorphicfunction,φχ1,χ2mustbeconstant.
Propertiesofφχ1,χ2Lemma1.
Thefunctionφχ1,χ2isindependentofz.
Proof.
SinceEχ1,χ2(γz,1)=ψ(γ)Eχ1,χ2(z,1)andEχ1,χ2(z,1)=fχ1,χ2(z)+χ2(1)fχ1,χ2(z),φχ1,χ2(γ,z)=χ2(1)φχ1,χ2(γ,z).
Sinceφχ1,χ2isaholomorphicfunctionandφχ1,χ2isanantiholomorphicfunction,φχ1,χ2mustbeconstant.
Fromnowon,wewillwriteφχ1,χ2(γ)insteadofφχ1,χ2(γ,z).
Propertiesofφχ1,χ2Lemma2.
Letγ1,γ2∈Γ0(q1q2).
Thenφχ1,χ2(γ1γ2)=φχ1,χ2(γ1)+ψ(γ1)φχ1,χ2(γ2).
Propertiesofφχ1,χ2Lemma2.
Letγ1,γ2∈Γ0(q1q2).
Thenφχ1,χ2(γ1γ2)=φχ1,χ2(γ1)+ψ(γ1)φχ1,χ2(γ2).
Proof.
Sinceψismultiplicative,φχ1,χ2(γ1γ2)=fχ1,χ2(γ1γ2z)ψ(γ1γ2)fχ1,χ2(z)=fχ1,χ2(γ1γ2z)ψ(γ1)ψ(γ2)fχ1,χ2(z)=fχ1,χ2(γ1γ2z)ψ(γ1)fχ1,χ2(γ2z)+ψ(γ1)fχ1,χ2(γ2z)ψ(γ1)ψ(γ2)fχ1,χ2(z)=φχ1,χ2(γ1)+ψ(γ1)φχ1,χ2(γ2).
MainTheoremTheorem.
Letγ=abcd∈Γ0(q1q2).
Thenφχ1,χ2(γ)=πiχ2(1)τ(χ1)j(modc)n(modq1)χ2(j)χ1(n)B1jcB1nq1ajc,whereB1(z)=zz12,z/∈Z0,otherwise,andτ(χ)=q1n=0χ(n)e2πinq,forχmoduloq.
CarnivalFunhouseProofofMainTheoremLetγ=abcd∈Γ0(q1q2).
Choosez=dc+ic2u∈Hforsomeu∈R,u=0.
Thenγz=ac+iu.
φχ1,χ2(γ)=limu→0+fχ1,χ2ac+iuψ(γ)fχ1,χ2dc+ic2uCarnivalFunhouseProofofMainTheoremLetγ=abcd∈Γ0(q1q2).
Choosez=dc+ic2u∈Hforsomeu∈R,u=0.
Thenγz=ac+iu.
φχ1,χ2(γ)=limu→0+fχ1,χ2ac+iuψ(γ)fχ1,χ2dc+ic2ulimu→0+fχ1,χ2dc+ic2u=0.
CarnivalFunhouseProofofMainTheoremLetγ=abcd∈Γ0(q1q2).
Choosez=dc+ic2u∈Hforsomeu∈R,u=0.
Thenγz=ac+iu.
φχ1,χ2(γ)=limu→0+fχ1,χ2ac+iuψ(γ)fχ1,χ2dc+ic2ulimu→0+fχ1,χ2dc+ic2u=0.
Thus,φχ1,χ2(γ)=limu→0+fχ1,χ2ac+iu.
CarnivalFunhouseProofofMainTheoremfχ1,χ2(z)=∞k=1∞l=1χ1(l)χ2(k)le2πiklz.
CarnivalFunhouseProofofMainTheoremfχ1,χ2(z)=∞k=1∞l=1χ1(l)χ2(k)le2πiklz.
Simplifyingfχ1,χ2andevaluatinglimu→0+fχ1,χ2ac+iu,wegetφχ1,χ2(γ)=χ2(1)∞l=1χ1(l)lj(modc)χ2(j)B1jce2πialjc.
CarnivalFunhouseProofofMainTheoremFromthetransformationpropertiesofEχ1,χ2,wehaveφχ1,χ2(γ)=12(φχ1,χ2(γ)χ2(1)φχ1,χ2(γ)).
Wesimplifythismoresymmetricversionofφχ1,χ2togetφχ1,χ2(γ)=πiχ2(1)τ(χ1)j(modc)n(modq1)χ2(j)χ1(n)B1jcB1nq1ajc.
SummaryofResultsWefounda"natural"proofforthegeneralizedDedekindsumformulawithDirichletcharacters.
SummaryofResultsWefounda"natural"proofforthegeneralizedDedekindsumformulawithDirichletcharacters.
WebeganwithanicerversionoftheEisensteinseries.
SummaryofResultsWefounda"natural"proofforthegeneralizedDedekindsumformulawithDirichletcharacters.
WebeganwithanicerversionoftheEisensteinseries.
WecalculatedthegeneralizedDedekindsumdirectlyfromtheFourierexpansionoftheEisensteinseries.
SummaryofResultsWefounda"natural"proofforthegeneralizedDedekindsumformulawithDirichletcharacters.
WebeganwithanicerversionoftheEisensteinseries.
WecalculatedthegeneralizedDedekindsumdirectlyfromtheFourierexpansionoftheEisensteinseries.
Withmoretime,wewouldliketocalculateareciprocitytheoremforourgeneralizedDedekindsum.
SummaryofResultsWefounda"natural"proofforthegeneralizedDedekindsumformulawithDirichletcharacters.
WebeganwithanicerversionoftheEisensteinseries.
WecalculatedthegeneralizedDedekindsumdirectlyfromtheFourierexpansionoftheEisensteinseries.
Withmoretime,wewouldliketocalculateareciprocitytheoremforourgeneralizedDedekindsum.
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