mndede标签
dede标签 时间:2021-02-28 阅读:()
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.
12hks(h,k)+12khs(k,h)=h2+k23hk+1References1.
T.
M.
Apostol,ModularFunctionsandDirichletSeriesinNumberTheory,Springer-VerlagNewYork,Inc.
,1976.
2.
B.
Berndt,CharacterTransformationFormulaeSimilartoThosefortheDedekindEta-Function,Proc.
Sym.
PureMath.
,No.
24,Amer.
Math.
Soc,Providence,(1973),9–30.
3.
M.
C.
Dagl,M.
Can,OnReciprocityFormulasforApostol'sDedekindSumsandtheirAnalogues,J.
IntegerSeq.
17(5)(2014),Article14.
5.
4,105–1244.
L.
Goldstein,DedekindSumsforaFuchsianGroup,I.
NagayaMath.
J.
50(1973),21–47.
5.
C.
Nagasaka,OnGeneralizedDedekindSumsAttachedtoDirichletCharacters,JournalofNumberTheory19(1984),no.
3,374–383.
6.
M.
Young,ExplicitCalculationswithEisensteinSeries.
arXiv:1710.
03624,(2017),1–37.
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.
12hks(h,k)+12khs(k,h)=h2+k23hk+1References1.
T.
M.
Apostol,ModularFunctionsandDirichletSeriesinNumberTheory,Springer-VerlagNewYork,Inc.
,1976.
2.
B.
Berndt,CharacterTransformationFormulaeSimilartoThosefortheDedekindEta-Function,Proc.
Sym.
PureMath.
,No.
24,Amer.
Math.
Soc,Providence,(1973),9–30.
3.
M.
C.
Dagl,M.
Can,OnReciprocityFormulasforApostol'sDedekindSumsandtheirAnalogues,J.
IntegerSeq.
17(5)(2014),Article14.
5.
4,105–1244.
L.
Goldstein,DedekindSumsforaFuchsianGroup,I.
NagayaMath.
J.
50(1973),21–47.
5.
C.
Nagasaka,OnGeneralizedDedekindSumsAttachedtoDirichletCharacters,JournalofNumberTheory19(1984),no.
3,374–383.
6.
M.
Young,ExplicitCalculationswithEisensteinSeries.
arXiv:1710.
03624,(2017),1–37.
digital-vm：VPS低至$4/月，服务器$80/月，10Gbps超大带宽，不限流量，机房可选：日本新加坡美国英国西班牙荷兰挪威丹麦
digital-vm，这家注册在罗马尼亚的公司在国内应该有不少人比较熟悉了，主要提供VPS业务，最高10Gbps带宽，还不限制流量，而且还有日本、新加坡、美国洛杉矶、英国、西班牙、荷兰、挪威、丹麦这些可选数据中心。2020年，digital-vm新增了“独立服务器”业务，暂时只限“日本”、“新加坡”机房，最高也是支持10Gbps带宽... 官方网站：https://digital-vm.co...
宝塔面板企业版和专业版618年中活动 永久授权仅1888元+
我们一般的站长或者企业服务器配置WEB环境会用到免费版本的宝塔面板。但是如果我们需要较多的付费插件扩展，或者是有需要企业功能应用的，短期来说我们可能选择按件按月付费的比较好，但是如果我们长期使用的话，有些网友认为选择宝塔面板企业版或者专业版是比较划算的。这样在年中大促618的时候，我们也可以看到宝塔面板也有发布促销活动。企业版年付899元，专业版永久授权1888元起步。对于有需要的网友来说，还是值...
青云互联：美国洛杉矶CN2弹性云限时八折,15元/月起,可选Windows/可自定义配置
青云互联怎么样?青云互联是一家成立于2020年6月的主机服务商，致力于为用户提供高性价比稳定快速的主机托管服务，目前提供有美国免费主机、香港主机、香港服务器、美国云服务器，让您的网站高速、稳定运行。美国cn2弹性云主机限时8折起，可选1-20个IP，仅15元/月起，附8折优惠码使用!点击进入：青云互联官方网站地址青云互联优惠码：八折优惠码：ltY8sHMh (续费同价)青云互联活动方案：美国洛杉矶...
dede标签为你推荐
-
可以发外链的论坛给几个可以发外链的论坛，还有分类信息网，不要有限制的哪种，收录不收录无所谓行业关键词为什么有些行业关键词竟价出价很低有些行业很高eset最新用户名密码ESET4.0最新用户名和密码ghostxp3GhostXP3电脑公司特别版V499怎么安装二叉树遍历写出二叉树的先序遍历、中序遍历、后序遍历。淘宝店推广给淘宝店铺推广有什么好处？苹果5怎么越狱苹果5怎么越狱？qq怎么发邮件用QQ怎样发送文件bluestackbluestacks安卓模拟器有什么用电子商务网站模板电子商务网站模板哪个好？电子商务网站模板免费建站怎么样？