On the Riemann zeta-function, Part II

1TheLaplacerepresentationof1/(sin(πs/4)∙2ξ(½+s)).OntheRiemannzeta-function,PartII.ByAnthonyCsizmaziaE-mail:[e-mail]hicfunctionf(s)isconstructedfromtheRiemannzeta-functionevaluatedatone-halfpluss.Wedeterminethetwo-sidedLaplacetransformrepresentationoff(s)onopenverticalstrips,V'(4w),disjointfromthe(translated)criticalstrip.V'(4w)consistsofallswithrealpart,Re(s),ofabsolutevaluegreaterthanone-halfandRe(s)betweensuccessivepoles4w,4(w+1)off(s),withwaninteger.ThecorrespondingLaplacedensityisrelatedtoconfluenthypergeometricfunctions.Thatdensityisshowntobepositivefornonzerowotherthan-1.Thoseresultsareobtainedwithoutrelyingonanyunprovenhypothesis.TheyareusedtogetherwiththeRiemannhypothesisandhypothesesadvancedbytheauthortoobtainconditionalresultsconcerningthezeta-function.ThoseresultsarepresentedinPartI.TheirproofsarederivedinPartsIII-V.AmetricgeometryexpressionofthepositivityoftheLaplacedensitiesarisingisestablishedinPartVI.Keywords:Riemannzeta-function;Laplacetransform;analytic/entire/meromorphic/function;confluenthypergeometricfunction;positivedefinitefunction;analyticcharacteristicfunction.MSC(MathematicsSubjectClassification).11MxxZetaandL-functions:analytictheory.11M06ζ(s)andL(s,χ).30xxFunctionsofacomplexvariable.44A10Laplacetransform.42xxFourieranalysis.42A82Positivedefinitefunctions.60E10Characteristicfunctions;othertransforms.33C15Confluenthypergeometricfunctions,Whittakerfunctions,1F1.Tableofcontents.Abstract.Keywords.MSC(MathematicsSubjectClassification).JournalofNumberTheoryclassifications.Indexofabbreviations.Indexofsymbols.ReviewofelementsofPartI.§1Theroleofsin(πs/4)inf(s):=1/(sin(πs/4)2ξ(½+s)).2Propertiesofζ(s).Propertyofζ(s).Integrabilityof|m(x+it,β)|int.TheLaplacerepresentationofm(z,β).TheLaplacerepresentationofπ/sin(πs).Translationprincipal.§2TheLaplacerepresentationonV0′off(s):=1/(b(s)ζ(½+s))viathatof1/b(s).§3StrategyfordeterminingtheLaplacerepresentationonV0′of1/b(s).Divergenceoftheformalpartialfractionexpansionof1/b(s).Integrabilityof|f(s)|onverticallinesoutsideofthecriticalstrip.Boundfor1/|ζ(z)|.Integrabilityof|N(z,¼)|onverticallines.DeterminationoftheLaplacerepresentationsoff0(s)and1/b(s)onV0′fromthatofF(z,¼)onV(¼,2).TranslationrelationforF(z,β).Translationrelationforf(z).TheMellintransformrepresentationofj(u,m).TheMellintransformrepresentationof1/(z)m.TheMellintransformrepresentationsof1/(S+2m-½),j(½+S,m)andf0(S).TheMellintransformrepresentationoff(s),forsonV0′+4w.Positivity.DeterminationoftheLaplacerepresentationoff(s)onV4w′.DeterminationoftheLaplacerepresentationofF(z,β)relativetozonV4w.Integrabilityof|F(z,β)|onverticallines.DivergenceoftheformalpartialfractionexpansionofF(z,β).SplittingF(u,β,1).Trigonometricidentity.ThesplittingofF(u,β,1)viaE(u,β,1)andE(u,β,2).Trigonometricasymptotics.Theasymptoticbehaviorof|E(u,β,1)|andof|E(u,β,2)|.Integrability.TheMellintransformrepresentationofE(u,β,2).TheMellintransformrepresentationofE(u,β,1).DefinitionandpropertiesofW(z,β).TranslationrelationforW(z,β).Mellintransformrepresentationof((π/2)/cos((π/2)(u+β)))∙Γ(u).DefinitionandpropertiesofB0(z,β).MellintransformrepresentationofE(u,β,1).TheMellintransformrepresentationofF(u,β,1).DefinitionandpropertiesofM(z,β).MellintransformrepresentationofF(u,β,1).DefinitionandpropertiesofI(p,z,u).ThedeterminationofW(z,1+β)fromI(p,z,u).ThedeterminationofI(p,z,u)fromI(p,z/u).ThedeterminationofW(z,1+β)fromI(-β,±iz).§4TheincompletegammafunctionsandconfluenthypergeometricfunctionsRelationbetweenI(p,z)andΓ(1–p,z).TheconfluenthypergeometricfunctionsM(a,B,z)andU(a,a,z).Kummer’sequation.Relationbetweenφ(1+β,z)andγ(β,z,*).TheLaplacerepresentationofφ(1+β,z).Relationbetweenφ(1+β,z)andφ(β,z).RelationbetweenI(-β,z)andφ(1+β,z).§5H(z,β)andtheMellintransformrepresentationofF(z,β).ThedeterminationofW(z,1+β)fromH(z,β).ThedeterminationofM(z,β)3fromW(z,1+β).ThedeterminationofM(z,β)fromH(z,β).Integrabilityof|F(z,β)|onverticallines.RepresentationofH(z,β).Uniformboundedness.Positivity.TranslationrelationforH(z,β).ThedeterminationofH(z,β)fromW(z,β-1).TherepresentationofW(z,β)viaR(z,β).TheboundednessofH(z,β).MellintransformrepresentationofF(z,β).Positivity.§6TheMellintransformrepresentationoff(s,β):=1/(sin(πs/4)∙2ξ(2β+s)).RelationsusedtodeterminetheMellintransformrepresentationoff(s,β).Integrabilityof1/|b(s,β)|onverticallines.TheMellintransformrepresentationoff0(s,β).TheMellintransformrepresentationof1/b(s,β).TheMellintransformrepresentationoff(s,β).ThedeterminationofP4wfromT0.TheorderofP0(r,β)inr.Resultswhenβ=¼.Mainunconditionaltheorem(1),(4)(i).Positivitywhenβandwarenonnegative.Resultswhenβ=¼.Mainunconditionaltheorem(4)(i),(ii).Metricnormsandanalyticcharacteristicfunctions.Metricresultwhenβ=¼.ReferencesIndexofabbreviations.§1Corollary1.1(Analyticcharacteristicfunction)ACF.Indexofsymbols(Complexplane)C.(Realline)R,.§1n(s,α),f(s,α).m(z,β).Qk(z).M(x).q(u,β),lk(y,β).§2P(z),g(r,j,q),g(j,q)(r),g(r,j,q,p).q0(T),j0(T).θT,p(r),ωT,p(r).§3n0(s),f0(s),hα.N(z,β),F(z,β).j(u,m),E(v,m)N(u,β,1),F(u,β,1).E(u,β,1),E(u,β,2),J(z).