A Sieve Auxiliary Function

ASieveAuxiliaryFunctionDavidBradleyDedicatedtoProfessorHeiniHalberstam,ontheoccasionofhisretirement.Abstract.InthesievetheoriesofRosser-IwaniecandDiamond-Halberstam-Richert,theupperandlowerboundsievefunctions(Fandf,respectively)satisfyacoupledsystemofdierential-dierenceequationswithretardedarguments.Toaidinthestudyofthesefunctions,Iwaniecintroducedaconjugatedierence-dierentialequationwithanadvancedargument,andgaveasolution,q,whichisanalyticintherighthalf-plane.Theanalysisoftheboundingsievefunctions,Fandf,isfacilitatedbyanadjointintegralinner-productrelationwhichlinksthelocalbehaviourofFfwiththatofthesieveauxiliaryfunction,q.Inaddition,qplaysafundamentalroleindeterminingthesievinglimitofthecombinatorialsieve,andhenceindeter-miningtheboundaryconditionsofthesievefunctions,Fandf.Thesieveauxiliaryfunction,q,hasbeentabulatedpreviously,butthesedatawerenotsupportedbynumericalanalysis,duetotheprohibitivepresenceofhigh-orderpartialderivativesarisingfromthenumericalquadraturemethodsused[15,17].Inthispaper,wedevelopadditionalrepresentationsofq.Certainoftheserepresentationsareamenabletodetailederroranalysis.Weprovidethiserroranalysis,andasaconsequence,weindicatehowq-valuesguaranteedtoatleastsevendecimalplacescanbetabulated.1.IntroductionInhisseminalpaper,Rosser’sSieve[11],Iwaniecintroducedapairofdierence-dierentialequationswhichhavebeenstudiedmorerecentlybyDiamond,Hal-berstam,andRichert[3{10],andbyWheeler[17,18].TheequationsappearasauxiliaryequationsinconnectionwiththeproblemofestimatingS(A;P;x):=#fa2A:gcd(a;Ypxp2Pp)=1g;wherePisasetofprimesandAisanitesetofintegers.InthesievetheoriesofRosser-IwaniecandDiamond-Halberstam-Richert,theequationstaketheform(1.1)(uq(u))0=q(u)+q(u+1)2DAVIDBRADLEYand(1.2)(up(u))0=p(u)p(u+1);whereu;arerealandpositive.Theparameterdenotesthedimensionofthesieve,orsiftingdensity,andisameasureoftheaveragenumberofresidueclassesperprimeinthesequencebeingsifted.Iwaniecgavesolutionsto(1.1)and(1.2)involvingtheso-calledcomplementaryexponentialintegral[14,p.40]denedbyEin(z):=Zz01ettdt:Thesolutionsare(1.3)q(u)=(2)2iZz2euzeEin(z)dzand(1.4)p(u)=Z10exuEin(x)dx;wherein(1.3),thecontourstartsat1,huggingthenegativerealaxis,thencirclestheorigininthepositivedirectionbeforereturningto1.Inthispaper,wefocusontheproblemspresentedbythefunctionq,sincep,beingtheLaplacetransformofapositivefunction,isrelativelysimpletodealwith.In[7],itisshownthatthesolutions(1.3),(1.4)areunique,subjecttomildpolynomial-likegrowthconditionsatinnity.InSection2below,weprovethat(1.3)istheuniquesolutioninaclassoffunctionsrepresentableasaLaplace/Mellintransform.InSection3,anasymptoticexpansionisderived,andafewpropertiesofthecoecientsareproved.InSection4,wegivearepresentationofqintermsofanoperatorthatarisesinothercontexts.Finally,itisofsomeinteresttohavevaluesofqtabulated.WetakeupthisprobleminSection5.Weremarkthatthispaperisbasedinsignicantpartontheauthor’sPh.D.thesis[2].2.TheFunctionq(u)Thedierence-dierentialequation(1.1)canberewrittenintheformu1q(u)0=uq(u+1);sothatthevalueofthefunctionatuisgivenbyanintegralinvolvingthefunctionatlargervaluesoftheargument.SinceintegrationisasmoothingASIEVEAUXILIARYFUNCTION3operation,oneexpectsrepeatedintegrationstoyieldaC1solution,givenonlymildassumptionsonthebehaviourofthefunctionatinnity.Infact,itiseasytoseethatIwaniec’ssolution(1.3)isanalyticintherighthalf-planeandthatq(u)asgivenby(1.3)isasymptotictou21asutendstoinnity.In[7],thesolution(1.3)isshowntobeuniqueintheclassofnormalizedpolynomial-likefunctions.Inotherwords,(1.3)istheuniquesolutionto(1.1)whichsatisesq(u)ubasu!1,forsomeconstantb(andhencewemusthaveb=21).Inthesequel,weshallproveauniquenessresultofasomewhatdierentkind,whichshowsthat(1.3)isuniqueinaclassoffunctionsrepresentableasaLaplace/Mellintransform.Forthistask,itisprotabletoviewq(s)asafunctionofthecomplexvariable,withslyingintherighthalf-plane,althoughforsieveapplications,weareprimarilyconcernedwithpositiverealvaluesoftheparameters.Butrst,weneedtorecast(1.3)asanintegraloverthepositiverealaxis.Proposition2.1.Letnbeanon-negativeinteger,andsuppose(n+12)0.Then(2.1)q(s)=(1)n(n+12)Z10xn2@@xnesxeEin(x)dx;(s)0:Remark.Ifisrealandsmallenoughsothatn=0orn=1ispermissible(i.e.1=2intheformercase,1inthelatter)thenonecanusetherepresentation(2.1)tocomputeq(u)quiteeasily.However,asnincreases,thehigherorderpartialderivativesrapidlybecomecumbersome,andsoforlargervaluesof,themethodofSection5ispreferrable.ProofSketch.Thecasen=0canbefoundinIwaniec[11,p.184].TeRiele[15,p.6]andWheeler[17,p.73]derive(2.1)fromthen=0casebyperformingrepeatedintegrationbypartsonthelatter.Onecanalsoobtain(2.1)directlyfrom(1.3),integratingbypartsntimes.TheintegratedtermsallvanishduetothepresenceofeszasafactorineveryderivativeofeszeEin(z).Onecanthencollapsethecontourontothenegativerealaxis,andaftersomeminorsimplications,(2.1)results.See[2,p.12]fordetails.Wearenowreadytoprovethat(2.1)istheuniquesolutionto(1.1)intheclassoffunctionsrepresentableasaLaplace/Mellin