The distribution of spacings between quadratic residues(3)

Abstract. We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among oth

THEDISTRIBUTIONOFSPACINGSBETWEENQUADRATICRESIDUES3Itiswellknown[15]thatthespacingstatisticsofthesuperpositionofseveralindependentspectraconvergestothePoissoncase-thespac-ingsstatisticsofuncorrelatedlevels.Thustheheuristicthat“primesareindependent”togetherwithDavenport’sresultindicatesthatthespac-ingstatisticsofthesquaresmoduloqshouldinthelimitasω(q)→∞bePoissonian,i.e.,thatinsomesensesquaresmoduloqbehavesasrandomnumbers.Itisourpurposetocon rmthisexpectation.

Inordertostudythelevelspacings,weproceedbystudyingther-levelcorrelationfunctions.ThesemeasureclusteringpropertiesofasequenceinR/Zonascaleofthemeanspacing.Theirde nitionandtheirapplicationtocomputingvariouslocalspacingsstatisticsarerecalledinappendixA.Inourcase,theseturnouttobegivenbybythefollowing:Forr≥2andaboundedconvexsetC Rr 1,letRr(C,q)=1#{xidistinctsquaresmodq:(x1 x2,...xr 1 xr)∈sC}.Nq

Thisisimmediatelytransformedinto

1(1.1)Rr(C,q)=N(h,q)Nqr 1h∈sC∩Z

whereN(h,q)isthenumberofsolutionsofthesystemofcongru-encesyi+1 yi=himodqwithy1,y2,...yrsquaresmoduloqandh=(h1,...hr 1)∈Zr 1.

TocomputethecorrelationsfordistinctxiweconsideronlysetsCwhicha-priorionlycontainvectors(xi xi+1)withdistinctcoordinates. 1Todothis,wede ne“roots”σijonRr 1fori<jbyσij(h)=j

k=ihk.

Thehyper-planes{σij=0} Rr 1arecalled“walls”,and(xi xi+1)doesnotlieinanyofthewallsifandonlyifallcoordinatesxiaredistinct.

OurmainresultshowsthatifCdoesnotintersectanywallthenRr(C,q)→vol(C)foranysequenceofsquare-freeqwithω(q)→∞:Theorem1.Letqbesquare-free,r≥2andC Rr 1aboundedconvexsetwhichdoesnotintersectanyofthewalls.Thenther-levelcorrelationfunctionsatis es

Rr(C,q)=vol(C)+O(s 1/2+ )

forall >0,wheresisthemeanspacing.

ThistheoremimpliesthatallspacingstatisticsarePoissonian(seeAppendixA).Forinstance,ifwedenotebys1,...,sN 1thenormalizeddi erencesbetweenneighboringsquares,thenwehaveass→∞

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新资格考试认证The distribution of spacings between quadratic residues(3)全文阅读和word下载服务。