The distribution of spacings between quadratic residues(2)

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

2

1.Introduction

Ourgoalinthispaperistostudythedistributionofspacings(orgaps)betweensquaresinZ/qZ,asq→∞.Inthecasethatqisprime,atheoremofDavenport[3,4,11,18]showsthattheprobabilityoftwoconsecutivequadraticresiduesmoduloaprimeqbeingspacedhunitsapartis2 h,asq→∞.Forourpurposes,wemayinterpretthisresultassayingthatwhenwenormalizethespacingstohaveunitmean,thenthedistributionofspacingasq→∞alongprimesisgivenby

∞ hP(s)=2 hδ(s )2h=1

thatis,asumofpointmassesathalf-integerswithexponentiallyde-creasingweights.

Inthispaperwestudythespacingdistributionofsquaresmoduloqwhenqissquare-freeandhighlycomposite,thatisthelimitingdistribu-tionofspacingsbetweenthesquaresmoduloqasthenumberofprimedivisors,ω(q),tendstoin nity.Foroddsquare-freeqthenumberNqofsquaresmoduloqequals

p+1Nq=2p|q

Thisisbecauseifpisanoddprime,thenumberofsquaresmodulopis(p+1)/2andforqsquare-free,xisasquaremoduloqifandonlyifxisasquaremodulopforallprimespdividingq.Thusforoddq,themeanspacingsq=q/Nequals

2ω(q)2ω(q)

sq==σ 1(q)p|q(1+1/p)

Forq=2q evenandsquare-free,itiseasilyseenthatsq=sq .Itfollowsthatsq→∞asω(q)→∞,unlikethecaseofprimeqwherethemeanspacingisessentiallyconstant.Thus,unlikeintheprimecasewherethelevelspacingdistributionwasforcedtobesupportedonalattice,inthehighlycompositecasethereisana-priorichanceofgettingacontinuousdistribution.

ArelevantstatisticalmodelforthedistributionofspacingsisgivenbylookingatrandompointsintheunitintervalR/Z.Forindependent,uniformlydistributednumbersinR/Z,thespacingstatisticsaresaidtobePoissonian.ThedistributionP(s)ofspacingsbetweenconsecutivepointswillbethatofaPoissonarrivalprocess,i.e.P(s)=e s(see

[6]).Moreover,thejointdistributionofkconsecutivespacingsistheproductofkindependentexponentialrandomvariables.

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(2)全文阅读和word下载服务。