LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS (2)

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1、LARGEGAPSBETWEENCONSECUTIVEPRIMENUMBERSKEVINFORD,BENGREEN,SERGEIKONYAGIN,ANDTERENCETAOABSTRACT.LetG(X)denotethesizeofthelargestgapbetweenconsecutiveprimesbelowX.AnsweringaquestionofErd˝os,weshowthatlogXloglogXloglogloglogXG(X)>f(X),(logloglogX)2wheref(X)isafunctiontendingtoinfinitywithX.Ourproofco

2、mbinesexistingargumentswitharandomconstructioncoveringasetofprimesbyarithmeticprogressions.Assuch,werelyonrecentworkontheexistenceanddistributionoflongarithmeticprogressionsconsistingentirelyofprimes.CONTENTS1.Introduction12.Onarithmeticprogressionsconsistingofprimes53.Mainconstruction94.Probabil

3、ityestimates145.Linearequationsinprimeswithlargeshifts176.ProofofLemma2.1(i)197.ProofofLemma2.1(ii)238.Furthercommentsandspeculations26AppendixA.Linearequationsinprimes26References311.INTRODUCTIONarXiv:1408.4505v1[math.NT]20Aug2014WriteG(X)forthemaximumgapbetweenconsecutiveprimeslessthanX.Itiscle

4、arfromtheprimenumbertheoremthatG(X)>(1+o(1))logX,astheaveragegapbetweentheprimenumberswhichare6Xis∼logX.In1931,Westzynthius[30]provedthatinfinitelyoften,thegapbetweenconsecutiveprimenumberscanbeanarbitrarilylargemultipleoftheaveragegap,thatis,G(X)/logX→∞asX→∞.Moreover,heprovedthequalitativebound1l

5、ogXlog3XG(X)≫.log4X1Asusualinthesubject,log2x=loglogx,log3x=logloglogx,andsoon.Theconventionsforasymptoticnotationsuchas≪ando()willbedefinedinSection1.2.12KEVINFORD,BENGREEN,SERGEIKONYAGIN,ANDTERENCETAOIn1935Erd˝os[9]improvedthistologXlog2XG(X)≫(log3X)2andin1938Rankin[25]madeasubsequentimprovement

6、logXlog2Xlog4XG(X)>(c+o(1))(log3X)2withc=1.Theconstantcwassubsequentlyimprovedseveraltimes:to1eγbySch¨onhage[27],thento32c=eγbyRankin[26],c=1.31256eγbyMaierandPomerance[23]and,mostrecently,c=2eγbyPintz[24].Ouraiminthispaperistoshowthatccanbetakenarbitrarilylarge.Theorem1.LetR>0.Thenforanysufficien

7、tlylargeX,thereareatleastlogXlog2Xlog4XR(log3X)2consecutivecompositenaturalnumbersnotexceedingX.Inotherwords,wehavelogXlog2Xlog4XG(X)>f(X)(log3X)2forsomefunctionf(X)thatgoestoinfinityasX→∞.Theorem1settlesintheaffirmative