Bounded_gaps_between_primes

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1、BoundedgapsbetweenprimesYitangZhangAbstractItisprovedthatliminf(pp)<7107;n+1nn!1wherepnisthen-thprime.OurmethodisarenementoftherecentworkofGoldston,PintzandYildirimonthesmallgapsbetweenconsecutiveprimes.AmajoringredientoftheproofisastrongerversionoftheBombieri-Vino

2、gradovtheoremthatisapplicablewhenthemoduliarefreefromlargeprimedivisorsonly(seeTheorem2below),butitisadequateforourpurpose.Contents1.Introduction22.Notationandsketchoftheproof33.Lemmas74.UpperboundforS1165.LowerboundforS2226.Combinatorialarguments247.Thedispersionmeth

3、od278.EvaluationofS3(r;a)299.EvaluationofS2(r;a)3010.AtruncationofthesumofS1(r;a)3411.EstimationofR1(r;a;k):TheTypeIcase3912.EstimationofR1(r;a;k):TheTypeIIcase4213.TheTypeIIIestimate:Initialsteps4414.TheTypeIIIestimate:Completion48References5511.IntroductionLetpndeno

4、tethen-thprime.Itisconjecturedthatliminf(pn+1pn)=2:n!1Whileaproofofthisconjectureseemstobeoutofreachbypresentmethods,recentlyGoldston,PintzandYildirim[6]havemadesignicantprogresstowardtheweakercon-jectureliminf(pn+1pn)<1:(1:1)n!1Inparticular,theyprovethatiftheprime

5、shavelevelofdistribution#=1=2+$foran(arbitrarilysmall)$>0,then(1.1)willbevalid(see[6,Theorem1]).Sincetheresult#=1=2isknown(theBombieri-Vinogradovtheorem),thegapbetweentheirresultand(1.1)wouldappeartobe,assaidin[6],withinahair'sbreadth.Untilveryrecently,thebestresulton

6、thesmallgapsbetweenconsecutiveprimeswasduetoGoldston,PintzandYildirim[7]thatgivespn+1pnliminfp<1:(1:2)n!1logpn(loglogpn)2Onemayaskwhetherthemethodsin[6],combinedwiththeideasinBombieri,Fried-landerandIwaniec[1]-[3]whichareemployedtoderivesomestrongerversionsoftheBombi

7、eri-Vinogradovtheorem,wouldbegoodenoughforproving(1.1)(seeQuestion1on[6,p.822]).Inthispaperwegiveanarmativeanswertotheabovequestion.Weadoptthefollowingnotationof[6].LetH=fh1;h2;:::;hk0g(1:3)beasetcomposedofdistinctnon-negativeintegers.WesaythatHisadmissibleifp(H)

8、oreveryprimep,wherep(H)denotesthenumberofdistinctresidueclassesmodulopoccupiedbythehi.Theorem1.SupposethatHisadmissiblewith