BOUNDED GAPS BETWEEN PRIMES

590 Pages·2021·5.55 MB·english

by  Kevin Broughan

BOUNDED GAPS BETWEEN PRIMES
Searching for small gaps between consecutive primes is one way to approach the twin
primes conjecture, one of the most celebrated unsolved problems in number theory. This
bookdocumentstheremarkabledevelopmentsofrecentdecades,wherebyanupperbound
ontheknowngaplengthbetweeninfinitenumbersofconsecutiveprimeshasbeenreduced
toatractablefinitesize.Thetextisbothintroductoryandcomprehensive:thedetailedway
inwhichresultsareprovedisfullysetoutandplentyofbackgroundmaterialisincluded.
Thereaderjourneysfromselectedhistoricaltheoremstothelatestbestresult,exploringthe
contributionsofavastarrayofmathematicians,includingBombieri,Goldston,Motohashi,
Pintz, Yildirim,Zhang, Maynard, Tao and Polymath8. The book issupported by alinked
and freely available package of computer programs. The material is suitable for graduate
students and of interest to any mathematician curious about recent breakthroughs in the
field.
kevin broughan is Emeritus Professor at the University of Waikato, New Zealand.
HecofoundedandisafellowoftheNewZealandMathematicalSociety.Broughanbrings
a unique set of knowledge and skills to this project, including number theory, analysis,
topology,dynamicalsystemsandcomputationalmathematics.Hepreviouslyauthoredthe
two-volume work Equivalents of the Riemann Hypothesis (Cambridge University Press,
2017) and wrote a software package which is part of Goldfeld’s Automorphic Forms and
L-FunctionsortheGroupGL(n,R)(CambridgeUniversityPress,2006).

BOUNDED GAPS BETWEEN PRIMES
The Epic Breakthroughs of the Early Twenty-First Century
KEVIN BROUGHAN
UniversityofWaikato
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Informationonthistitle:www.cambridge.org/9781108836746
DOI:10.1017/9781108872201
©KevinBroughan2021
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Firstpublished2021
PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCatalouging-in-PublicationData
Names:Broughan,KevinA.(KevinAlfred),1943–author.
Title:Boundedgapsbetweenprimes:theepicbreakthroughsoftheearly
twenty-firstcentury/KevinBroughan.
Description:NewYork:CambridgeUniversityPress,2021.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2020043768(print)|LCCN2020043769(ebook)|
ISBN9781108836746(hardback)|ISBN9781108799201(paperback)|ISBN9781108872201(epub)
Subjects:LCSH:Numbers,Prime.|Numbertheory.
Classification:LCCQA246.B7432021(print)|LCCQA246(ebook)|DDC512.7/3–dc23
LCrecordavailableathttps://lccn.loc.gov/2020043768
LCebookrecordavailableathttps://lccn.loc.gov/2020043769
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DedicatedtoJackie,JudeandBeck
Pintztookacloserlookattheflawedproofandcameupwithakeyinsightfortheultimate
fix. He contacted Goldston in December 2004, and the three number theorists, Goldston,
PintzandYildirim,hadacompleteproofbyearlyFebruary.Theycirculatedthemanuscript
toahandfulofexperts...Oneofthese,Motohashi,foundashortcutthatledtoasurprising
shortproofofthebasicqualitativeresult.
–ScienceMagazine
In April 2013, a lecturer at the University of New Hampshire submitted a paper to the
AnnalsofMathematics.Withinweekswordspread–alittle-knownmathematician,withno
permanentjob.Therestishistory.
–IMDb
InNovember2013,inspiredbyZhang’sextraordinarybreakthrough,JamesMaynarddra-
maticallyslashedthebound (foraninfinitenumber ofprimepairs)to 600, byasubstan-
tiallyeasiermethod.BothMaynardandTerryTao,whohadindependentlydevelopedthe
sameidea,wereabletoextendtheirproofstoshowthatforanygivenintegerm≥1there
exists a bound such that there are infinitely many intervals of that length containing at
leastmdistinctprimes.IfZhang’smethodiscombinedwiththeMaynard–Taosetup,then
itappearsthattheboundcanbefurtherreducedto246.Ifallofthesetechniquescouldbe
pushedtotheirlimit,thenwewouldobtainaboundof12(orarguably6),sonewideasare
stillneededtohaveafeasibleplanforprovingthetwinprimeconjecture.
–AndrewGranville
Theconquestoftheboundedgapsisanhistoricaleventthatwillcontinuetoattractatten-
tionlongintothefuture.
–YoichiMotohashi
Contents
Preface pagexi
Acknowledgements xiv
1 Introduction 1
1.1 WhyThisStudy? 1
1.2 SummaryofThisChapter 2
1.3 HistoryandOverviewofTheseDevelopments 2
1.4 PolymathProjectsandMembersofPolymath8 5
1.5 TimelineofDevelopments 8
1.6 PrimePatternsandtheHardy–LittlewoodConjecture 9
1.7 JumpingChampions 13
1.8 ThevonMangoldtFunction 15
1.9 TheBombieri–VinogradovTheorem 18
1.10 AdmissibleTuples 21
1.10.1 Introduction 21
1.10.2 BoundsforH(k) 24
1.10.3 TheSecondHardy–LittlewoodConjecture 26
1.11 ABriefGuidetotheLiterature 31
1.12 EndNotes 33
2 TheSievesofBrunandSelberg 35
2.1 Introduction 35
2.2 SummaryofThisChapter 36
2.3 Brun’sPureSieve 38
2.4 Brun’sPureSieveAddendum 46
2.5 TheSelbergSieve 47
2.6 MakingtheConstantExplicit 60
2.7 AnApplicationtoaBrun–TitchmarshInequality 70
2.8 Brun’s,Selberg’sandOtherSieves 72
vii
viii Contents
2.9 ABriefReader’sGuidetoSieveTheory 82
2.10 EndNote:TwinAlmostPrimesandtheSieveParityProblem 83
3 EarlyWork 89
3.1 Introduction 89
3.2 ChapterSummary 90
3.3 Erdo˝sandtheFirstUnconditionalStep 91
3.4 TheBeautifulMethodofBombieriandDavenport 93
3.5 Maier’sMatrixMethod 112
3.6 EndNotes 113
4 TheBreakthroughofGoldston,Motohashi,PintzandYildirim 114
4.1 Introduction 114
4.2 OutlineoftheGPYMethod 116
4.3 DefinitionsandSummary 120
4.4 GeneralPreliminaryResults 127
4.5 SpecialPreliminaryResults 135
4.6 TheEssentialTheoremofGallagher 158
4.7 TheMainGPYTheorem 167
4.8 TheSimplifiedProof 168
4.9 GPY’sConditionalBoundedGapsTheorem 180
4.10 EndNotes 183
5 TheAstoundingResultofYitangZhang 184
5.1 Introduction 184
5.2 SummaryofZhang’sMethod 187
5.3 Notation 189
5.4 ChapterSummary 189
5.5 VariationsontheBombieri–VinogradovEstimates 190
5.6 PreliminaryLemmas 192
5.7 UpperBoundfortheSumS 200
1
5.8 LowerBoundfortheSumS 210
2
5.9 Zhang’sPrimeGapResult 216
5.10 EndNotes 217
6 Maynard’sRadicalSimplification 219
6.1 Introduction 219
6.2 Definitions 221
6.3 ChapterSummary 223
6.4 Selberg’sSieveLemmas 226
6.5 OtherPreliminaryLemmas 238
6.6 FundamentalLemmas 251
Contents ix
6.7 IntegrationFormulas 261
6.8 Maynard’sAlgorithm 265
6.9 MainTheorems 266
6.10 EndNotes 271
7 Polymath’sRefinementsofMaynard’sResults 272
7.1 Introduction 272
7.2 Definitions 274
7.3 ChapterSummary 275
7.4 PreliminaryResults 279
7.5 Polymath’sAlgorithmforM 300
k
7.6 LimitstoTheseTechniques:UpperBoundforM 302
k
7.7 Bogaert’sKrylovBasisMethod 305
7.8 Bogaert’sAlgorithm 312
7.9 HowtheGapBoundpn+1−pn ≤ 246IsDerived 314
7.10 LimitstoThisApproachforM 320
k,(cid:2)
7.11 EndNotes 323
8 VariationsonBombieri–Vinogradov 327
8.1 Introduction 327
8.2 SpecialNotationsandDefinitions 328
8.3 ChapterSummary 332
8.4 PreliminaryResults 337
8.5 MultipleDenseDivisibility 343
8.6 ImprovingZhang 347
8.7 AFundamentalTechnicalResult 377
8.8 UsingHeath-Brown’sIdentity 385
8.9 One-DimensionalExponentialSums 399
8.10 Polymath’sTypeIandIIEstimates 433
8.11 ApplicationtoPrimeGaps 444
8.12 EndNotes 446
9 FurtherWorkandtheEpilogue 451
9.1 Introduction 451
9.2 AssumingElliott–Halberstam’sConjecture 451
9.3 AssumingtheGeneralizedElliott–HalberstamConjecture 452
9.4 GapsbetweenAlmostPrimes 452
9.5 AffineFormsandClustersofPrimesinIntervals 454
9.6 LimitPointsofNormalizedConsecutivePrimeDifferences 455
9.7 Artin’sPrimitiveRootConjecture 456
9.8 ConsecutivePrimesinAPwithaFixedCommonDifference 457
x Contents
9.9 PrimeIdealsandIrreduciblePolynomials 457
9.10 CoefficientsofModularForms 458
9.11 EllipticCurves 459
9.12 Epilogue 460
AppendixA BesselFunctionsoftheFirstKind 465
AppendixB ATypeofCompactSymmetricOperator 473
AppendixC SolvinganOptimizationProblem 480
AppendixD ABrun–TitchmarshInequality 492
AppendixE TheWeilExponentialSumBound 502
AppendixF ComplexFunctionTheory 516
AppendixG TheDispersionMethodofLinnik 522
AppendixH OneThousandAdmissibleTuples 528
AppendixI PGpackMinimanual 531
References 555
Index 567