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254Aannouncement:Analyticprimenumbertheory13November,2014in254Aanalyticprimenumbertheory,admin

Inthewinterquarter(startingJanuary5)IwillbeteachingagraduatetopicscourseentitledAnintroductiontoanalyticprimenumbertheory.Asthenamesuggests,thisisacoursecoveringmanyoftheanalyticnumbertheorytechniquesusedtostudythedistributionoftheprimenumbers .IwilllistthetopicsIintendtocoverinthiscoursebelowthefold.Aswithmypreviouscourses,Iwillplacelecturenotesonlineonmybloginadvanceofthephysicallectures.

ThetypeofresultsaboutprimesthatoneaspirestoprovehereiswellcapturedbyLandausclassicallistofproblems:

1. EvenGoldbachconjecture:everyevennumber greaterthantwoisexpressibleasthesumoftwoprimes.

2. Twinprimeconjecture:thereareinfinitelymanypairs whicharesimultaneouslyprime.3. Legendresconjecture:foreverynaturalnumber ,thereisaprimebetween and .4. Thereareinfinitelymanyprimesoftheform .

AllfourofLandausproblemsremainopen,butwehaveconvincingheuristicevidencethattheyarealltrue,andineachofthefourcaseswehavesomehighlynontrivialpartialresults,someofwhichwillbecoveredinthiscourse.Wealsonowhavesomeunderstandingofthebarrierswearefacingtofullyresolvingeachoftheseproblems,suchastheparityproblemthiswillalsobediscussedinthecourse.

Oneofthemainreasonsthattheprimenumbers aresodifficulttodealwithrigorouslyisthattheyhaveverylittleusablealgebraicorgeometricstructurethatweknowhowtoexploitforinstance,wedonothaveanyusefulprimegeneratingfunctions.Oneofcoursecancreatenonusefulfunctionsofthisform,suchastheorderedparameterisation thatmapseachnaturalnumber tothe prime ,oronecouldinvokeMatiyasevichstheoremtoproduceapolynomialofmanyvariableswhoseonlypositivevaluesareprime,butthesesortsoffunctionshavenousablestructuretoexploit(forinstance,theygivenoinsightintoanyoftheLandauproblemslistedaboveseealsoRemark2below).Thevariousprimalitytestsintheliterature,whileusefulforpracticalapplications(e.g.cryptography)involvingprimes,havealsoproventobeoflittleutilityforthesesortsofproblemsagain,seeRemark2.Infact,inordertomakeplausibleheuristicpredictionsabouttheprimes,itisbesttotakealmosttheoppositepointofviewtothestructuredviewpoint,usingasastartingpointthebeliefthattheprimesexhibitstrongpseudorandomnesspropertiesthatarelargelyincompatiblewiththepresenceofrigidalgebraicorgeometricstructure.Wewilldiscusssuchheuristicslaterinthiscourse.

Itmaybeinthefuturethatsomeusablestructuretotheprimes(orrelatedobjects)willeventuallybelocated(thisisforinstanceoneofthemotivationsindevelopingarigoroustheoryofthefieldwithoneelement,althoughthistheoryisfarfrombeingfullyrealisedatpresent).Fornow,though,analyticandcombinatorialmethodshaveproventobethemosteffectivewayforward,astheycanoftenbeusedeveninthenearcompleteabsenceofstructure.

Inthiscourse,wewillnotdiscusscombinatorialapproaches(suchasthedeploymentoftoolsfromadditivecombinatorics)indepth,butinsteadfocusontheanalyticmethods.Thebasicprinciplesofthisapproachcanbesummarisedasfollows:

http://en.wikipedia.org/wiki/Field_with_one_elementhttp://terrytao.wordpress.com/category/teaching/254a-analytic-prime-number-theory/http://en.wikipedia.org/wiki/Pseudorandomnesshttp://en.wikipedia.org/wiki/Goldbach%27s_conjecturehttp://en.wikipedia.org/wiki/Formula_for_primes#Formula_based_on_a_system_of_Diophantine_equationshttp://en.wikipedia.org/wiki/Legendre%27s_conjecturehttp://en.wikipedia.org/wiki/Arithmetic_combinatoricshttp://en.wikipedia.org/wiki/Twin_prime_conjecturehttp://en.wikipedia.org/wiki/Landau's_problemshttp://en.wikipedia.org/wiki/Formula_for_primeshttp://terrytao.wordpress.com/category/non-technical/admin/https://ccle.ucla.edu/course/view/15W-MATH254A-1http://en.wikipedia.org/wiki/Primality_test

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1. Ratherthantrytoisolateindividualprimes in ,oneworkswiththesetofprimes inaggregate,focusinginparticularonasymptoticstatisticsofthisset.Forinstance,ratherthantrytofindasinglepair

oftwinprimes,onecanfocusinsteadonthecount oftwinprimesuptosomethreshold .Similarly,onecanfocusoncountssuchas ,

,or ,whicharethenaturalcountsassociatedtotheotherthreeLandauproblems.InallfourofLandausproblems,thebasictaskisnowtoobtainanontriviallowerboundsonthesecounts.

2. Ifonewishestoproceedanalyticallyratherthancombinatorially,oneshouldconvertallthesecountsintosums,usingthefundamentalidentity

(orvariantsthereof)forthecardinality ofsubsets ofthenaturalnumbers ,where istheindicatorfunctionof (and rangesover ).Thuswearenowinterestedinestimating(andparticularlyinlowerbounding)sumssuchas

or

3. Onceoneexpressesnumbertheoreticproblemsinthisfashion,wearenaturallyledtothemoregeneralquestionofhowtoaccuratelyestimate(or,lessambitiously,tolowerboundorupperbound)sumssuchas

ormoregenerallybilinearormultilinearsumssuchas

or

forvariousfunctions ofarithmeticinterest.(Importantly,oneshouldalsogeneralisetoincludeintegralsaswellassums,particularlycontourintegralsorintegralsovertheunitcircleorrealline,butwepostponediscussionofthesegeneralisationstolaterinthecourse.)Indeed,ahugeportionofmodernanalyticnumbertheoryisdevotedtopreciselythissortofquestion.Inmanycases,wecanpredictanexpectedmaintermforsuchsums,andthenthetaskistocontroltheerrortermbetweenthetruesumanditsexpectedmainterm.Itisoftenconvenienttonormalisetheexpectedmaintermtobezeroor

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negligible(e.g.bysubtractingasuitableconstantfrom ),sothatoneisnowtryingtoshowthatasumofsignedrealnumbers(orperhapscomplexnumbers)issmall.Inotherwords,thequestionbecomesoneofrigorouslyestablishingasignificantamountofcancellationinonessums(alsoreferredtoasagainorsavingsoverabenchmarktrivialbound).Ortophraseitnegatively,thetaskistorigorouslypreventaconspiracyofnoncancellation,causedforinstancebytwofactorsinthesummand exhibitinganunexpectedlylargecorrelationwitheachother.

4. Itisoftendifficulttodiscerncancellation(ortopreventconspiracy)directlyforagivensum(suchas)ofinterest.However,analyticnumbertheoryhasdevelopedalargenumberoftechniquesto

relateonesumtoanother,andthenthestrategyistokeeptransformingthesumintomoreandmoreanalyticallytractableexpressions,untilonearrivesatasumforwhichcancellationcanbedirectlyexhibited.(Notethoughthatthereisoftenashorttermtradeoffbetweenanalytictractabilityandalgebraicsimplicityinatypicalanalyticnumbertheoryargument,thesumswillgetexpandedanddecomposedintomanyquitemessylookingsubsums,untilatsomepointoneappliessomecrudeestimationtoreplacethesemessysubsumsbytractableonesagain.)Therearemanytransformationsavailable,rangingsuchbasictoolsasthetriangleinequality,pointwisedomination,ortheCauchySchwarzinequalitytokeyidentitiessuchasmultiplicativenumbertheoryidentities(suchastheVaughanidentityandtheHeathBrownidentity),Fourieranalyticidentities(e.g.Fourierinversion,Poissonsummation,ormoreadvancedtraceformulae),orcomplexanalyticidentities(e.g.theresiduetheorem,Perronsformula,orJensensformula).Thesheerrangeoftransformationsavailablecanbeintimidatingatfirstthereisnoshortageoftransformationsandidentitiesinthissubject,andifoneappliesthemrandomlythenonewilltypicallyjusttransformadifficultsumintoanevenmoredifficultandintractableexpression.However,onecanmakeprogressifoneisguidedbythestrategyofisolatingandenhancingadesiredcancellation(orconspiracy)tothepointwhereitcanbeeasilyestablished(ordispelled),oralternativelytoreachthepointwherenodeepcancellationisneededfortheapplicationathand(orequivalently,thatnodeepconspiracycandisrupttheapplication).

5. Oneparticularlypowerfultechnique(albeitonewhich,ironically,canbehighlyineffectiveinacertaintechnicalsensetobediscussedlater)istouseonepotentialconspiracytodefeatanother,atechniqueIrefertoastheduelingconspiraciesmethod.Thistechniquemaybeunabletopreventasinglestrongconspiracy,butitcansometimesbeusedtopreventtwoormoresuchconspiraciesfromoccurring,whichisparticularlyusefulifconspiraciescomeinpairs(e.g.throughcomplexconjugationsymmetry,orafunctionalequation).Arelated(butmoreeffective)strategyistotrytodisperseasingleconspiracyintoseveraldistinctconspiracies,whichcanthenbeusedtodefeateachother.

Asstatedbefore,theabovestrategyhasnotbeenabletoestablishanyofthefourLandauproblemsasstated.However,theycancomeclosetosuchproblems(andwenowhavesomeunderstandingastowhytheseproblemsremainoutofreachofcurrentmethods).Forinstance,byusingthesetechniques(andalotofadditionaleffort)onecanobtainthefollowingsamplepartialresultsintheLandauproblems:

1. Chenstheorem:everysufficientlylargeevennumber isexpressibleasthesumofaprimeandanalmostprime(theproductofatmosttwoprimes).Theproofproceedsbyfindinganontriviallowerboundon ,where isthesetofalmostprimes.

2. Zhangstheorem:Thereexistinfinitelymanypairs ofconsecutiveprimeswith.Theproofproceedsbygivinganonnegativelowerboundonthequantity

forlarge andcertaindistinctintegers between and.(Thebound hassincebeenloweredto .)

http://en.wikipedia.org/wiki/Poisson_summation_formulahttp://annals.math.princeton.edu/2014/179-3/p07http://terrytao.wordpress.com/2009/09/24/the-prime-number-theorem-in-arithmetic-progressions-and-dueling-conspiracies/http://en.wikipedia.org/wiki/Vaughan's_identityhttp://terrytao.wordpress.com/2014/07/20/variants-of-the-selberg-sieve-and-bounded-intervals-containing-many-primes/http://en.wikipedia.org/wiki/Chen%27s_theoremhttp://en.wikipedia.org/wiki/Residue_theoremhttp://en.wikipedia.org/wiki/Jensen%27s_formulahttp://en.wikipedia.org/wiki/Fourier_inversion_theoremhttp://en.wikipedia.org/wiki/Perron%27s_formulahttp://en.wikipedia.org/wiki/Almost_prime

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3. TheBakerHarmanPintztheorem:forsufficientlylarge ,thereisaprimebetween and .Provenbyfindinganontriviallowerboundon .

4. TheFriedlanderIwaniectheorem:Thereareinfinitelymanyprimesoftheform .Provenbyfindinganontriviallowerboundon .

Wewilldiscuss(simplerversionsof)severaloftheseresultsinthiscourse.

Ofcourse,fortheabovegeneralstrategytohaveanychanceofsucceeding,onemustatsomepointusesomeinformationabouttheset ofprimes.Asstatedpreviously,usefullystructuredparametricdescriptionsof donotappeartobeavailable.However,wedohavetwootherfundamentalandusefulwaystodescribe :

1. (Sievetheorydescription)Theprimes consistofthosenumbersgreaterthanone,thatarenotdivisiblebyanysmallerprime.

2. (Multiplicativenumbertheorydescription)Theprimes arethemultiplicativegeneratorsofthenaturalnumbers :everynaturalnumberisuniquelyfactorisable(uptopermutation)intotheproductofprimes(thefundamentaltheoremofarithmetic).

Thesievetheoreticdescriptionanditsvariantsleadonetoagoodunderstandingofthealmostprimes,whichturnouttobeexcellenttoolsforcontrollingtheprimesthemselves,althoughthereareknownlimitationsastohowmuchinformationontheprimesonecanextractfromsievetheoreticmethodsalone,whichwewilldiscusslaterinthiscourse.Themultiplicativenumbertheorymethodsleadone(aftersomecomplexorFourieranalysis)totheRiemannzetafunction(andotherLfunctions,particularlytheDirichletLfunctions),withthedistributionofzeroes(andpoles)ofthesefunctionsplayingaparticularlydecisiveroleinthemultiplicativemethods.

Manyofourstrongestresultsinanalyticprimenumbertheoryareultimatelyobtainedbyincorporatingsomecombinationoftheabovetwofundamentaldescriptionsof (orvariantsthereof)intothegeneralstrategydescribedabove.Incontrast,moreadvanceddescriptionsof ,suchasthosecomingfromthevariousprimalitytestsavailable,have(untilnow,atleast)beensurprisinglyineffectiveinpracticeforattackingproblemssuchasLandausproblems.Onereasonforthisisthatsuchtestsgenerallyinvolveoperationssuchasexponentiation

orthefactorialfunction ,whichgrowtooquicklytobeamenabletotheanalytictechniquesdiscussedabove.

Togiveasimpleillustrationofthesetwobasicapproachestotheprimes,letusfirstgivetwovariantsoftheusualproofofEuclidstheorem:

Theorem1(Euclidstheorem)Thereareinfinitelymanyprimes.

Proof:(Multiplicativenumbertheoryproof)Supposeforcontradictionthattherewereonlyfinitelymanyprimes .Then,bythefundamentaltheoremofarithmetic,everynaturalnumberisexpressibleastheproductoftheprimes .Butthenaturalnumber islargerthanone,butnotdivisiblebyanyoftheprimes ,acontradiction.

(Sievetheoreticproof)Supposeforcontradictionthattherewereonlyfinitelymanyprimes .Then,bytheChineseremaindertheorem,thesetofnaturalnumbers thatisnotdivisiblebyanyofthehasdensity ,thatistosay

http://en.wikipedia.org/wiki/Riemann_zeta_functionhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetichttp://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theoremhttp://en.wikipedia.org/wiki/Dirichlet_L-functionhttp://www.ams.org/mathscinet-getitem?mr=1851081http://en.wikipedia.org/wiki/Primality_testhttp://en.wikipedia.org/wiki/Euclid%27s_theorem

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Inparticular, haspositivedensityandthuscontainsanelementlargerthan .Buttheleastsuchelementisonefurtherprimeinadditionto ,acontradiction.

Remark1OnecanalsophrasetheproofofEuclidstheoreminafashionthatlargelyavoidstheuseofcontradictionseethispreviousblogpostformorediscussion.

Bothproofsinfactextendtogiveastrongerresult:

Theorem2(Eulerstheorem)Thesum isdivergent.

Proof:(Multiplicativenumbertheoryproof)Bythefundamentaltheoremofarithmetic,everynaturalnumberisexpressibleuniquelyastheproduct ofprimesinincreasingorder.Inparticular,wehavetheidentity

(bothsidesmakesensein aseverythingisunsigned).Sincethelefthandsideisdivergent,therighthandsideisaswell.But

and ,so mustbedivergent.

(Sievetheoreticproof)Supposeforcontradictionthatthesum isconvergent.Foreachnaturalnumber,let bethesetofnaturalnumbersnotdivisiblebythefirst primes ,andlet bethesetofnumbersnotdivisiblebyanyprimein .Asinthepreviousproof,each hasdensity .Also,since containsatmost multiplesof ,wehavefromtheunionboundthat

Since isassumedtobeconvergent,weconcludethatthedensityof convergestothedensityof thus hasdensity ,whichisnonzerobythehypothesisthat converges.Ontheotherhand,sincetheprimesaretheonlynumbersgreaterthanonenotdivisiblebysmallerprimes, isjust ,whichhasdensityzero,givingthedesiredcontradiction.

Remark2WehaveseenhoweasyitistoproveEulerstheorembyanalyticmethods.Incontrast,theredoesnotseemtobeanyknownproofofthistheoremthatproceedsbyusinganysortofprimegeneratingformulaoraprimalitytest,

http://terrytao.wordpress.com/2010/10/18/the-no-self-defeating-object-argument-revisited/

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whichisfurtherevidencethatsuchtoolsarenotthemosteffectivewaytomakeprogressonproblemssuchasLandausproblems.(ButtheweakertheoremofEuclid,Theorem1,cansometimesbeprovenbysuchdevices.)

ThetwoproofsofTheorem2givenaboveareessentiallythesameproof,asishintedatbythegeometricseriesidentity

OnecanalsoseetheRiemannzetafunctionbegintomakeanappearanceinbothproofs.OnceonegoesbeyondEulerstheorem,though,thesievetheoreticandmultiplicativemethodsbegintodivergesignificantly.Ononehand,sievetheorycanstillhandletosomeextentsetssuchastwinprimes,despitethelackofmultiplicativestructure(onesimplyhastosieveouttworesidueclassesperprime,ratherthanone)ontheother,multiplicativenumbertheorycanattainresultssuchastheprimenumbertheoremforwhichpurelysievetheoretictechniqueshavenotbeenabletoestablish.Thedeepestresultsinanalyticnumbertheorywilltypicallyrequireacombinationofbothsievetheoreticmethodsandmultiplicativemethodsinconjunctionwiththemanytransformsdiscussedearlier(and,inmanycases,additionalinputsfromotherfieldsofmathematicssuchasarithmeticgeometry,ergodictheory,oradditivecombinatorics).

1.Topicscovered

Analyticprimenumbertheoryisavastsubject(the615pagetextofIwaniecandKowalski,forinstance,givesagoodindicationastoitsscope).Iwillthereforehavetobesomewhatselectiveindecidingwhatsubsetofthisfieldtocover.Ihavechosenthefollowingcoretopicstofocuson:

Elementarymultiplicativenumbertheory.Heuristicrandommodelsfortheprimes.ThebasictheoryoftheRiemannzetafunctionandDirichletLfunctions,andtheirrelationshipwiththeprimes.ZerofreeregionsforthezetafunctionandtheDirichetLfunction,includingSiegelstheorem.Theprimenumbertheorem,theSiegelWalfisztheorem,andtheBombieriVinogradovtheorem.Sievetheory,smallandlargegapsbetweentheprimes,andtheparityproblem.Exponentialsumestimatesovertheintegers,andtheVinogradovKorobovzerofreeregion.Zerodensityestimates,Hohieselstheorem,andLinnikstheorem.Exponentialsumestimatesoverfinitefields,andimproveddistributionestimatesfortheprimes.(Iftimepermits)Exponentialsumestimatesovertheprimes,thecirclemethod,andVinogradovsthreeprimestheorem.

Inordertocoverallthismaterial,Iwillfocusonmorequalitativeresults,asopposedtothestrongestquantitativeresults,inparticularIwillnotattempttooptimisemanyofthenumericalconstantsandexponentsappearinginvariousestimates.Thisalsoallowsmetodownplaytheroleofsomekeycomponentsofthefieldwhicharenotessentialforestablishingthecoreresultsofthiscourseatsuchaqualitativelevel:

Iwillminimisetheuseofalgebraicnumbertheorytools(suchastheclassnumberformula).Iwillavoiddeployingthefunctionalequation(orrelatedidentities,suchasPoissonsummation)iftheyare

http://en.wikipedia.org/wiki/Prime_number_theoremhttp://www.ams.org/mathscinet-getitem?mr=2061214http://en.wikipedia.org/wiki/Riemann_zeta_function

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unnecessaryataqualitativelevel(thoughIwillnotewhenthefunctionalequationcanbeusedtoimprovethequantitativeresults).Asitturnsout,allofthecoreresultsmentionedabovecaninfactbederivedwithouteverinvokingthefunctionalequation,althoughoneusuallygetspoorernumericalexponentsasaconsequence.Somewhatrelatedtothis,Iwillreducetherelianceoncomplexanalyticmethodsascomparedtomoretraditionalpresentationsofthematerial,relyinginsomeplacesinsteadonFourieranalyticsubstitutes,oronresultsaboutharmonicfunctions.(ButIwillnotgoasfarasdeployingtheprimarilyrealvariablepretentiousapproachtoanalyticnumbertheorycurrentlyindevelopmentbyGranvilleandSoundararajan,althoughmyapproachheredoesaligninspiritwiththatapproach.)Thediscussiononsievemethodswillbesomewhatabridged,focusingprimarilyontheSelbergsieve,whichisagoodgeneralpurposesieveforqualitativeapplicationsatleast.Iwillalmostcertainlyavoidanydiscussionofautomorphicformsmethods.Similarly,Iwillnotcovermethodsthatrelyonadditivecombinatoricsorergodictheory.

Ofcourse,manyoftheseadditionaltopicsarewellcoveredinexistingtextbooks,suchastheabovementionedtextofIwaniecandKowalski(or,forthefinerpointsofsievetheory,thetextofFriedlanderandIwaniec).OthergoodtextsthatcanbeusedforsupplementaryreadingareDavenportsMultiplicativenumbertheoryandMontgomeryVaughansMultiplicativenumbertheoryI..Asforprerequisites:someexposuretocomplexanalysis,Fourieranalysis,andrealanalysiswillbeparticularlyhelpful,althoughwewillreviewsomeofthismaterialasneeded(particularlywithregardtocomplexanalysisandthetheoryofharmonicfunctions).Experiencewithotherquantitativeareasofmathematicsinwhichlowerbounds,upperbounds,andotherformsofestimationareemphasised(e.g.asymptoticcombinatoricsortheoreticalcomputerscience)willalsobeuseful.Knowledgeofalgebraicnumbertheoryorarithmeticgeometrywilladdavaluableadditionalperspectivetothecourse,butwillnotbenecessarytofollowmostofthematerial.

2.Notation

Inthiscourse,allsumswillbeunderstoodtobeoverthenaturalnumbersunlessotherwisespecified,withtheexceptionofsumsoverthevariable (orvariantssuchas , ,etc.),whichwillbeunderstoodtobeoverprimes.

Wewilluseasymptoticnotationintwocontexts,oneinwhichthereisnoasymptoticparameterpresent,andoneinwhichthereisanasymptoticparameter(suchas )thatisgoingtoinfinity.Inthenonasymptoticsetting(whichisthedefaultcontextifnoasymptoticparameterisexplicitlyspecified),weuse , ,or todenoteanestimateoftheform ,where isanabsoluteconstant.Insomecaseswewouldliketheimpliedconstant todependonsomeadditionalparameterssuchas ,inwhichcasewewilldenotethisbysubscripts,forinstance denotestheclaimthat forsome dependingon .

Insomecasesitwillinsteadbeconvenienttoworkinanasymptoticsetting,inwhichthereisanexplicitlydesignatedasymptoticparameter(suchas )goingtoinfinity.Inthatcase,allmathematicalobjectswillbepermittedtodependonthisasymptoticparameter,unlesstheyareexplicitlyreferredtoasbeingfixed.Wethenuse , ,or todenotetheclaimthat forsomefixed .Notethatinslightcontrasttothenonasymptoticsetting,theimpliedconstant hereisallowedtodependonotherparameters,solongastheseparametersarealsofixed.Assuch,theasymptoticsettingcanbeaconvenientwaytomanagedependenciesofvariousimpliedconstantsonparameters.Intheasymptoticsettingwealsouse todenotetheclaimthat ,where isaquantitywhichgoestozeroastheasymptoticparametergoestoinfinity.

http://www.ams.org/mathscinet-getitem?mr=2378655http://www.ams.org/mathscinet-getitem?mr=2647984http://www.ams.org/mathscinet-getitem?mr=1790423

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24comments Commentsfeedforthisarticle13November,2014at8:09pmKumarAshok

Thankyouforgivingmeopportunityforthemostpreciouscomment!Since1901insecondweekSeptembereveryyearallMathematiciansfeelembarrassedandslappednottonominatedforNobelPrize.Why?Notthestoryoflove

floatedforA.Nobel.ButforthereasonMathematiciandonotdothemathematicsthatsuitmankindbutdothatsuittothem.Mostofthemdonotrecognizegoodworkforothers,soitwouldbedifficulttoselectthebestTodaymanyproblemsareremainevenaftersomepeoplehavesolvedthem,butafterduetonottocooperatenatureofmostofmathematiciansandrubbishpoliciesofJournalsmostoftheproblemsareunsolvedevenaftergettingsolved.Thesemathematicianareprintingrubbishsubjectmaterialaburdentocominggenerationwithoutanyapplicationtomankind.Mathematicianneedtothinkmanytimesiftheyareworkingfoemankindorforself.Differentiationbetweenthenisdifficultastheyallareidenticalsotheythemselvesrecognizeworkamongthem.Youallhaveaccepted100+pagesolutionofFermateTheorem(realsolis78page),Goldbachproblemissolved(sol1012page),TwinprimeProblemisslved(sol10+page),Riemannproblemissolved(sol810

Remark3Inlaterpostswewillmakeadistinctionbetweenimpliedconstantsthatareeffective(theycanbecomputed,atleastinprinciple,bysomeexplicitmethod)andthoseatareineffective(theycanbeproventobefinite,butthereisnoalgorithmknowntocomputetheminfinitetime).

Weuse todenotetheassertionthat divides ,and todenotetheresidueclassof modulo .

Weuse todenotetheindicatorfunctionofaset ,thus when and otherwise.Similarly,foranymathematicalstatement ,weuse todenotethevalue when istrueand when isfalse.Thusforinstance istheindicatorfunctionoftheevennumbers.

Weuse todenotethecardinalityofaset .

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2 56 RateThis

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16November,2014at2:30pmChandanSinha

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13November,2014at9:01pmBoJacoby

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13November,2014at11:11pmmathtuition88

14November,2014at6:59amarch1@gmail.com

page).aaaaannnn..dddddmanymore.

BUTNONEUTRALJOURNALISFOUNDTHISISMOSTTEDIOUSANDDIFFICULTPROBLEMTOBESOLVED.Firstmathematicianshavetosolvethisproblemotherwillgetsolvedimmediately.

ThankstoeveryonewhoreadandacttothisproblemKumarAshok

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Yourcommentwasquiteinsightfulsiranditreallystatestheproblemswithmathematicianstoconsiderableextentbutyoushouldalsotakeintoaccounttheapplicabilityofadvancement.Thoughtheroleoftodaysmathematicaldevelopment

forthewelfareofhumanityisnotapparentbutitdoesaffectininconspicuouswaysforthebetterment.Alsoweshouldseethattheresearcheshavebeendonesincealmostthebeginningofcivilizationandithascometoofar.LetssayIntegrationitispartofmathematicsandwouldbeconsideredasinventedbymathematicianitself(NewtonorLeibnitzwhomeveryouconsider)butitsfindsitsapplicationasatoolinalmosteveryareawhichenhancesthelivingstandard.ThoughitwasthecontributingsubjectbutsodoeseverythingelseforwhichNobelprizeisgiven.Thecombinationofallthesebringstheapplicableresulthopeyougetmypoint!AnywayitissaidthatMathematicsisthequeenofscienceandittoowouldhavearoleinrunningthewholeempire:)

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Remark2WehaveseenhoweasyitistoproveEulerstheoremreadRemark2WehaveseenhoweasyitistoproveEuclidstheorem

[Actually,myintenthereisindeedtohighlighttheeaseofproofofEulerstheoremifoneusesanalytictechniques.T.]

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RebloggedthisonSingaporeMathsTuitionandcommented:BookmarkTerenceTaossiteifyouareinterestedinhisnotesonAnalyticnumbertheory!Hewillbeplacinglecturenotesonlineonhisblog.

ThisisaoneinalifetimechancetolearnAnalyticNumberTheoryfromaMasterFieldsMedallistTerrenceTao.

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ThereferencedLandausProblemsWikipediaarticlereportsonlypartialprogressonGoldbachsweak(3primes)conjecture,whiletheGoldbachsweakconjecturearticlereportsa2013solution.Perhapsoneofyouwhoknowsthereal

storycanfix.PS.Atleastthe1starticlespellsD.H.J.Polymathcorrectly:)

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14November,2014at9:35amarch1

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14November,2014at2:54pmAnonymous

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15November,2014at12:47pmarch1

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1)Holymoly.*Thanks*fortheselecturenotes.2)usingsomecombinationintothegeneralstrategy:using>incorporating

[Corrected,thanksT.]

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:star::star::star:onceagainbigcongratulationsonyour$3Mbreakthruprizeandappearingoncolbert!soyourblogoverflowswithdenseimpersonalmath,butitwouldbeveryinterestingtohearyouropinionoftheawards,amsuremanywould

like/appreciateit,apersonalreactiontotheceremony,etc.egdidyougettoflirtwithcamerondiazorkatebeckinsdaleatalllol:pasamodelconsiderGowerscoverageoftheIMU2014meetinginkoreainhisblog.alas,thinkingyouarenotgonnagoforit,soasavastlyinferiorsubstituteforyourreaders,seeegmorecoverage/commentaryetcatthislink,starstudded,cashoverflowing2015breakthruawards

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ArethereanyreferencesfortherealvariableapproachtakenbyGranvilleandSoundararajan?

[Thereisabookinpreparation,untilthenonehastogototheoriginalarticles,suchashttp://arxiv.org/abs/math/0503113T.]

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NaiveQ:IsitreallythecasethattheproofofChenstheoremproceedsbylowerboundingtheverysamesumthatoneseekstolowerboundwhenattackingtheGoldbachconjectureitself?OrinthedescriptionofChenstheorem,

shouldoneoftheindicatorfunctionsinthesumindicatealmostprimalityratherthanprimality?

[Corrected,thanksT.]

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Terry,therefyoucite(andtheMathworldarticle*it*cites)saythatP2isthesetof2almostprimes(exactly2primefactors,countingmultiplicity).ShouldtheP2subscriptinsteadbePunionP2?

[Corrected,thanksT.]

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24November,2014at7:50pmMikeRoss

Theresanextra(inthelastsentenceofthesecondparagraphupfromTheorem1:withthedistributionofzeroes(andpolesofthesefunctionsplayingaparticularlydecisiveroleinthemultiplicativemethods.

[Corrected,thanksT.]

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RebloggedthisonZHANGRONG.

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IamstillbusyreadingyourotherMath254A

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Perri:

Thislookslikearareopportunityanonlinecourseonthistopic,givenbyarealmaster.IComingtoacomputernearyouinJanuary.

David

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Thisseemslikeafantasticcourse,IwishIcouldtakeit.Bestofluck!

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Willyoubepostingthevideolecturesalsoonanysite?

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RebloggedthisonDivine_Lifezandcommented:Mathematics!!!

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IsortoffollowedtheinterestingarticleconcerningLandausProblems.Theonemostobviousthingthatwasnotsaidaboutthemistheyareallrelatedtoperfectsquares.ThethirdandfourthLegendresconjectureandArethereinfinitelymany

primespsuchthatp1isaperfectsquare?areovertlyso.ThesecondistheTwinPrime,andtheconnectionhereisthattheproductofpandp+2is(X^2)1.(ThatleavesGoldbachsconjecture,wheretheconnectionrequireslookingattheproductsofthepartitionsinrelationtoperfectsquares.)Thatthetwinprimesarealwaysthefactorsof(x^2)1,actuallyx1andx+1,showsaveryprecisegeometricrelationshipwithsquarenumbersthatIthinkiseasytooverlook.Reversingthelogicandconsideringcompositesfirstandtheirprimefactors

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3January,2015at7:30pmAnonymous

7January,2015at9:14amAnonymous

secondcanbringunexpectedclaritytothesefourproblems.

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Thisshowsthatthetwinprimeconjectureisequivalenttotheconjecturethatthereareinfinitelymanysemiprimesoftheform .

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Aninterestingpropertyofevenperfectsquaresminus1isthetrivialityoftheirsmallestprimefactorunlesstheyaretwinprimecomposites.Thismakesitextremelyfasttofactorthemandeasytodeterminetheinstancesoftwinprimes(simply

byelimination).TheruleisthatthesmallestprimefactorofanontwinprimeX^21compositecannotbegreaterthanthesquarerootofitssquarerootandusuallymuchsmaller.Ifsuchafactorisnotfound,thecompositemustbetheproductoftwinprimes.Thusthelargestofthesenontwinprimefactorslessthan1millionis991for999836006723.

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Ienjoyedreadingthisentry.Lookingforwardtofollowingalongtherestoftheselecturenotes.

Ifoundtwotypos:1)Oneofcoursecreateismissingcan.2)Ahaspositivedensityandisthuscontains.hasanextrais.

[Corrected,thanksT.]

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RebloggedthisonMathOnlineTomCircleandcommented:Thisisexcellentlecturenotes.ThanksProf.TerenceTao.!

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Whatdoyoumeanhere?Theredoesnotseemtobeanyknownproofofthistheoremthatproceedsbyusinganysortofprimegeneratingformulaoraprimalitytest.Ithinkthefirstproofisfromaround1300or1400.Whichmakesmethinkweare

notcommunicatingwell.

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IntheSievetheoreticproofofTheorem2,shouldthesentenceinsteadreadAsinthepreviousproof,each hasdensity asopposedto ?

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[Corrected,thanksT.]

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