A Study of the Anatomy of the Integers via Large Prime Factors and an Application to Numerical Factorization

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A Study of the Anatomy of the Integers via Large Prime Factors and an Application to Numerical Factorization
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Molnar, Todd W
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Master's ( M.S.)
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University of Florida
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Mathematics
Committee Chair:
Alladi, Krishnaswa
Committee Members:
Rosalsky, Andrew J
Sin, Peter

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anatomy -- factorization -- factors -- integers -- large -- prime
Mathematics -- Dissertations, Academic -- UF
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Mathematics thesis, M.S.
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Abstract:
Ostensibly, this thesis attempts to discuss some important aspects of the theory surrounding the anatomy of the integers (to borrow a term from de Koninck, Granville, and Luca), by which I mean the theory surrounding the prime factorization of an integer, and the properties of these prime factors. The anatomy of the integers is a very dense topic which has attracted the attention of theoretical and applied mathematicians for many years, due largely to the fact that the questions involved are often difficult and can be approached from many different angles. Virtually every branch of mathematics has benefited from the study of the prime factorization of the integers including (but certainly not limited to) number theory, combinatorics, algebra, and ergodic theory (to name but a few). This thesis will concern itself with both the theoretical and computational aspects of this study, and will use that theory to understand the algorithmic factorization technique introduced by Knuth and Pardo in 1976. To fully understand this algorithmic procedure, we will discuss a good deal of theory related to the distribution of prime numbers and the largest prime factor of an integer. The thesis is divided into four chapters, Chapter 1 is an introduction which gives a brief overview of the motivation and rich history surrounding these deep problems. Chapter 2 sketches two different proofs of the Prime Number Theorem, which is an indispensable tool in addressing problems related to integer factorization, as well as containing several comments related to (currently) unproven results concerning the distribution of primes (in particular, the Riemann hypothesis). One of the proofs of the Prime Number Theorem we have included in this section appears novel, although it undoubtedly could be deduced by any individual with sufficient knowledge of analytic number theory. Following the work of Alladi, Erdos, Knuth, and Pardo, Chapter 3 develops the necessary theoretical results concerning the largest prime factors of an integer. To this end, proofs of the average value of the Alladi-Erdos functions, the average value of the largest prime factor, a proof of a special case of Alladi’s duality principle, and certain density estimates are supplied. The proofs presented here related to the Alladi-Erdos functions differ from their original proofs in that we use the theory of complex variables, whereas in their original paper Alladi and Erdos derive their results elementarily. This allows us to improve their estimates for bounded functions, as well as showing the connection of these estimates with the Riemann hypothesis. The proof of Alladi’s duality principle is also derived in an analytic fashion, although not in its most general form. This differs from Alladi’s original treatment of the problem which is entirely elementary; however, in using analytic techniques we need to impose certain bounds on the arithmetic functions in question, whereas the elementary approach holds unconditionally. Further results on the largest prime factors of integers are also included, but they are due to Knuth and Pardo. The final section, Chapter 4, formally introduces the Knuth-Pardo factorization algorithm and includes their proof that the probability of a random integer between 1 and N with k -th largest prime factor < N^x , for any x greater than 0, approaches a limiting distribution; furthermore, we quote several results from the paper of Knuth and Pardo that will prove useful for studying their algorithm.
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In the series University of Florida Digital Collections.
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Includes vita.
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by Todd W Molnar.
Thesis:
Thesis (M.S.)--University of Florida, 2012.
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Adviser: Alladi, Krishnaswa.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-12-31

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ASTUDYOFTHEANATOMYOFTHEINTEGERSVIALARGEPRIMEFACTORSANDANAPPLICATIONTONUMERICALFACTORIZATIONByTODDMOLNARATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2012

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c2012ToddMolnar 2

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ACKNOWLEDGMENTS Therearefartoomanypeoplewithoutwhichthisthesiscouldnothavebeenwritten;however,IowespecialthankstomyparentsWilliamandDeborahMolnarandmybrothersBradleyandAndrewMolnar.Withouttheirsupportthisthesissimplywouldnothavebeenpossible.IwouldalsoliketothankDr.PeterSinandDr.AndrewRosalskyfortheirmanyusefulcomments,andinparticularIwouldliketothankmyadvisorDr.KrishnaswamiAlladiforhissuperbguidanceandexceptionaladvice. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 ABSTRACT ......................................... 5 CHAPTER 1INTRODUCTIONANDHISTORY .......................... 7 2THEDISTRIBUTIONOFPRIMESANDPRIMEFACTORS ........... 35 2.1NotationandPreliminaryObservations .................... 35 2.2ThePrimeNumberTheorem ......................... 44 2.3TheHardy-RamanujanTheorem ....................... 53 2.4Remarks .................................... 66 3ARITHMETICFUNCTIONSINVOLVINGTHELARGESTPRIMEFACTOR ... 69 3.1TheAlladi-ErdosFunctions .......................... 69 3.2(x,y) ..................................... 87 3.3GeneralizedAlladi-Duality ........................... 92 4THEKNUTH-PARDOALGORITHM ........................ 101 REFERENCES ....................................... 109 BIOGRAPHICALSKETCH ................................ 111 4

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceASTUDYOFTHEANATOMYOFTHEINTEGERSVIALARGEPRIMEFACTORSANDANAPPLICATIONTONUMERICALFACTORIZATIONByToddMolnarDecember2012Chair:KrishnaswamiAlladiMajor:MathematicsOstensibly,thisthesisattemptstodiscusssomeimportantaspectsofthetheorysurroundingtheanatomyoftheintegers(toborrowatermfromdeKoninck,Granville,andLucain[ 15 ]),bywhichImeanthetheorysurroundingtheprimefactorizationofaninteger,andthepropertiesoftheseprimefactors.Theanatomyoftheintegersisaverydensetopicwhichhasattractedtheattentionoftheoreticalandappliedmathematiciansformanyyears,duelargelytothefactthatthequestionsinvolvedareoftendifcultandcanbeapproachedfrommanydifferentangles.Virtuallyeverybranchofmathematicshasbenetedfromthestudyoftheprimefactorizationoftheintegersincluding(butcertainlynotlimitedto)numbertheory,combinatorics,algebra,andergodictheory(tonamebutafew).Thisthesiswillconcernitselfwithboththetheoreticalandcomputationalaspectsofthisstudy,andwillusethattheorytounderstandthealgorithmicfactorizationtechniqueintroducedbyKnuthandPardoin1976.Tofullyunderstandthisalgorithmicprocedure,wewilldiscussagooddealoftheoryrelatedtothedistributionofprimenumbersandthelargestprimefactorofaninteger.Thethesisisdividedintofourchapters,Chapter1isanintroductionwhichgivesabriefoverviewofthemotivationandrichhistorysurroundingthesedeepproblems.Chapter2sketchestwodifferentproofsofthePrimeNumberTheorem,whichisanindispensabletoolinaddressingproblemsrelatedtointegerfactorization,aswellascontaining 5

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severalcommentsrelatedto(currently)unprovenresultsconcerningthedistributionofprimes(inparticular,theRiemannhypothesis).OneoftheproofsofthePrimeNumberTheoremwehaveincludedinthissectionappearsnovel,althoughitundoubtedlycouldbededucedbyanyindividualwithsufcientknowledgeofanalyticnumbertheory.FollowingtheworkofAlladi,Erdos,Knuth,andPardo,Chapter3developsthenecessarytheoreticalresultsconcerningthelargestprimefactorsofaninteger.Tothisend,proofsoftheaveragevalueoftheAlladi-Erdosfunctions,theaveragevalueofthelargestprimefactor,aproofofaspecialcaseofAlladi'sdualityprinciple,andcertaindensityestimatesaresupplied.TheproofspresentedhererelatedtotheAlladi-Erdosfunctionsdifferfromtheiroriginalproofsinthatweusethetheoryofcomplexvariables,whereasintheiroriginalpaperAlladiandErdosderivetheirresultselementarily.Thisallowsustoimprovetheirestimatesforboundedfunctions,aswellasshowingtheconnectionoftheseestimateswiththeRiemannhypothesis.TheproofofAlladi'sdualityprincipleisalsoderivedinananalyticfashion,althoughnotinitsmostgeneralform.ThisdiffersfromAlladi'soriginaltreatmentoftheproblemwhichisentirelyelementary;however,inusinganalytictechniquesweneedtoimposecertainboundsonthearithmeticfunctionsinquestion,whereastheelementaryapproachholdsunconditionally.Furtherresultsonthelargestprimefactorsofintegersarealsoincluded,buttheyareduetoKnuthandPardo.Thenalsection,Chapter4,formallyintroducestheKnuth-Pardofactorizationalgorithmandincludestheirproofthattheprobabilityofarandomintegerbetween1andNwithk-thlargestprimefactorNx,foranygivenx0approachesalimitingdistribution;furthermore,wequoteseveralresultsfromthepaperofKnuthandPardothatwillproveusefulforstudyingtheiralgorithm. 6

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CHAPTER1INTRODUCTIONANDHISTORYThisthesiswillconcernitselfprimarilywithquestionsrelatedtonumericalfactorization,thatis,thestudyoftheprimedecompositionofintegers-bothfromatheoreticalangle,andfromapracticalpointofview-namely,thenumericalfactorizationofintegersintoprimesandprimepowers,andtherunningtimeforsuchaprocedure.WewilltakethebasicfactorizationalgorithmintroducedbyKnuthandPardoasaprototypicalexampleduetoitsrudimentarynatureandintuitiveappeal.Astudyofthisalgorithmprovidestheoreticalinsightsintothemultiplicativestructureoftheintegers,andleadstoausefulapplicationofthealgorithmitself.FollowingtheworkofKnuthandPardo[ 14 ],theiralgorithmwillbetreatedinamoreformalfashioninChapter4;however,abriefoverviewofthemethodwillbeenlighteningasanintroductiontofactorizationtechniques.Foragivenintegern,wetestforaprimedivisorofnamongthenumbers1
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withthepublishedliterature,aswellasofferingausefulintroductiontothosewhoarenot.Tofullyappreciatethedensetheorysurroundingthelargestprimefactorofaninteger,thedistributionoftheprimenumbers,andthealgorithmictechniquesdevelopedbyKnuthandPardo,wewilltakeabriefdigressioninthischaptertodiscussthehistoryandmotivationforstudyingthequestionsofthisthesis.Itisnotclearwhentheclassicationofintegersintothetwodistinctcategoriesofbeingprimeorcompositewasrstconsidered.Primenumbersarethoseintegersnsuchthatifn=abthena=1ora=n,compositenumbersarethoseforwhichthereexistintegersaandbsuchthatn=abwitha,b>1.Notethatitisatrivialfactthatthereexistsinnitelymanycompositenumbers;however,itisanimportantandhighlynon-trivialfactthatthereexistinnitelymanyprimenumbers.Again,itisunclearwhorstprovedthisobservationrigorouslyasitwasmadesolongagothatnorecordssurvive.However,thisresultwasknowntotheGreekmathematicianEuclidwhoincludeditsproofinBookVIIofhistextbookTheElementsataround300B.C.andthereforethisresultistypicallyattributedtoEuclidbymodernauthors[ 5 ].Whilepreviousmathematiciansmayhavebeenawareofthisresult,itwasEuclidwhorstrecognizedthenecessitytosystematicallydevelopthetheoryofnumbersandgeometryfrombasicaxiomsanditisforthisreasonthatEuclidisconsideredthefatherofbothgeometryandnumbertheory.Euclid'sproofsarealsobrilliantintheirsimplicity;forexample,Euclid'sproofthatthereexistinnitelymanyprimeseasilygeneralizestoprovethatanyinnitecommutativeringwithunitycontainsinnitelymanyprimeideals(fortheproofofthisfactseetheintroductorytext[ 13 ]byI.MartinIsaacs).Hence,thisbasicresultofEuclidanticipatedtheworkoflatermathematiciansbyalmosttwomillennia;butEuclidwasalsoagreatdisseminatorofmathematicalknowledge,andhismagnumopus,TheElements,isconsideredbymanytobethesinglemostinuentialtextbookeverpublishedintheeldofmathematics.Thiscanhardlybe 8

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understated,TheElementsisthesecondmostpublishedtextinhistory(secondonlytotheBible),wasusedbyArabandEuropeanmathematiciansthroughouttheancientandmedievalworld,wasoneoftherstmathematicaltextstobesetintype(byVenetianprintersin1482),andisstillwidelyreferencedinourmodernage[ 5 ].Forexample,thephysicistSirIsaacNewtonlearnedgeometryfromEuclid'stextsaslateas1667,andthephilosophersImmanuelKant(whodied1804)andArthurSchopenhauer(whodiedin1860)vehementlydefendedEuclid'sgeometryintheirwritings.There-examinationofEuclid'sworkinthe18thand19thcenturiesinthehandsofN.I.Lobachevsky,J.Bolyai,C.F.Gauss,andB.Riemannledtothediscoveryofsocallednon-Euclideangeometries.Furthermore,itshouldbenotedthatwhileovertwothousandyearshaveelapsedbetweenthetimeofitsoriginalpublication,virtuallyeveryintroductorynumbertheorytextbookstillusesEuclid'sprooftoshowtheinnityofprimenumbers.AroundaboutEuclid'slifetimetherelivedanothermathematicianknownasEratosthenesofCyrene,whoinadditiontobeingamathematicianwasalsoacelebratedpoet,athlete,geographer,andastronomer.EratosthenesisperhapsbestrememberedforbeingtherstpersontoaccuratelymeasuredthecircumferenceoftheEarth.Sometimearound270B.C.Eratosthenesdevelopedwhatmaybeconsideredtherstalgorithmfordeterminingtheprimenumberslessthanagiveninteger.Thismethod,knownasthesieveofEratosthenes,restsonthefollowingsimpleobservation(whichwewillusethroughoutthisthesis):ifn=abanda,b>p nthenn=ab>n,whichisacontradiction;hence,ifn=abandabthenap nb.Therefore,ifniscomposite,thenitmustalwayscontainaprimefactorlessthanorequaltop n.Eratosthenesnotedthatifonelistsallthenumbersfrom1toN,thenstartingbycirclingtheprimetwo,strikeoffeverysecond(i.e.even)numberbeyondit,thengotocircletheprimethreeandstrikeoffeverythirdnumberbeyondit,andcontinuethisprocessuntilyoureachthep N,thenallthenumberswhichhavebeencrossedoffarecomposite.AllthenumberswhichhavenotbeencrossedoffarepreciselytheprimenumberslessthanorequaltoN.This 9

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processissuccessfulpreciselybecauseweareeliminatingthosenumberslessthanorequaltoNwhicharecomposite,i.e.sinceeachofthesecompositenumbersmusthaveaprimefactorpsuchthatpp N,allofthesenumbershavebeeneliminated.Thus,wehaveaneffectivemethodfordeterminingprimenumbers.ItisinterestingtocomparethetwomethodsutilizedbyEuclidandEratosthenestoaddressthetheoryofprimenumbers.Euclid'sproofisveryusefulfromatheoreticalstandpoint,asitalwaysguarantees(withincertainbounds)theexistenceofaprimenumber,andinfactadirectapplicationofhisproofshowsthatthenumberofprimespxisgreaterthanloglog(x),forx>2,andaslightlymoresophisticatedargumentallowsustoimprovethisresulttolog(x)=2log(2)(seePropositions2.4.1and2.4.2in[ 12 ]).However,despiteitstheoreticalutilityanditsabilitytoderivenontriviallowerbounds,Euclid'smethoddoesnotofferanyhopeofexplicitlycomputingtheprimenumberspx.Bycontrast,Eratosthenesmethodallowsonetoexplicitlycomputeallprimespx;nevertheless,atpresentnosievingprocedure(beitthesieveofEratosthenesoritsmoremoderngeneralizationssuchastheLegendresieveorBrun'ssieve)canevenprovetheinnitudeofprimenumbers(whichiseasilyprovedusingEuclid'stheorem).Thishighlightsamajorthemewhichwillfollowusfortheremainderofthethesis:theoreticalresults,howeverpowerful,areoftenuselesswhenattemptingtoanswerquestionsrelatedtotheexplicitcomputationoftheprimefactorsofintegers.However,thesequestionsmaybeansweredquiteeasilybyusingcomputationalmethods(suchastheSieveofEratosthenesortheKnuth-Pardoalgorithm).Ofcourse,computationalmethodsrarelyaddressquestionsrelatedtothedistributionofprimes,fortunately,thisispreciselywheretheoreticalresultsshowtheirmettle.Hence,ifwetrulywishtounderstandtheprimefactorizationofanintegerwemustfamiliarizeourselveswiththeallieddisciplinesofcomputationalandtheoreticalnumbertheory.AfterthefallofthePtolemaicDynastyatthehandsoftheRomanEmpire,theeraofGreekmathematicsinwhichEuclidandEratostheneslivedcametoanend.The 10

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waningofthegreatGreekEmpiresbroughtwithitastagnationinthestateofnumbertheory,andparticularlythetheoryofprimenumbers.Formanyyearsnumbertheory,algorithms,andalgebra(relyinguponpersonalresearchandtranslationsofGreekmathematicalworks)wastheexclusivedomainofEasternmathematicianslivingintheArabworld,anditisforthisreasonthatthewordsAlgebraandAlgorithmhavetheirrootsintheArabandFarsilanguages.Itwasnotuntilafterthemiddleagesthatmodernwesternauthors,suchasP.FermatandL.Euler,notedhowfundamentaltheprimesnumbersweretobasicquestionsofarithmetic,andbegantoreexaminetheirproperties.Itisoftenthecaseinnumbertheorythatonestudiesagivenarithmeticfunctionf(n),whereifn=abandaandbsharenocommonprimefactors(i.e.arerelativelyprime)thenf(n)=f(ab)=f(a)f(b).Arithmeticfunctionswiththispropertyarecalledmultiplicativefunctions;andmanyfundamentalarithmeticfunctionsareindeedmultiplicative.AlthoughEuclidessentiallyprovedinantiquitythatallintegerscanbewrittenuniquelyastheproductofprimepowers,thisstatementwasnotrigorouslyproveduntilthe18thcenturybythemathematicianC.F.Gauss[ 5 ],andisofsuchimportancetothetheoryofnumbersthatitisoftenreferredtoastheFundamentalTheoremofArithmetic.Henceifoneisgivenamultiplicativefunctionf,andasdistinctprimepowersarerelativelyprime,itfollowsthatifwecandeterminethevaluesoffattheprimepowersthenwemayalwaysdeterminethevalueoff(n)whereniscomposite.Thisfactalonemotivatesustodelvedeeperintothepropertiesoftheprimenumbers.AninterestingdigressionfollowsfromnotonlystudyingtheprimesintheringZbutbystudyingprimesinageneralcommutativeringR.Ofcourse,asshouldbeexpected,thereareafewtechnicalitieswhendealingwithRinsteadofZ.IfIisasubmoduleoverRsuchthatIRthenIiscalledanideal(althoughRsatisesthispropertyitiscustomarytoonlylookatidealsI6=R,whicharecalledproperideals).Anelementu2Riscalledaunitifu)]TJ /F7 7.97 Tf 6.59 0 Td[(12R,i.e.uisaunitifitisinvertible.Anonunita2Rissaidtobeirreducibleifwhenevera=bc,withb,c2Rtheneither 11

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borcisaunit(notethatthisdenitionofirreducibilitywastheoriginaldenitionofbeingprime,untillaterdevelopmentsinalgebraseparatedthetwoideas).Anonunita2Rissaidtobeprimeiftheidealgeneratedbya,(a)=P,hasthepropertythatifbc2Ptheneitherb2Porc2P.Ofcourse,inZthetwoconceptsofirreducibleelementsandprimeelementscoincide;however,ingeneralitisnotnecessarythatirreducibleelementsmustbeprime.Furthermore,Gauss'sobservationthatintegersfactoruniquelyintoprimepowersdoesnotholdinringsingeneral,astheexampleof6=(2)(3)=(1+p )]TJ /F3 11.955 Tf 9.3 0 Td[(5)(1)]TJ 12.33 9.46 Td[(p )]TJ /F3 11.955 Tf 9.3 0 Td[(5)demonstratesthatinZ[p )]TJ /F3 11.955 Tf 9.3 0 Td[(5]theinteger6maybefactoredintotheproductoftwoirreducibleelementsintwodifferentways.Nevertheless,theconceptofuniquefactorizationcanbegeneralizedtoidealsinRbyimposingtheconditionthatsuchidealsfactoruniquelyastheproductofthepowersofprimeideals.ThisleadstotheconceptofDedekindDomains,whichhaveacentralplaceinthemoderntheoryofcommutativealgebra(unfortunatelyitwouldtakeustoofaraeldtodiscussthesealgebraicstructuresintheirfullgenerality).Essentially,itisourhopethatthereaderwillrecognizehowimportantGauss'stheoremonuniquefactorizationis.ItisoffurtherinteresttonotethatwhiletheGreekandArabmathematiciansofantiquitywerewellawareoftheinnityofprimenumbers,andthattheywereallabletoprovetheoremsconcerningprimenumbers,thereisnorecordedevidencethatanymathematicianfromthiseraconsideredthequestionofhowmanyprimenumberspxor,essentially,howaretheprimesdistributedthroughouttheintegers?Furthermore,thereisnoreferencetotheelementarilyequivalentquestionofhowlargepi(theithprimenumber)is?Whileonecaneasilyspeculateastowhythesequestionswerenotasked(namely,duetohindsight,wenowknowthesequestionsareverydeep),itisentirelypossiblethattheancient'slackofmoreadvancedmathematicaltechniquesmadeitnearlyimpossibleforthemtoaddresssuchquestions.Also,itisonlynaturalthattheGreekswouldhavepublishedonlywhattheycouldprove;hence,whilethereisnoevidenceofanyonetryingtoproveresultsconcerningthedistributionofprimenumbers 12

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inantiquity,thisshouldnotbetakenasevidencethatthequestionwasofnointeresttosomeoneintheancientworld.Theresolutionofthisquestionwouldoccupytheattentionofmanymathematiciansforseveralcenturies,beforeultimatelybeingresolvedattheendofthe19thcentury.Let(x)denotethenumberofprimeslessthanorequaltox.Inthelate18thcentury,theFrenchmathematicianA.M.Legendre,usingextensivenumericalevidence,conjecturedthat(x)wouldbeaboutx log(x))]TJ /F5 7.97 Tf 6.59 0 Td[(AforsomeconstantA.HealsofurtherconjecturedthatA1.08366....Thisconstantappearstohavebeenchosenlargelysothattheestimatewouldtthedataavailable.AtaroundthesametimeLegendremadethisconjecture,theprodigiousGermanmathematicianC.F.Gauss(alsousingextensivenumericaldata)speculatedthat(x)wouldbeabout li(x)=Zx2dt log(t)=x log(x)+x log2(x)+...(1)Notethatwhiletheratiolimx!1li(x)(log(x))]TJ /F4 11.955 Tf 11.95 0 Td[(A) x=1foranyvalueofA,thefunctionli(x)isafarstrongerapproximationthanwhatwasconjecturedbyLegendre.AnotherreasonwhyGauss'sestimateissuperiortothatofLegendre'sisthat,hadthetwoindividualshadtheextensivelistsofprimenumbersthatwehavetoday,theywouldhavenotedthatjli(x))]TJ /F6 11.955 Tf 11.96 0 Td[((x)j
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(forfurtherresultssee[ 7 ]).Forlargervaluesofxthisdiscrepancyonlybecomeslarger,andbysimplenumericalestimatesitbecameapparentthatli(x)wasthebetterapproximationto(x).Theonlyproblemwasthatforalloftheirbrilliance,neitherGaussnorLegendrecouldprovethisconjecture,thereafterreferredtoastheGauss-LegendreConjecturebymanynotableauthors,andlatercametobecalledthePrimeNumberTheorem.WhiletheproofofthePrimeNumberTheoremeludedmathematiciansforthemajorityofthe19thcentury,itsutilitycouldnotbeignored.Aswasstatedbefore,manyusefulfunctionsinnumbertheoryaremultiplicative;therefore,ifonecoulddeterminetheapproximatedistributionoftheprimenumbersthenmanyotherquestionsinnumbertheorycouldalsobesolved.Thus,aproofoftheprimenumbertheorembecameasortofholygrailfor19thcenturymathematicians.Thenextnotablecontributiontothestudyof(x)camefromtheprolicRussianmathematicianP.L.Tchebyschev.Around1850Tchebyschevprovedthat 0.89x log(x)<(x)<1.11x log(x),(1)sothattheorderofmagnitudeof(x)conjecturedbyGaussandLegendrewasindeedcorrect.TchebyschevalsoprovedthatifthereexistsanAsuchthattherelativeerrorof(x))]TJ /F4 11.955 Tf 36.95 8.09 Td[(x log(x))]TJ /F4 11.955 Tf 11.96 0 Td[(Aisminimized(i.e.ifthereexistsabestpossibleAintheGauss-Legendreestimate),thenA=1,disprovingLegendre'sobservationthatA=1.083....Furthermore,Tchebyschevdemonstratedthatifthelimit limx!1(x)log(x) x=C(1)exists,thenC=1(proofsofalloftheseresultscanbefoundin[ 7 ]).ThisresultillustratesthesubtletieswhicharisewhenattemptingtoprovethePrimeNumberTheorem,asTchebyschev'sresultstatesthattheGauss-Legendreconjectureiscorrect 14

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ifandonlyif(x)log(x) xapproachesalimit.ItistemptingtotakethisresulttoofarandconcludethePrimeNumberTheorem,butrecallthatthereisnoreason(giventheresultsknowntomathematiciansbefore1850)that(x)log(x) xmustapproachalimitatall!Tchebyschev'smethodsweresubsequentlyrenedandgivensharperboundsintheyearsfollowingthepublicationofhismemoiron(x),withthebestknownresultduetothemathematicianJ.J.SylvesterwhoimprovedTchebyschev'sestimateto.956x log(x)<(x)<1.045x log(x);unfortunately,neitherthebrillianceofSylvester,northatofanyofhiscontemporaries,wascapableofimprovingtheTchebyschevestimatetodemonstratethevalidityoftheGauss-Legendreconjecture.ThisfactwaslamentedbySylvester,whoconcludedhisarticleimprovingTchebyschev'sestimate[ 28 ]withthestatementthatinordertoprovethePrimeNumberTheorem...weshallprobablyhavetowaituntilsomeoneisbornintotheworldasfarsurpassingTchebyschevininsightandpenetrationasTchebyschevhasprovedhimselfsuperiorinthesequalitiestotheordinaryrunofmankind.InmanywaysthedreamofJ.J.Sylvesterwasalreadyrealizedseveraldecadesearlierbythenextmajoradvancementinthetheoryofprimenumbers,whichcamefromtheGermanmathematicianG.B.Riemann.Riemann'sapproachtothePrimeNumberTheoremdifferedsignicantlyfromthatofhispredecessor'sinthathebegantounleashthepowerfultechniquesofcomplexanalysistoanswerthequestion.Ifwelet(s)=1Xn=11 nsthenawellknownidentityduetoL.Eulershowsthatfor<(s)>1, 1Xn=11 ns=Yp1)]TJ /F3 11.955 Tf 15.57 8.09 Td[(1 ps)]TJ /F7 7.97 Tf 6.59 0 Td[(1.(1) 15

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Hence,theconnectionbetweentheaboveseriesandtheprimenumberswasknownfromthetimeofEuler,andwasusedtogreateffectbyEulerandTchebyschev;however,theyappeartohaveonlyconsideredthisseriesasafunctionofarealvariables2R,[ 5 ].Riemann'sgreatinsightwasthatbyconsideringtheaboveseriesasafunctionofacomplexvariableonecouldthenextendittoafunctionforanys2C,s6=1(atechniquewhichinmodernterminologyisknownasanalyticcontinuation);hence,thisfunctionisnowcalledtheRiemannZetafunctioninhishonor,andfollowingRiemann'snotationisdenoted(s)fors2C,s6=1.UsingMellininversion(avariationofFourier'sinversiontechnique)Riemannobtainedananalyticexpressionforthefunction(x),andgivesthissolutionas (x)+1 2(x1=2)+...=li(x))]TJ /F12 11.955 Tf 12.35 11.36 Td[(X=>0li(x)+li(x1)]TJ /F8 7.97 Tf 6.59 0 Td[())]TJ /F3 11.955 Tf 10.25 0 Td[(log(2)+Z1xdt t(t2)]TJ /F3 11.955 Tf 11.96 0 Td[(1)log(t),(1)forx>1,whererepresentsacomplexzeroofthefunction(s)[ 7 ].Thisequalitydemonstratestheveryimportantconnectionbetweenthecomplexzerosofthezetafunctionandthedistributionoftheprimenumbers,inparticular,itdemonstratesthatifonecanshowthat<()<1forallcomplexzeros,thenthePrimeNumberTheoremcouldbeproved.Inaddition,Riemannspeculatedthat<()=1 2forallcomplexzerosofthezetafunction.ThislaststatementisthestillunprovenRiemannHypothesis,andisconsideredbymanytobeoneofthemostimportantunsolvedproblemsinmathematics(see[ 7 ]).Riemannsummarizedallofhisresultsconcerningthedistributionofprimenumbersinhisinuential8-pagepaperUeberdieAnzahlderPrimzahlenuntereinergegebenenGrosse(Onthenumberofprimeslessthanagivenmagnitude)whichhesubmittedtothePrussianAcademyin1859inthanksforhisinductionthere.RiemannstatesinitsrstparagraphthathewishestosharesomeofhisobservationswiththeAcademy,anditisforthisreasonperhapsthatRiemanndoesnotincluderigorousproofsof 16

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manyoftheresultswhichhederives.Sadly,Riemanndiedshortlyafterthepublicationofthismemoir,sowewillnevertrulyknowifthesestatementswereconjecturesorwell-reasonedtheoremswhoseproofsweresimplyomittedforbrevity.Nevertheless,Riemann'spaperessentiallyoutlinedforfuturegenerationsawaytoprovethePrimeNumberTheorem,andoverthecourseofthenext40-yearsseveralmathematicianswouldllthesegapsandresolvetheconjectureofGaussandLegendre.Thelastdecadeofthe19thcenturysawgreatstridesinthetheoryoffunctionsofacomplexvariable.TheseresultsallowedtheGermanmathematicianH.vonMangoldttorigorouslyproveidentity(1-5)in1894[ 7 ],aswellasseveralotherassertionmadebyRiemannconcerningthecomplexzerosof(s).TheFrenchmathematicianJ.Hadamardalsosucceededinprovingseveralnecessaryresultsonthefunction(s)toresolvesomeofRiemann'sstatements;however,in1896thetwomathematiciansJ.HadamardandC.delaValleePoussinnallydemonstratedthat<()<1,anecessaryresultwhichtheyusedtosuccessfullyprovethePrimeNumberTheorem(formoreonthehistoryandproofsofthesetheoremssee[ 7 ]).Ofthetwoproofs,delaValleePoussin'sisthedeeperandwillbesketchedinsection2.2,whileHadamard'sisthesimpler.Hadamard'sproofdemonstratesthat(x)=li(x)+R(x),where(R(x)=li(x))!0asx!1,whereasdelaValleePoussin'sproofdemonstratesthatwemaytakeR(x)tobesomefunctionwhichgrowsnofasterthanaconstanttimesxe)]TJ /F5 7.97 Tf 6.59 0 Td[(Cp log(x),forsomeconstantC>0,i.e. (x)=li(x)+O(xe)]TJ /F5 7.97 Tf 6.59 0 Td[(Cp log(x)).(1)DelaValleePoussinestablishedthisboundbyshowingthatif=+iwith,2R,then 17

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>1)]TJ /F4 11.955 Tf 25.5 8.08 Td[(c log(),(1)forsomeconstantc>0;hence,(s)hasnozerosinsomeregionabouttheline<(s)=1.However,ifonecouldproveRiemann'shypothesisthat<()=1 2thentheerrortermcouldbeimprovedsubstantially,thatis,onemaytakeR(x)tobeafunctionwhichgrowsnofasterthanaconstanttimesx1=2log(x).Therefore,optimizingtheerrorterminthePrimeNumberTheoremrequiredknowledgeofthecomplexzerosof(s)whichwerenotavailabletomathematiciansofthe19thcentury.Largelyfromthismotivation,numbertheoristsofthe20thcenturybegantofurtherexplorethepropertiesoffunctionsofacomplexvariable.Theadventofthe20thcenturybroughtwithitavastincreaseintheapplicationsandworldlinessofnumbertheoryresearch.Advancesintechnologymadecommunicationamongstmathematicianseasierandresultscoulddisseminatemorequicklythaninthepast.OnemaynotethatthemajorityoftheworkdoneonthePrimeNumberTheoremduringthe19thcenturywasaccomplishedbymathematiciansworkingincontinentalEurope.England,atthetime,wasstillstymiedinitsreverenceforthepast(thenthecornerstoneoftheEnglisheducationalsystem)andlaggedinthetheoryoffunctionsofacomplexvariable.Asanexample,studentsinCambridgeUniversityin1910wouldstillprefertheuseNewton'scumbersomenotationfordifferentiationoverthatofLeibniz,andwithallduerespecttoNewton'smathematicaltalents,Leibniz'snotationisclearlysuperior.However,thersttwodecadesofthe20thcenturybroughtEnglishmathematiciansintothecontinentaldiscussionoverthedistributionoftheprimenumbers,withgreateffect.DuringthistimeG.H.HardyandJ.E.Littlewoodadvancedthetheoryoffunctionsofacomplexvariable,andevensucceededingivingtheirownproofsofthePrimeNumberTheoremwhichwerefarsimplerthantheoriginalssuppliedbyHadamardanddelaValleePoussin.TheyalsodemonstratedthelogicalequivalenceofthePrimeNumberTheoremwiththestatementthat(s)6=0where<(s)=1.G.H. 18

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Hardyalsosucceededinshowingthatthereareinnitelymanyzerosof(s)suchthat<()=1 2,andwentontogivemorespecicestimatesofhowmanyzerosof(s)lieontheline<(s)=1 2(fortheseresults,andtheirextensions,see[ 7 ]).TothisdayG.H.Hardy'sworkissomeofthestrongestevidencethatRiemann'sHypothesisistrue,althoughhisworkdoesnotprovideaproofofthisstatement.TheworkofHardyandLittlewoodonfunctionsofacomplexvariable,whilemotivatedprimarilybyquestionsrelatedtonumbertheory,extendsfarbeyondtheimplicationsofthesetheoremsforarithmeticfunctions.AclassicalresultduetothecelebratedNorwegianmathematicianN.H.Abelstatesthatif f(z)=1Xk=0bkzk(1)isapowerserieswithradiusofconvergence1,convergingforz=1suchthatallbi0and1Xk=0bk<+1,then limz!1)]TJ /F4 11.955 Tf 8.25 5.82 Td[(f(z)=1Xk=0bk=f(1)(1)ThistheoremhasananalogueforDirichletseries,whicharebasicobjectsinthestudyofanalyticnumbertheory;therefore,itbecameapparentthatifaDirichletseriescouldbeshowntosatisfycertainconditions,thenAbel'stheoremwouldimplyimportantpropertiesaboutthearithmeticfunctionsgeneratingtheseDirichletseries.OneirritatingfactformathematicianswholivedduringAbel'stimeisthatthefullconverseoftheabovetheoremisnottrue,andinfactthisoversightledeventheverytalentedAbeltopublishsomeerroneousresultsconcerninginniteseries.Itwouldnotbeuntil1897thatapartialconversetoAbel'stheoremwouldbediscoveredbytheGermanmathematician 19

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A.Tauber.Tauber'stheoremstatesthatifweletf(z)bethepowerseriesinequation(1-8)withradiusofconvergence1andsupposethatthereexistsan`2Csuchthatlimz!1f(z)=`,where0z<1.Furthermore,ifXnxbnn=o(x)then, limz!1f(z)=1Xk=0bk=`<+1.(1)Thistheorem,whichallowsonetosolveforthesumofthecoefcientsofapowerseriesbasedsolelyontheanalyticnatureofthegivenpowerseries,ledtoanewandpowerfulanalyticapproachtoansweringquestionsinthetheoryofnumbers(thoughitshouldbenotedthatTauberwasnotanumbertheorist,andtheapplicationofhistheoremtoprimenumbertheorywouldhavetowaitfortheworkoflaterauthors).HardyandLittlewood,recognizingtheutilityofthisapproach,generalizedtheresultsofA.Tauberinordertodeducetheirownproofsoftheprimenumbertheorem;furthermore,theyaddedanewwordtothemathematicallexiconbyreferringtoresultsofthistypeasTauberianTheorems,inhonoroftheworkofAlfredTauber.Informally,anyresultthatdeducespropertiesofafunctionfromtheaverageofthatfunctionwewillrefertoasaTauberiantheorem;andconversely,anyresultwhichdeducespropertiesoftheaverageofafunctionfromthepropertiesofthatfunctionwewillrefertoasanAbeliantheorem.Ingeneral,TauberiantheoremsareofgreaterinteresttonumbertheoriststhanareAbeliantheoremsbecausethelatterclassoftheoremsisessentiallyanexercise(oneisgivenafunctionandsoonemustcalculateitsaverage)whereastheformerclassof 20

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theoremsattemptstoimposerestrictionsuponwhatfunctions,ifany,canpossessagivenaverage.In1913HardyandLittlewoodshowedthattheresultsofTauber'stheoremwouldfollowfromfarmoregeneralassumptions.In1931theSerbianmathematicianJ.KaramatasucceededingivingamuchsimplerproofoftheHardy-LittlewoodTauberiantheorem,anditisforthisreasonthattheTauberiantheoremofHardy-LittlewoodisoftenreferredtoastheHardy-Littlewood-Karamatatheorem.Unfortunately,evenusingtheHardy-Littlewood-Karamatatheoremitisnotparticularlystraightforwardhowonecandeducetheprimenumbertheoreminasimplemanner,anditwasnotuntil1971thatLittlewoodsuppliedaquickproofoftheprimenumbertheoremusingthismethod;however,itshouldbenotedthatLittlewood'sproofrequiresseveraldeepresultsontheanalyticnatureoftheRiemannZetafunction(see[ 29 ]).OnedisadvantageoftheHardy-Littlewood-Karamatatheoremisthatitcanonlydealwithsingularitiesofastandardtype,i.e.singularitiesoftheform(s)]TJ /F4 11.955 Tf 12.15 0 Td[(a))]TJ /F5 7.97 Tf 6.59 0 Td[(b,wherea2Randb2Z.Thisdeciencywasovercomebythecombinedworkofseveralprominentmathematicians.In1931theAmericanmathematicianN.WienerandtheJapanesemathematicianS.Ikehara(anerstwhilestudentofWiener's)extendedtheworkofHardy,Littlewood,andKaramatatoincludesingularitiesofthetypes)]TJ /F8 7.97 Tf 6.59 0 Td[(!)]TJ /F7 7.97 Tf 6.59 0 Td[(1,where!>)]TJ /F3 11.955 Tf 9.3 0 Td[(1isanyrealnumber([ 29 ]).Theirapproach,initsmodernformulation,alsoowesmuchtotheworkoftheEnglishmathematicianA.Ingham.ItisalsointerestingtonotethattheTauberiantheoremofWiener-Ikehara-Inghamallowsonetodeducetheprimenumbertheoremwithonlyasingleminimalassumption,namely,thattheRiemannZetafunction(s)isnonzeroforanycomplexnumberssuchthat<(s)=1;hence,theprimenumbertheoremandthenon-vanishingof(s)for<(s)=1arelogicallyequivalent([ 29 ]).Asthenon-vanishingof(s)for<(s)=1isanentirelycomplex-analyticpropertyof(s),thisresultledmanyprominentmathematicians(suchasG.H.Hardy)tobelievethattheprimenumbertheoremcouldnotbededucedwithoutusing(eitherimplicitly 21

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orexplicitly)thetheoryofcomplexvariables.Infact,HardywassosurethatthePrimeNumberTheoremcouldnotbeprovedwithoutthetheoryofcomplexfunctionsthathefamouslystatedthatifsuchanelementaryproofcouldbefound,thenallofournumbertheorytextbookswouldhavetobetakenoffoftheshelvesandrewritten.Itthereforecameasquiteasurprisewhen,in1948,P.ErdosandA.SelbergsucceededinprovingtheprimenumbertheoremwithouttheuseoftheRiemannZetafunctionorthepropertiesofcomplexfunctions.Theirproof,whileelementaryinthesensethatitdoesnotrequiretheuseofcomplexfunctiontheory,isnotlackinginsubtlety;hence,itwouldbeunwisetoconfuseelementarywithsimple(forthemotivatedreader,theproofmaybefoundin[ 7 ]).AroundthesametimethatTauber,Hardy,andLittlewoodwereconductingtheirinvestigationsintoTauberiantheorems,theFinnishmathematicianH.MellinbeganconsideringanintegraltransformnowreferredtoastheMellintransform.NotethatifwearegivenaconvergentDirichletseriesD(s)=1Xn=1dn ns,anddenoteS(x)=Xnxdn,whereS(x)=O(x)thenwemayinterpretD(s),for<(s)>astheStieltjesintegral: D(s)=Z11dS(t) ts=sZ11S(t) ts+1,(1)obtainedbyusingintegrationbyparts(andusingthefactthat<(s)>).Identity(1-11)canalsobeviewedasaspecialcaseofthefollowingmucholderformuladuetoAbel,andwhichistypicallyreferredtoastheAbelsummationformula.If 22

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SN=NXn=0anbnandBn=nXk=0bkthen SN=aNBN)]TJ /F5 7.97 Tf 11.96 14.95 Td[(N)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xn=0Bn(an+1)]TJ /F4 11.955 Tf 11.96 0 Td[(an).(1)Itisinterestingtonotethat(1-11)canbededucedfrom(1-12).Theintegralin(1-11),however,isnowaMellintransform,anditisanastoundinglyfortunatepropertythatmanyDirichletseriescaneasilybeviewedastheMellintransformofcertaininterestingarithmeticfunctions,anditisevenmorefortunatethatthetheoryoftheMellintransformhasachievedsuchalevelofmaturityinourmodernage(beingageneralizationoftheworkofseveralfamousmathematicians).Asearlyas1744theSwissmathematicianLeonardEulerconsidered(inaratherun-rigorousmanner)whatwasessentiallyaMellintransform,andhisworkwasexpandeduponbyJosephLouisLagrange(agreatadmirerofEuler's).Themoderntheoryofintegraltransformsbeganin1785withtheFrenchmathematicianandphysicistPierre-SimonmarquisdeLaplace,whogavethetheoryamoresolidtheoreticalbasisandmadetwoimportantobservationsregardingintegraltransforms.Firstly,LaplaceshowedthatbyapplyingaspecicMellintransformtoagivenfunction(calledtheLaplacetransform)onecoulddeduceimportantpropertiesaboutthederivativesofthetransformedfunction,makingtheLaplacetransformanimportanttoolinthestudyofdifferentialequations.Laplace'ssecondobservationwasthatonecouldrecovertheoriginalfunctionfromitsLaplacetransformbyapplyinganotherintegraltransform,whichforobviousreasonsiscalledtheinverseLaplacetransform.Anotherimportantcontributortothetheoryofsuchinversion 23

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techniqueswastheFrenchscientistJeanBaptisteJosephFourier,who'smostpopularpublicationTheorieanalytiquedelachaleurappearedin1822.ThekeyobservationofLaplace,Fourier,andotherswasthat(subjecttocertainconditionsonthefunctionbeingtransformed)onecouldapplyanintegraltransformtoafunction,deduceusefulpropertiesaboutthatfunction,andtheninverttheproceduretorecovertheoriginalfunction;therebyelucidatingnewpropertiesaboutthefunctioninquestion.Itisalsointerestingtonotethatthistechniquewasdevelopedlargelytoanswerquestionsinheatconduction,wavepropagation,celestialmechanics,andprobability.Perhapsowingtotheemphasisofintegraltransformsintheappliedsciences,Mellinwaslessinterestedinthenumbertheoreticramicationsofhisworkthanthefunctiontheoreticimplicationsofhisinversionprocedure.WhiletheapplicationofthisinversionprocedurehadessentiallyalreadybeenusedbyHadamardanddelaVallee-PoussinintheirproofsofthePrimeNumberTheorem,Itwasnotuntilthe1908paper[ 19 ]whentheGermanmathematicianO.PerronappliedMellin'sinversionproceduretoageneralDirichletseriesthatveryimportantarithmeticconsequencesofMellin'sworkcouldbeappreciated.Perronsucceededinderivingaformula(nowreferredtoasPerron'sformula)whichequatedthepartialsumsofthecoefcientsofagivenDirichletserieswiththeinverseMellintransformofthisDirichletseries.Moreprecisely,ifaDirichletseriesgivenbyD(s)=1Xn=1an nsisconvergentforalls2Csuchthat<(s)>c,and>max(0,c),then S(x)=1 2iZ+i1)]TJ /F5 7.97 Tf 6.58 0 Td[(i1D(s)xss)]TJ /F7 7.97 Tf 6.59 0 Td[(1ds,(1)wherex>0andS(x)=Xnxanifx2R)]TJ /F10 11.955 Tf 12.29 0 Td[(N,whileS(x)=Xn
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Formula(1-13),whichwewillusethroughoutthethesis,isaremarkablyusefultoolforderivingresultsinanalyticnumbertheory,andisoneofthemostusefulAbeliantheoremscurrentlyknown(seeChapterII.2in[ 29 ]).Infact,theformulaofPerronwouldeasilysupersedemostTauberiantheoremsifitweren'tforthefactthat,ingeneral,itisaratherdifculttasktoevaluatetheinverseMellintransformofaDirichletseries.In1953theIndianmathematicianL.G.Sathesucceededinderivinganasymptoticformulaforthenumberofintegersnxwhosedistinctprimefactorswereequaltok,denotedk(x).Whileothermathematicians(suchasE.Landau)werecapableofderivingthisasymptoticforxedk,Sathedemonstratedthatthisresult(equation(2-32))helduniformlyink.Sathe'soriginalproofofthisresultusestheprincipalofmathematicalinduction,andisveryinvolved(see[ 24 ]).Asanexample,Sathe'soriginalproofspannedover70pagesandwasinfactsoinvolvedthatoneyearlater,in1954,Sathesawitttopublishasimpliedaccountofhisoriginalproofwhich,althoughshorterthantheoriginal,stillinvolves54pagesofverydifcultmathematics.InthewordsofthehighlyinuentialNorwegianmathematicianAtleSelberg(whoinfactwastherefereeofSathe'ssecondpaper),WhiletheresultsofSathe'spaperareverybeautifulandhighlyinteresting,thewaytheauthorhasproceededinordertoprovetheseresultsisarathercomplicatedandinvolvedone,andthisbynecessitysincetheproofbyinduction...presentsoverwhelmingdifcultiesinkeepingtrackoftheestimatesoftheremainderterms...ItisneverthelesstoSathe'screditthathecouldderivehisresultsinductively,howevercomplicatedhisargumentsmayhavebeen.SelbergnotedthatSathe'sresultscouldbederived,andexpandedupon,byattackingtheproblemfromamoreclassicalapproach(i.e.usingMellin'sinversiontheorem,whichdoesnotrequiretheuseofSathe'scomplicatedinductiveargument,tobediscussedinsection2.3).Inasomewhatunorthodoxmoment,Selbergauthoredashortnote,[ 26 ],onSathe'spaperwhichappearedinthesameissueoftheJournaloftheIndianMathematicalSocietyasSathe'ssecondpaper(andrecallthatSelberg 25

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wastherefereeforSathe'spaper!).SelbergsuccessfullyapproximatedthepartialsumsofthecoefcientsofDirichletseriesofthetypeF(s)=G(s;z)(s)z,where(s)istheRiemannZetafunction,<(s)>1,z2Cisanarbitrarycomplexnumber,andG(s;z)isafunctionwhichisanalyticin<(s)>1=2satisfyingrathermodestgrowthconditions(seeChapterII.5of[ 29 ],orthediscussionoftheSathe-SelbergtechniqueinSection2.3);moreover,theseresultseitherreprovedorexpandeduponallofSathe'stheorems.Sadly,perhapsowingtothelengthofhisownpapers,orthebrevityofSelberg'sarguments,Satheneverpublishedanotherpaperintheeldofmathematics.Essentially,theSathe-SelbergmethodallowsustotreatDirichletserieswithsingularitiesoftheform(s)]TJ /F3 11.955 Tf 13.15 0 Td[(1)z,z2C.However,itisoftenthecaseinanalyticnumbertheorythatwemustconsidersingularitiesofadifferenttype,suchaslogarithmicsingularitiesoftheform:log(1 s)]TJ /F7 7.97 Tf 6.58 0 Td[(1).In1954theFrenchmathematicianH.DelangeexpandedupontheworkofSatheandSelbergtoprovideatheoremwhichappliedtoallDirichletseriessatisfyingmodestconditionsandpossessingsingularitiesofthetype(s)]TJ /F3 11.955 Tf 12.88 0 Td[(1))]TJ /F8 7.97 Tf 6.58 0 Td[(!logk(1 s)]TJ /F7 7.97 Tf 6.59 0 Td[(1),where!2Randk2Zwithk0(Seenote5.1attheendofChapterII.5in[ 29 ]).AsaresultofthecombinedeffortsofSathe,Selberg,andDelange,themethodutilizedtoderiveresultsofthistypeistypicallyreferredtoastheSathe-SelbergorSelberg-Delangemethod(seeChapterII.5TheSelberg-Delangemethodin[ 29 ]).InTheorem2.2.1ofChapterIIwewillstateamajorgeneralizationoftheWiener-Ikehara-InghamtheoremrstprovedbyDelangein1954.InmanywaysthisresultcanbeviewedasthepinnacleofknownTauberiantheorems,asitappliesinanastoundinglygeneralsetting(thatis,forsufcientlyregularDirichletseriespossessingsingularitieswhicharebothmonomialandlogarithmic).InChaptersIandIIitwillfurtherbedemonstratedthatmostofourresultsfollowfromsimpleapplicationsoftheSelberg-DelangemethodintheguiseofTheorem2.2.1.AtthesametimethatG.H.HardyandJ.E.Littlewoodwereconductingtheirresearchinanalyticnumbertheory,thelargelyself-taughtIndianmathematicianS. 26

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RamanujandraftedalettertoHardyinanattempttogainrecognitionofhiswork.Hardy,sooverwhelmedwiththebrillianceofRamanujan'swork,arrangedfortheyoungmathematiciantocomeworkwithhimatCambridge.Thisbeganasignicantcollaborationwhichinspired,amongstotherthings,thestudyofadditivearithmeticfunctions.Recallthatanarithmeticfunctionf(n)ismultiplicativeifn=abwithaandbrelativelyprimeimpliesthatf(n)=f(ab)=f(a)f(b).Anadditivefunctionisverysimilar,ifn=abwithaandbrelativelyprime,thenanarithmeticfunctionisadditiveiff(n)=f(ab)=f(a)+f(b).Oneexampleofanadditivefunctionislog(n).Amoreinterestingnumbertheoreticexampleisthefunction!(n),whichisusedtodenotethenumberofdistinctprimedivisorsofanintegern;similarly,thenumberofprimepowerdivisorsofanintegern,denoted(n),isalsoadditive.HardyandRamanujanprovedthattheratiolimx!1Pnx!(n) Pnx(n)=1,andthat(aswillbediscussedinsection2.3)theaveragesofboth!(n)and(n)fornxhaveorderofmagnitudeloglog(x)(see[ 11 ]or[ 29 ]).Thisinitselfisarathersurprisingresult,asitstatesthatonaveragemostprimefactorsoccursquare-free.Thisresultwasalsooneoftherstresultsconcerningtheaveragevalueofanadditivearithmeticfunction.Notethat!(n)=Xpjn1andthat(n)=Xpjjn, 27

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wherepjjnimpliesthatpjnbutp+1doesnotdividen.WiththesedenitionsitisonlynaturaltoaskaboutthesimilarfunctionsA(n)=XpjnpandA(n)=Xpjjnp,whicharethesumofthedistinctprimedivisorsofnandthesumoftheprimedivisorsofnweightedaccordingtomultiplicity,respectively.Thesefunctions,justlike!(n)and(n),arealsoadditiveandareintimatelyconnectedwiththesefunctions.Itisthereforesomesurprisethatittookover50yearsfromthetimeofHardyandRamanujan'sworkbeforethesefunctionswerestudiedinearnest.ThetheoryofthesefunctionsisalsothestoryofacollaborationbetweenaneminentEuropeanandayoungIndianmathematician,andasthesefunctionshaveacentralroleinthefollowingthesiswewillgointosomedetailaboutthiscollaboration,anditsultimateresults.TheprolicHungarianmathematicianP.ErdosvisitedCalcuttain1974wherehemetthetheoreticalphysicistandeducatorAlladiRamakrishnan.Ramakrishnan'sson,KrishnaswamiAlladi,hadbeeninvestigatingthefunctionsA(n)andA(n)(denedbelowastheAlladi-Erdosfunctions)independentlyandhadobtainedseveralinterestingresults,aswellasraisingmanydeeperquestionswhichhewasunabletoanswer.Erdos,evereagertocollaboratewithyoungmathematicians,reroutedhisightfromCalcuttatoAustraliatostopinMadrassothathecouldvisittheyoungK.Alladi,whowasatthetimestillanundergraduate.Aftertheirdiscussions,manyofwhichwereconductedwhilewalkingalongthebeach,ErdosandAlladipublishedseveralpaperswhichproved,amongstotherthings,thatPnxA(n)andPnxA(n)bothhaveorderofmagnitude2 12x2 log(x),[ 2 ](theAlladi-Erdoscollaborativeencounterisalsonicelyrecounted 28

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inChapter1ofBruceSchechter'sbiographyonErdosentitled:MyBrainisOpen,[ 25 ]).NotethattheresultsofAlladiandErdosfurthervalidatetheobservationsfromtheHardy-RamanujanTheoremthatmostintegersdonothavelargeprimepowerdivisors.AlladiandErdos'sworkalsoconcerneditselfwiththelargestprimefactorsofaninteger.LetP1(n)bethelargestprimefactorofn,thenclearly XnxP1(n)XnxA(n);(1)however,itisafact(rstshownbyAlladiandErdos)thatPnxP1(n)alsohasorderofmagnitude2 12x2 log(x),whichissomewhatsurprising.Infact,thisresultshowsthatthemajorityofthevalueofA(n)isaccountedforbyP1(n)(see[ 2 ]).Inalaterpaper(see[ 3 ])AlladiandErdosproceedtoevaluatethesumXnx(A(n))]TJ /F4 11.955 Tf 11.96 0 Td[(P1(n))]TJ /F3 11.955 Tf 11.96 0 Td[(...)]TJ /F4 11.955 Tf 11.96 0 Td[(Pk)]TJ /F7 7.97 Tf 6.58 0 Td[(1(n)),and(rathersurprisingly)demonstratethat: limx!1Xnx(A(n))]TJ /F4 11.955 Tf 11.96 0 Td[(P1(n))]TJ /F3 11.955 Tf 11.95 0 Td[(...)]TJ /F4 11.955 Tf 11.95 0 Td[(Pk)]TJ /F7 7.97 Tf 6.59 0 Td[(1(n)) XnxPk(n)=1.(1)ThisresultstatesthatnotonlyisthemajorityofthecontributionofA(n)accountedforbyP1(n),butthemajorityofthecontributionofA(n))]TJ /F4 11.955 Tf 12.22 0 Td[(P1(n)isaccountedforbyP2(n),i.e.thesecondlargestprimedivisor,andsoforth.AnotherusefuloutcomefromtheAlladi-Erdoscollaboration,wasthatErdosdrewAlladi'sattentiontothefunction(x,y),whichisdenedasthenumberofpositiveintegersnxwhoseprimefactorsaresmallerthany,i.e.P1(n)y.Thisfunction,whoserstpublishedreferencecanbefoundintheworkofR.Rankintoinvestigatethedifferencesbetweenconsecutiveprimenumbers,hasbeenstudiedextensivelybymanymathematicians.ItplaysanimportantroleintheAlladi-Erdospapers,andwe 29

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willhavereasontorefertoitsbasicpropertiesinthisthesisaswell.Whiletheoriginalstudyofthefunction(x,y)istypicallyattributedtothepaper[ 23 ]byRankin(withcontemporaneousinvestigationsbeingconductedbyA.Buchstab[ 6 ],K.Dickman[ 10 ]andV.Ramaswami[ 22 ])itisinterestingthatthestudyof(x,y)(andrelatedfunctions)actuallyappearsinthemuchearlierworkofS.Ramanujan.AsthestorysurroundingthisareaofRamanujan'sresearchisparticularlyfascinating,wewilltakeabriefpausetoappreciatehislifeandtherediscoveryofhisworkrelatedto(x,y).Aswasnotedpreviously,Ramanujanwasalargelyself-taughtIndianmathematicianwhocollaboratedlaterinlifewithG.H.Hardy.Ramanujanhadatremendousloveformathematicsandwouldnotonlystudyitinhisfreetime,butwouldalsogoontoderivenumerousidentitieswhichherecordedinhispersonalpapers.Unfortunately,Ramanujanwasverypoor,andhispovertymadepaperapreciouscommodity,soasaresultRamanujanwouldoftensimplywritedownaformulawithoutanyreferencetohowhehadarrivedattheresultor(moreimportantly)howonecouldprovehisclaim.Ramanujanlledhisnoteswithliterallythousandsofidentitieswhichrunthegamutfromsimplegeometricobservations(whichcanbeproveninapageortwo)toabsurdlycomplicatedidentitieswhoseproofrequireshundredsofpagestoestablish.Sadly,perhapsduetooverwork,livingintheforeignclimateofCambridge,England,orasaresultofgeneralill-health(typicallyattributedtoamoebicdysentery,malnutrition,ortuberculosis)Ramanujandiedin1920attheageof32.AfterRamanujanpassedaway,manyofhispersonalpapersweregiventotheUniversityofMadrasbyhiswife,andtheUniversitypassedalargenumberofRamanujan'snotebookstohisfriendandcollaboratorG.H.Hardy.AfterHardyreceivedRamanujan'spapers,heandhiscontemporariesspentyearsattemptingtoproveRamanujan'sclaims,andafterHardy'sdeathin1947,manyothermathematiciansresumedtheworkofprovingRamanujan'sidentities.MostoftheposthumouslypublishedworkofRamanujanwasalreadyprovenandwell-knownto 30

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mathematiciansby1976,whenthefabledlostnotebookofRamanujanwasdiscoveredbytheAmericanmathematicianG.AndrewsattheWrenLibraryatTrinityCollege,Cambridge.Apparently,sometimebetween1934and1947G.H.HardyhandedanumberofRamanujan'smanuscriptpagestotheEnglishmathematicianG.N.Watson,wholostthesepagesintheseaofclutteredmathematicalpaperswhichoccupiedhisofce.AfterWatson'sdeath,thesepagesweresavedfromincineration(butnotobscurity)bythemathematiciansJ.M.WhittakerandR.RankinwhohadthemstoredintheWrenLibrary.TheyremainedthereuntilAndrewscausedasensationinthemathematicalworldwhenhelocatedthemin1976.Andrews'accountofthestoryissomewhatlessapocryphal,anddenitelylessswashbuckling;apparentlyhehadagoodideaofwheretolookfortheselostnotesinthelibrary,althoughhewasnotcertainabouttheircontentorexactlocation.AmongstthemanytopicsstudiedbyRamanujanduringthelastyearsofhislifewasthefunction(x,y),aswellasarelatedfunctioncalledtheDickmanFunction.Ramanujanmanagedtodeducemanyresultsconcerningthesefunctionsdecadesbeforetheindividualsforwhomtheyarenamed,andwereitnotforthemisplacementofhispapersmaybenamedforhim.Ramanujan'smathematicalabilitiesweretrulyamazing,andofthe1600-orsoidentitiescontainedinhislostnotebook,itseditorB.Burntspeculatesthatlessthanveareincorrect.Asanexampleofthisself-taughtindividual'sinsight,in1930K.Dickmanre-derivedafunctionrelatedto(x,y)calledtheDickmanfunction,denoted(u)whereu=log(x) log(y),xy2.TheDickmanfunctionisrelatedto(x,y)bythefactthatuniformlyforxy2wehave(x,x1=u)sx(u)and,inadditiontothisimportantconnection,(u)satisessomerathersurprisingidentities.Forinstance,(u)iscontinuousatu=1,differentiableforu>1,andforu>1satisesthedifference-differentialequation 31

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u0(u)+(u)]TJ /F3 11.955 Tf 11.95 0 Td[(1)=0,whichwereallknowntodeBruijn[ 9 ].DeBruijnusedthedifferencedifferentialequationtodemonstratethat(u)decreasesveryrapidly(seesection3.2forexplicitbounds).Unknowntoanyone,manyoftheresultsrelatedtotheDickmanfunctionhadalreadybeendeducedbyRamanujanalmost20-yearsearlier(formoredetailsconcerningthere-discoveryoftheseinterestingidentitiessee[ 27 ]).SevenyearsafterDickmanpublishedhisworkA.Buchstabderivedafunctionalequationfor(x,y),andtwodecadesafterBuchstab'swork,N.G.deBruijnsucceededinsignicantlyimprovingontheresultsofRankin,Ramaswami,andDickmanconcerningthisfunction.Forthisreason(x,y)isoftenreferredtoastheBuchstab-deBruijnfunc-tion,asopposedtotheRamanujan(orsomeothermoreappropriatelynamed)function.AroundthetimethatBuchstab,Rankin,Dickman,anddeBruijnwereinvestigating(x,y)(aperiodfromabout1930-1960)theworldenteredwhatmaybetermedthecomputerage.Computingmachinesbegantooutpacehumancalculatingskillstothepointthatinaveryshortperiodoftimemathematicianshadaccesstoamountsofdatawhichwerefarinexcessofwhattheirpredecessorshadknown(orevencouldhaveknown).Inthepresent,mostofususetheinternet,apersonalcomputer,orsomesortofmodularcomputingdeviceonadailybasis,anditwouldbedifcultforustoimagineaworldwheredigitalcommunicationdidnotexist.Thisrelativelycheap,reliable,andinstantaneousformofcommunicationhasonemajorproblem,simplyput,itisdifculttodeterminewhenthemessageissentinasecurefashion.Tosolvethisproblem,manycomputerprogrammersandmathematicianshavedevelopedsophisticatedmeansofencryptingmessagessothatonlytheintendedrecipientcanreadthetransmittedmessage.Theseencryptiontechniqueshavebecomeverysophisticated;however,mostrelyupononesimpleprinciple,namely,thatittakesarelativelyshortamountoftimeforacomputertomultiplyagivensetofintegersbutittakesanextraordinarilylargeamount 32

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oftimetofactoranintegerintoprimenumbers.Forexample,somenumbersusedincomputerencryptiontakeonlyfractionsofasecondtomultiply,butrequirebillionsofyearsusingourbestknownfactoringalgorithmsandsupercomputerstofactorintoprimenumbers.Hence,numericalfactorizationisaveryimportantapplicationofmathematicstoourmodernworld.In1976,whileAlladiandErdoswereconductingtheirtheoreticalinvestigationsintothefunctionsA(n)andA(n),D.KnuthandL.T.Pardopublishedapaper[ 14 ]wheretheyinvestigatedtherunningtimeofwhatisperhapsthesimplestwaytofactoranintegeralgorithmically,namely,thealgorithmicproceduredescribedintherstparagraphofthisintroduction(andwewillcallthismethodtheKnuth-Pardofactorizationalgorithm).Thisfamiliarprocedureforfactoringanintegeristaughttomostofusinpre-calculus(albeitnotinanalgorithmicfashion)andisaveryintuitivewaytofactoraninteger.However,itdoesraisethequestion:howfastisthisprocedure?KnuthandPardoessentiallyanswerthisquestionin[ 14 ].Theconnectionbetween(x,y)andintegerfactorizationalgorithmsisquitetransparent,ifwecouldguaranteethatanintegernxhadallprimefactorslessthanorequaltoy,i.e.P1(n)yx,thenweneedonlycheckforprimedivisorsofnwhicharelessthanorequaltoyx(whichwouldobviouslydecreasethenumberoftrialdivisions,andhencetheefciency,oftheKnuth-Pardoalgorithm,providedthisyisappreciablysmallerthanxofcourse).Wewillhavemuchmoretosayabouttheconnectionbetween(x,y)andtherunningtimeoftheKnuth-Pardoalgorithminthebodyofthethesis.AsimilarobservationholdsifweknewthelargestprimefactorP1(n)ofanintegern,thenwecouldsimplyapplythealgorithmtom=n=P1(n)n,whichcannowbefactoredfasterusingtheKnuthPardoalgorithm(asmn).BylettingPk(n)denotethekthlargestprimefactorofanintegern,itisfairlyclearthatwecoulddecreasetherunningtimeoftheKnuth-PardoalgorithmevenfurtherifwehadknowledgeofnotonlyP1(n),butalsoPk(n),fork2.ForthisreasonKnuthand 33

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PardoinvestigatethemeanandvarianceofPk(n)intheirpapertheoretically,aswellassupplyingextensivetablesofhowthiseffectstherunningtimeoftheiralgorithm(see[ 14 ]). 34

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CHAPTER2THEDISTRIBUTIONOFPRIMESANDPRIMEFACTORS 2.1NotationandPreliminaryObservationsWebeginbyprovingtheclassicalresultthatthereexistinnitelymanyprimesinaperhapsunfamiliarfashion.Thefollowingproof,duetoPaulErdos[ 8 ],isparticularlybeautifulinthatitnotonlyshowsthestrongerresultthatP1 pdiverges,butusestheunderlyingstructure(oranatomy)oftheintegerstodoso. Theorem2.1.1. Xp1 pdiverges.Proof(Erdos):Letp1,p2,...bethesequenceofprimes,listedinincreasingorder.Assume,bywayofcontradiction,thatXp1 pconverges.ThentheremustexistanaturalnumberksuchthatXik+11 pi<1 2.Calltheprimesp1,p2,...,pkthesmallprimesandpk+1,pk+2,...thelargeprimes.LetNbeanarbitrarynaturalnumber.ThenXik+1N pi
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theproductofdifferentsmallprimes;hence,thereareprecisely2kdifferentsquare-freeparts.Furthermore,asbnp np N,weseethattheremustbeatmostp Ndifferentsquareparts,andsoNs2kp N;therefore,N=Ns+Nb1theRiemannZetafunctionisdenedas(s)=1Xn=11 ns.ThezetafunctionwasrststudiedbyEulerwhoderivedthefollowingproductformula,referredtobysomeauthorsasananalyticformofthefundamentaltheoremofarithmetic.Eulernotedthatfors>1 1Xn=11 ns=Yp1)]TJ /F3 11.955 Tf 15.57 8.09 Td[(1 ps)]TJ /F7 7.97 Tf 6.59 0 Td[(1,(2) 36

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andinfactthatidentityholdsforallcomplexssuchthat<(s)>1,[ 7 ].Thisproductformulaisthekeytounderstandinghowonemaydeducepropertiesoftheprimenumbersfromthoseof(s).Bysimplytakingthelogarithmof(s),for<(s)>1,andnotingthattheTaylorseriesexpansionof)]TJ /F3 11.955 Tf 11.29 0 Td[(log(1)]TJ /F4 11.955 Tf 11.97 0 Td[(x)isvalidforx2Csuchthatjxj1,weeasilyderivethefollowingidentity, log(s)=)]TJ /F3 11.955 Tf 11.29 0 Td[(logYp1)]TJ /F3 11.955 Tf 15.57 8.09 Td[(1 ps=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xplog(1)]TJ /F4 11.955 Tf 11.95 0 Td[(p)]TJ /F5 7.97 Tf 6.58 0 Td[(s)(2)=Xp1 ps+Xp1 2p2s+...=Xp1 ps+h(s),whereh(s)isboundedfor<(s)>1 2,andidentity(2-2)holdsfor<(s)>1.Hence,thelogarithmofthezetafunctionessentiallygivesusthesumofthereciprocalsoftheprimes,raisedtothespower.Thekeytounravelingtheasymptoticdistributionoftheprimenumberswillbetoapplysomesortofinversiontechniquetolog(s);ideallywewouldliketoevaluatethenitesumsXpx1 psats=1,butthisisnotinthedomainofabsoluteconvergenceoflog(s)sowecannotsimplysets=1intheaboveidentityandconcludeanymeaningfulresult.Instead,motivatedbytheapplicationsofFourierinversion,wemayconsider(s)tobeafunctionofacomplexvariables2C,thenapplyingaversionofFourier'sinversiontheoremcalledtheMellininversiontheoremwemayevaluatenitesumssuchasXpx1.Thissumisofsuchcentralimportancetonumbertheorythatitwarrantsitsowndenitionasafunction: Denition2.1.3. Wecallthesum(x)=Xpx1theprimecountingfunction,thatis,(x)isthenumberofprimeslessthanorequaltox,x2R.Furthermore,dene 37

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(x)=1Xn=1(x1=n) ntobethespecialprimecountingfunction.Notethataslimn!1x1=n=1,and(1)=0thespecialprimecountingfunctionisactuallyanitesum.IthasbeendiscoveredinthepersonalpapersofEuler,whowasthersttonotetheconnectionbetweentheprimenumbersandthezetafunction,thatheattemptedtoextend(s)tobeafunctionofacomplexnumbers.Euler,however,wasonlypartiallysuccessfulinthisattempt([ 7 ]).Asaninterestinghistoricalanecdote,itisnowknownthattheEuler-Maclaurinsummationformula(whichEulerwascertainlyawareof)canbeusedtoanalyticallycontinuethefunction(s)totheentirecomplexplane(see[ 30 ]);unfortunately,thismarksoneofthoserareinstanceswhenEulerdidnotexhausthismathematicaltalentstoattacktheproblemhewasaddressing.However,astraightforwardapplicationofStieltjesintegrationwillallowustoanalyticallycontinue(s)beyondtheline<(s)=1.Considertheintegralrepresentation(s)=Z11)]TJ /F4 11.955 Tf 13.04 16.61 Td[(d[t] ts=sZ11[t] ts+1dt.Uponreplacing[t]=t)-221(ftgwethenobtain sZ11[t] ts+1dt=sZ111 ts)]TJ 14.95 8.09 Td[(ftg ts+1dt=s s)]TJ /F3 11.955 Tf 11.95 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(sZ11ftg ts+1dt(2)andasfxgisbounded,itcaneasilybeseenthatthenalintegralin(2-3)isboundedfor<(s)>0.Therefore,(2-3)givestheanalyticcontinuationofthefunction(s)totheregion<(s)>0,s6=1[ 30 ].About50yearsafterEuler'stime,Riemannsucceededinanalyticallycontinuing(s),s6=1,totheentirecomplexplane,andinhisseminalmemoirsuppliedseveralproofsofthiscontinuation.Althoughitwillnotbenecessaryfor 38

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thepurposesofthispaper,itisinterestingtoseethesecomplexanalyticpropertiesof(s).Ifwelet\(s)bethegammafunctionofEulerandset (s)=1 2s(s)]TJ /F3 11.955 Tf 11.96 0 Td[(1))]TJ /F5 7.97 Tf 6.58 0 Td[(s=2)]TJ /F12 11.955 Tf 8.77 13.27 Td[(s 2(s),(2)thenRiemanndemonstratedthat (s)=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(s),(2)andthusfrom(2-4)and(2-5)(s)isdenedforallcomplexnumberss,suchthat<(s)<0and<(s)>1.Equation(2-3)suppliestheanalyticcontinuationfortheremainingcomplexnumbersssuchthat0<(s)1,s6=1;thesecomplexnumbersconstitutewhatisreferredtoasthecriticalstripofthefunction(s).Forseveralproofsof(2-3),(2-4),and(2-5)consult[ 29 ],[ 7 ],and[ 30 ].Riemannalsostated,withoutproof,thatthefunction(s)satisedaninniteproductoverthezerosofthezetafunction.Althoughhewascorrect,thisproductrepresentationwasnotprovedrigorouslyuntilseveraldecadeslaterasaconsequenceofHadamard'sproofofhismoregeneralfactorizationtheorem,andforthisreasonidentity(2-6)istypicallyreferredtoastheHadamardproduct.Foralls6=1,thereexistsaconstantbsuchthat: (s)=ebs 2(s)]TJ /F3 11.955 Tf 11.96 0 Td[(1)\(s 2+1)Y1)]TJ /F4 11.955 Tf 13.27 8.09 Td[(s es=(2)wherethearecomplexzerosof(s),i.e.thezerosof(s)withinthecriticalstrip.Itfollowsfrom(2-3),Riemann'sfunctionalequation(2-5),andthepropertiesof\(s)that(s)]TJ /F3 11.955 Tf 10.32 0 Td[(1)(s)isananalyticfunctionforalls2C;hence,(s)isameromorphicfunctioninCwithasolesingularityats=1.However,inordertoanswerquestionsconcerningtheprimenumbers,wemustconsidertheprincipalbranchofthefunctionlog(s),whichbytheaboveobservationsisnowafunctionofthecomplexvariables,withabranch 39

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pointsingularityextendingtotheleftof1inazero-freeregionof(s).Furthermore,byapplyingStieltjesintegrationwemayderivetheusefulidentityfor<(s)>1: log(s)=sZ12(x) xs+1dx,(2)whichjustiestheneedtodenetwodifferentprimecountingfunctions,as(x)ismoreeasilyhandledusinganalytictechniquesthan(x).Thetwofunctionsareneverthelessveryclosetooneanother,asj(x))]TJ /F6 11.955 Tf 11.95 0 Td[((x)jp xlogA(x)forsomeA1.Giventhat(x)isoftheorderofmagnitudeofx=log(x),thisdifferenceisactuallyquitesmall.Onedifcultywithlog(s)asacomplexfunctionisthatitmayhaveothersingularities,whichwillcorrespondtothezerosof(s).Forthisandotherreasons,thezerosof(s)areofgreatinteresttonumbertheorists.ThezerosoftheRiemannZetafunctionareamongthemostintriguingobjectsinallofmathematics;however,forthepurposeofprovingtheprimenumbertheoremweneedonlyshowthat(s)6=0for<(s)=1.Thefollowingtheorem,rstprovedbydelaVallee-Poussin,whileveryinterestingisthereforemorethanisrequiredfortheproofofthePrimeNumberTheorem.Thisresult,whencombinedwithsuitableboundsfor0(s) (s),willbesufcienttoproveamuchstrongerformoftheprimenumbertheorem.Thenextidentityfollowsfromtakingthelogarithmicderivativeofequation(2.1.6),andisvalidforalls2Cwheres6=1,)]TJ /F3 11.955 Tf 9.3 0 Td[(2n)]TJ /F3 11.955 Tf 12.14 0 Td[(2forn2Z+,or(acomplexzeroof(s)) 0(s) (s)=log(2))]TJ /F3 11.955 Tf 23.36 8.09 Td[(1 s)]TJ /F3 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2)]TJ /F11 7.97 Tf 6.77 4.34 Td[(0(s 2+1) \(s 2+1)+X1 s)]TJ /F6 11.955 Tf 11.96 0 Td[(+1 (2) 40

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Itisclearthatequation(2-8)holdsfor<(s)>1,thatitholdsforageneralcomplexs6=1,)]TJ /F3 11.955 Tf 9.3 0 Td[(2n)]TJ /F3 11.955 Tf 12.17 0 Td[(2,followsfromMittag-Lefer'stheorem.Therestrictionthats6=)]TJ /F3 11.955 Tf 9.3 0 Td[(2n)]TJ /F3 11.955 Tf 12.17 0 Td[(2comesfromthepolesofthelogarithmicderivativeoftheGammafunction. Theorem2.1.4. Lets=+iwith,2R.Thenthereexistsanabsoluteconstantc>0suchthat(s)hasnozerosintheregion1)]TJ /F4 11.955 Tf 39.26 8.09 Td[(c log(jj+2).Proof:For>1wehave<0(s) (s)=Xp,mlog(p) pmcos(mtlog(p)).Hence,for>1andanyreal,)]TJ /F3 11.955 Tf 9.3 0 Td[(30(s) (s))]TJ /F3 11.955 Tf 11.95 0 Td[(4<0(+i) (+i))-221(<0(+2i (+2i) =Xp,mlog(p) pm[3+4cos(mlog(p))+cos(2mlog(p))](2)andbythesimpletrigonometricidentity3+4cos+cos2=2(1+cos)20,itfollowsthat(2-9)isgreaterthanorequalto0.Now )]TJ /F6 11.955 Tf 13.15 8.09 Td[(0(s) (s)<1 )]TJ /F3 11.955 Tf 11.96 0 Td[(1+O(1).(2)andby(2-8))]TJ /F6 11.955 Tf 10.5 8.09 Td[(0(s) (s)=O(log(t)))]TJ /F12 11.955 Tf 11.95 11.36 Td[(X1 s)]TJ /F6 11.955 Tf 11.96 0 Td[(+1 ,where=+irunsthroughthecomplexzerosof(s).Hence,<0(s) (s)=O(log(t)))]TJ /F12 11.955 Tf 11.95 11.36 Td[(X)]TJ /F6 11.955 Tf 11.96 0 Td[( ()]TJ /F6 11.955 Tf 11.95 0 Td[()2+(t)]TJ /F6 11.955 Tf 11.96 0 Td[()2+ 2+2.Now,aseveryterminthelastsumispositive,itfollowsthat 41

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)-222(<0(s) (s)<
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singularities(althoughinthiscaseTheorem2.2.1actuallyappliesifwearedealingwithalogarithmicormonomialsingularity).Therefore,followingtheclassicalapproachtoprovingthePrimeNumberTheorem,wemayavoidthisdifcultybydifferentiatinglog(s)toobtain0(s) (s),whichisnowasinglevaluedfunctionwhosepolesareallremovablesingularities(ascanbeseenbyequation(2-8)),andhenceameromorphicfunctioninC.Thismotivatesthefollowingdenition: Denition2.1.5. Letnbeaninteger,n=p,wherepisaprimenumber,thendene(n)=log(p);ifanintegernisnotthepowerofaprimethen(n)=0.Thefunction(n)iscalledthevonMangoldtfunction.Furthermore,dene (x)=Xnx(n),calledtheTchebyschevfunction.ThemotivationforthesedenitionsbecomesapparentwhenweconsidertheDirichletseriesgeneratedby(n),thatis,for<(s)>1, 1Xn=1(n) ns=)]TJ /F6 11.955 Tf 10.49 8.08 Td[(0(s) (s)(2)whichissimplythenegativeofthelogarithmicderivativeof(s).ThevonMangoldtfunctioniscloselyrelatedtothefunction(x)as (x)=Xnx(n) log(n),(2)andapplyingformula(1-10)(theAbelsummationformula)givestheidentity (x)= (x) log(x)+Zx2 (t) tlog2(t)dt(2)relatingtheTchebyschevfunction (x)to(x).Notethatfromequation(2-15)theratio(x)log(x) xapproachesalimitasx!1ifandonlyif (x) xapproachesthesamelimit.ByatheoremofTchebyschev's(mentionedintheintroduction)ifthislimitexistsitmustequal1.ThePrimeNumberTheoremisthestatementthatthislimitdoesinfactexist; 43

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however,ratherthandealingwiththefunction(x)directlywewillprovethisstatementintheelementarilyequivalentform limx!1 (x) x=1.(2)ItwillbeusefultoapplyAbelsummationtotheDirichletseriesgeneratedby(n)toobtainthefollowingequalitieswhichwewilluseintheproofofthePrimeNumberTheorem,andwhichholdforallcomplexssuchthat<(s)>1: )]TJ /F6 11.955 Tf 13.15 8.08 Td[(0(s) (s)=1Xn=1(n) ns=sZ11 (t) ts+1dt;(2)furthermore,byTheorem2.1.4thisfunctionmaybeanalyticallycontinuedtoaneighborhoodofanypointontherealline<(s)=1,s6=1(asthe(s)inthedenominatorwillbenonzero). 2.2ThePrimeNumberTheoremFirstIwillremindthereaderofLandau'sbig-Oandlittle-onotation.Letg(x)andh(x)>0betworealvaluedfunctions.IfthereexistssomepositiveconstantCsuchthatforallsufcientlylargexjg(x)jCh(x)thenwesaythatg(x)=O(h(x)),org(x)<
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c1h(x)g(x)c2h(x)thenwewriteg(x)h(x).WenowrequireonlyoneadditionaltooltoprovethePrimeNumberTheorem,andthatisanupperboundonthefunction)]TJ /F6 11.955 Tf 10.49 8.09 Td[(0(s) (s)inthezero-freeregionof(s)suppliedbyTheorem2.1.4.Equation(2.1.11)givesasatisfactoryupperboundfortherealpartofthisfunction;however,itisafact(whichwillbenecessarytoobtainlaterestimates)thatfors=+itwhere1)]TJ /F4 11.955 Tf 25.01 8.09 Td[(c log(t)andt3: )]TJ /F6 11.955 Tf 13.15 8.09 Td[(0(s) (s)=O(log(t)),(2)andinfactthisresultcanbeimprovedtoO(log3=4(t)loglog3=4(t))byusingthemethodsofI.M.Vinogradov,[ 30 ].WemaynowproceedtoprovethePrimeNumberTheorem.TherstproofofthestrongformofthePrimeNumberTheorem(thatis,aproofofthePrimeNumberTheoremwithanontrivialerrorterm)wassuppliedbytheBarondelaVallee-Poussin.ThisclassicalproofalsocloselyparallelstheproofofTheorem3.1.7inSection3.1(althoughTheorem3.1.7hasabranchpointsingularity,whichrequiresgreatercare).Asthisproofissowellknownweomitsomedetails,howeverafulljusticationofanycommentscanbefoundin[ 4 ],[ 7 ],[ 29 ],or[ 30 ]. Theorem2.2.1. (delaVallee-Poussin)Thereexistsaconstanth>0suchthat (x)=x+O(xe)]TJ /F5 7.97 Tf 6.59 0 Td[(hp log(x))Proof:RecallthatPerron'sformula(equation(1-13))givesfor>1: (x)=1 2iZ+i1)]TJ /F5 7.97 Tf 6.58 0 Td[(i1)]TJ /F6 11.955 Tf 10.5 8.09 Td[(0(s) (s)xs sds.Wenowdeformthestraightlinecontourasfollows:xTandleta=1+c log(T)andb=1)]TJ /F4 11.955 Tf 28.1 8.08 Td[(c log(T),withctheconstantinTheorem2.1.4.Thendeformthepathofintegrationbetweena)]TJ /F4 11.955 Tf 12.07 0 Td[(iTanda+iTtogohorizontallyfroma)]TJ /F4 11.955 Tf 12.07 0 Td[(iTtob)]TJ /F4 11.955 Tf 12.07 0 Td[(iT,vertically 45

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fromb)]TJ /F4 11.955 Tf 12.53 0 Td[(iTtob+iTandhorizontallyfromb+iTtoa+iT.Takingintoaccountthesimplepoleats=1withresidue1of)]TJ /F6 11.955 Tf 10.5 8.08 Td[(0(s) (s),weobtain:1 2iZa+iTa)]TJ /F5 7.97 Tf 6.58 0 Td[(iT)]TJ /F6 11.955 Tf 10.49 8.09 Td[(0(s) (s)xs sds=x+1 2iZb)]TJ /F5 7.97 Tf 6.59 0 Td[(iTa)]TJ /F5 7.97 Tf 6.59 0 Td[(iT+Zb+iTb)]TJ /F5 7.97 Tf 6.59 0 Td[(iT+Za+iTb+iT=x+I1+I2+I3;consequently, (x)=x+1 2iZa)]TJ /F5 7.97 Tf 6.58 0 Td[(iTa)]TJ /F5 7.97 Tf 6.59 0 Td[(i1+Za+i1a+iT)]TJ /F6 11.955 Tf 10.49 8.09 Td[(0(s) (s)+I1+I2+I3=x+I4+I5+I1+I2+I3=x+R(x).Itmaythenbedemonstratedusingtheboundsofequation(2-18)thatthefollowingestimateshold:I1+I3=Oxa1 Tlog(T)I2=O(xblog2(T))I4+I5=Oxa T1 a)]TJ /F3 11.955 Tf 11.95 0 Td[(1+x1)]TJ /F5 7.97 Tf 6.58 0 Td[(alog(x)log(T)withthedominanttermsarisingfrom(xalog(T))=Tandxblog2(T).Tobalancetheseeffectsweset:xalog(T) T=xblog2(T)thatis, 46

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a)]TJ /F4 11.955 Tf 11.96 0 Td[(b=log(T) log(x).SolvingforTweobtainforsomeconstantc1>0:T=ec1p log(x),thusR(x)=O(xblog(T))=O(xe)]TJ /F5 7.97 Tf 6.58 0 Td[(c2log1=2(x)+c3loglog(x)),=O(xe)]TJ /F5 7.97 Tf 6.59 0 Td[(hp log(x)).Therefore,weobtaintheestimatethatforsomeh>0, (x)=x+O(xe)]TJ /F5 7.97 Tf 6.58 0 Td[(hlog1=2(x)),andinparticular:limx!1 (x) x=1,thus (x)sx,implyingthat(x)sx log(x).Applyingpartialsummationtotheresultsoftheabovetheoremweobtaintheimmediatecorollary: Corollary2.2.2. (x)=Zx2dt log(t)+O(xe)]TJ /F5 7.97 Tf 6.59 0 Td[(hp log(x))Itisalsoworthnotingthattheabovecorollary,dueoriginallytodelaVallee-Poussin,wasthebestestimatefor(x)foroverthirtyyears.ItshouldbesomewhatobviousfromtheaboveproofthattheerrorterminthePrimeNumberTheoremisdirectlyrelatedtohowfarwemayenlargethezero-freeregionof(s).SomenotableimprovementsincludethoseofLittlewood,whoin1922demonstratedthatthereexistsak>0suchthat(s)6=0foranys=+itwhere>1)]TJ /F4 11.955 Tf 11.96 0 Td[(kloglog(jtj+3) log(jtj+3).Thiscorrespondstoanimprovedestimateof 47

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(x)=Zx2dt log(t)+O(xe)]TJ /F8 7.97 Tf 6.59 0 Td[(p log(x)loglog(x)),theproofcanbefoundin[ 30 ],althoughitissignicantlydeeperthanTheorem2.1.4.Stillbetterestimatesareknown,althoughtheresultsareverydifculttoestablish.Thebestestimateknowntodate(see[ 29 ])is(x)=Zx2dt log(t)+O(xe)]TJ /F8 7.97 Tf 6.59 0 Td[(log3=5(x)loglog)]TJ /F15 5.978 Tf 5.75 0 Td[(1=5(x)),andisobtainedusingthezero-freeregionofKorobovandVinogradov.Thisdiscussionshouldgiveonetheimpressionthatimprovingthezero-freeregionof(s)appearstobeaverydifcultproblem.Forexample,itisstillunknownwhetherthereexistsasingle>0suchthat(s)6=0for<(s)>1)]TJ /F6 11.955 Tf 11.95 0 Td[(.Itshouldalsobenotedthatifisanupperboundfortherealpartsofthezerosof(s)then(x)=Zx2dt log(t)+O(xlog(x)).However,thisformulaisworthlessif=1,and(astheabovecommentsnote)thusfareveryzero-freeregionwhichhasbeenestablishedfor(s)cannotguaranteethatwemaychoose=1)]TJ /F6 11.955 Tf 11.95 0 Td[(forany>0.HavingdemonstratedtheproofofthePrimeNumberTheoremalongclassicallines,Iwillnowstatethe1954theoremofDelangealludedtointheintroduction.Thisresultitselfutilizesverycarefulestimatesof(s)whicharemuchstrongerthanwhatisrequiredtoprovethePrimeNumberTheorem,therefore,Delange'stheoremishardlytheeasiestwayinwhichonecouldconcludethePrimeNumberTheoremfromthepropertiesof(s).But,Delange'stheoremwillallowustoevaluatethenitesumsofamuchbroaderclassoffunctionsthan(n)-especiallyadditivefunctions.Delangewasactuallymotivatedtoestablishhisgeneraltheoreminordertoestimatethemomentsof 48

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certainadditivefunctions,anindoingsotogiveaproofoftheErdos-Kactheorembythemethodofmoments.However,wearegoingtousethistheoremtostudythemomentsof(n).Ofparticularinterestwillbethefunction2(n)whicharisesintheSelbergasymptoticformula,usedbyErdosandSelbergtogivetherstelementaryproofsoftheprimenumbertheorem.ItshouldalsobenotedthatDelange'stheoremisanexceedinglydeepresultwhichreliesuponcomplicatedmethodsofcontourintegration,andcaninmanywaysbeviewedasageneralizationoftheSathe-Selbergmethod(tobeoutlinedingreaterdetailshortly).AsDelange'stheoremappliestoDirichletserieswhichmayhavebranchpointsingularities(ats=a,say)itisnecessarytodeformthepathofintegrationbeyondtheline<(s)=ainsomezero-freeregionoftheseriesandevaluateso-calledHankelcontours.Thesecontoursareformedbythecirclejs)]TJ /F4 11.955 Tf 13 0 Td[(aj=rexcludingthepoints=a)]TJ /F4 11.955 Tf 12.78 0 Td[(r,togetherwiththeline(,a)]TJ /F4 11.955 Tf 12.78 0 Td[(r](forsome0<
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method(forafullproof,andamorethoroughdiscussionoftheSelberg-DelangemethodconsultChaptersII.5andII.7of[ 29 ]). Theorem2.2.3. LetF(s):=1Xn=1bn nsbeaDirichletserieswithnon-negativecoefcients,convergingfor>a>0.SupposethatF(s)isholomorphicatallpointslyingontheline=aotherthans=aandthat,intheneighborhoodofthispointandfor>a,wehaveF(s)=(s)]TJ /F4 11.955 Tf 11.95 0 Td[(a))]TJ /F8 7.97 Tf 6.59 0 Td[(!)]TJ /F7 7.97 Tf 6.59 0 Td[(1qXj=0gj(s)logj1 s)]TJ /F4 11.955 Tf 11.96 0 Td[(a+g(s)where!issomerealnumber,gj(s)andg(s)arefunctionsholomorphicats=a,thenumbergq(a)beingnonzero.If!isanon-negativeinteger,thenwehave:B(x):=Xnxbnsgq(a) a\(!+1)xalog!(x)loglogq(x)where\(s)isEuler'sGammafunction.If!=)]TJ /F4 11.955 Tf 9.3 0 Td[(m)]TJ /F3 11.955 Tf 11.96 0 Td[(1forsomenon-negativeintegermandifq1then:B(x)s()]TJ /F3 11.955 Tf 9.3 0 Td[(1)mm!qgq(a) axalog)]TJ /F5 7.97 Tf 6.59 0 Td[(m)]TJ /F7 7.97 Tf 6.58 0 Td[(1(x)loglogq)]TJ /F7 7.97 Tf 6.58 0 Td[(1(x)Notethatthistheoremappliessolongasthefunctionisholomorphicatallpointsontheline=a,s6=a.Itisasubtlepoint,butworthnoting,thatthisdoesnotimplythatthefunctionneedbeholomorphicinanyopenneighborhoodoftheentireline.ForourpurposeswewillbemostinterestedinapplyingtheSelberg-Delange'smethodtovariousmanifestationsoflog(s),andatpresentitisnotknownforanyconstant>0whetherlog(s)isholomorphicintheregion<(s)=>1)]TJ /F6 11.955 Tf 12.05 0 Td[(,s6=1;however,itcanbeshownthatlog(s)isholomorphicateverypointontheline=1,s6=1.Withthistheoreminhand,theprimenumbertheoremfollowseasilyfromtheknownpropertiesof(s),theRiemannZetaFunction.As(s)isabsolutelyconvergentin>1,hasasimplepoleats=1withresidue1,andisholomorphicandnonzeroforallpointslyingontheline=1fors6=1,wemayapplytheSelberg-Delange's 50

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methodto)]TJ /F6 11.955 Tf 10.49 8.08 Td[(0(s) (s),whichisalsoanabsolutelyconvergentDirichletseriesin>1,hasasimplepoleats=1withresidue1,hasnon-negativecoefcients,andisholomorphicforallpointslyingontheline=1,s6=1(foramoredetaileddiscussionofthesefactssee[ 7 ]or[ 29 ]).Thisimmediatelyimpliestheprimenumbertheoremintheform: (x)=x+o(x).Ofcourse,aswasshownabove,thereisnoneedtoappealtosuchadeeptheoremforsucharesult.Derivingtheprimenumbertheoremviathefunction (x)istheclassicalapproachtosolvingthisproblem,andwasusedbybothHadamardanddelaValleePoussinintheiroriginalproofsofthistheorem.Inthefollowingdiscussion,wewilltakeaslightlydifferentapproachfromthistraditionalwayofprovingtheprimenumbertheorem.ThisapproachisimplicitintheworkofHadamard,delaVallee-Poussin,Selberg,andDelange(tonameafew),althoughIhaveneverseentheresultstressedinthepublishedliterature.Let 2(x)=Xnx2(n).Ifwedene 2(x)=Xnx2(n) log2(n),(2)thenwemayviewtheabovesumasaStieltjesintegral,andapplyingtheintegrationbypartsformulayields, 2(x)=Zx2d 2(t) log2(t)(2)= 2(x) log2(x))]TJ /F6 11.955 Tf 16.92 8.09 Td[( 2(2) log2(2)+2Zx2 2(t) tlog3(t)dtThesecondterminequation(2-20)iseasilyseentobe 2(2) log2(2)=log2(2) log2(2)=1.ByaclassicalresultduetoTchebyschev([ 7 ]) (x)=O(x),implyingthat 51

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2(x)=Xnx2(n)log(x)Xnx(n)= (x)log(x)=O(xlog(x)).UsingTchebyschev'sestimatewemayconcludethatthethirdterminequation(2-20)isthus 2Zx2 2(t) tlog3(t)dt=OZx2tlog(t) tlog3(t)dt(2)=OZx2dt log2(t)=Ox log2(x),obtainedbyafurtherapplicationoftheintegrationbypartsformulatoZx2dt log2(t);hence,takingintoaccount(2-20)andtheaboveestimateswemayinferthat: 2(x)= 2(x) log2(x)+Ox log2(x)+1= 2(x) log2(x)+ox log(x).(2)Notingthatthespecialprimecountingfunction(x)isverycloseto2(x)(as(x))]TJ /F3 11.955 Tf -440.45 -23.91 Td[(2(x)p xlog(x) 4)thenifwecouldshowthat 2(x)=xlog(x)+o(xlog(x))itwouldfollowthat2(x)=x log(x)+ox log(x)andhence(x)=x log(x)+ox log(x).Webeginbydifferentiating0(s) (s)=)]TJ /F11 7.97 Tf 15.69 14.94 Td[(1Xn=1(n) ns=)]TJ /F3 11.955 Tf 20.7 8.09 Td[(1 s)]TJ /F3 11.955 Tf 11.96 0 Td[(1+...toobtain: 00(s) (s))]TJ /F12 11.955 Tf 11.95 16.86 Td[(0(s) (s)2=1Xn=1(n)log(n) ns=1 (s)]TJ /F3 11.955 Tf 11.96 0 Td[(1)2+O(1).(2)Moreover, 1Xn=1(n)log(n) ns=1Xn=12(n) ns+h(s)(2)whereh(s)isanalyticins2C,<(s)>1 2.ByapplyingpartialsummationtothisDirichletseriesfor<(s)>1wendthat: 00(s) (s))]TJ /F12 11.955 Tf 11.96 16.86 Td[(0(s) (s)2=sZ12 2(x) xs+1dx,(2) 52

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andastheDirichletseries00(s) (s))]TJ /F12 11.955 Tf 13.59 16.86 Td[(0(s) (s)2hasnon-negativecoefcients,isholomorphicatallpointslyingontheline<(s)=1,s6=1,withasolesingularityats=1ofmultiplicity2andacoefcientof1,wemayinvokeTheorem2.2.3(Delange'sTheorem)toconcludethat: 2(x)=xlog(x)+o(xlog(x)),(2)insertingequation(2-26)intoequation(2-22)yields2(x)=x log(x)+ox log(x)andtherefore,(x)=x log(x)+ox log(x),whichisthePrimeNumberTheorem.ItisagainstressedthatweneednotapproachtheproblemusingtheSelberg-Delangemethod,astheproofofTheorem2.2.1showsthatwecanevaluatesingularitiesofthetype1 (s)]TJ /F3 11.955 Tf 11.95 0 Td[(1),andthismethodiseasilyadaptedtothesingularity1 (s)]TJ /F3 11.955 Tf 11.95 0 Td[(1)2intheabovetheorem. 2.3TheHardy-RamanujanTheoremThefunctions!(n)and(n),whichdenotethenumberofdistinctprimedivisorsofnandthenumberofdistinctprimepowersdividingn,respectively,wererststudiedbyHardyandRamanujanintheir1917paper[ 11 ].Notonlyaretheyinterestingadditivefunctionsintheirownright,buttheyalsomotivatedthestudyofprobabilisticnumbertheoryandtheAlladi-Erdosfunctions(whichwillbediscussedinthenextchapter).HardyandRamanujandemonstratedthat Xnx!(n)=xloglog(x)+c1x+Ox log(x)(2) 53

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and Xnx(n)=xloglog(x)+c2x+Ox log(x)(2)wherec1=0.261497...andc2=1.0345653...[ 29 ].Allthatisneededtoprove(2-27)and(2-28)isthefactthat Xpx1 p=loglog(x)+c1+O1 log(x);(2)however,wenotethatDelange'sTheoremyields(2-27)and(2-28)analytically.Withthismethodofproofinmind,wewilllaterintroduceacompanionfunctionto(n)and!(n)which,whentakenwithawell-knownresultofHardyandRamanujan,willgivetheresultwithlittleeffort.BeforeproceedingtotheHardy-RamanujanTheorem,werequireoneadditionalresultfrom[ 29 ]:Xnx((n))]TJ /F6 11.955 Tf 11.96 0 Td[(!(n))=Cx.forsomewell-knownconstantC=Xp1 p(p)]TJ /F3 11.955 Tf 11.95 0 Td[(1)0.773156. Theorem2.3.1. Xnx!(n)=xloglog(x)+c1x+O1 log(x)Proof:Theclassicalapproachtothistheoremisgivenbythesimpleobservationthat:Xnx!(n)=XnxXpjn1=Xpmx1=Xpxx p=xXpx1 p+O((x))fromequation(2-29),andbyTchebyschev'sestimate(orthePrimeNumberTheorem),Xnx!(n)=xloglog(x)+c1x+ox log(x). 54

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Therefore,weobtainequation(2-27),andbythecommentsprecedingthistheoremweobtainequation(2-28).Letk(x)denotethenumberofintegersnxsuchthat!(n)=k,andletNk(x)denotethenumberofintegersnxsuchthat(n)=k.Byusinginductiononk,HardyandRamanujansucceededinprovingthatthereexistsaconstantCsuchthat: k(x)=Ox(loglog(x)+C)k)]TJ /F7 7.97 Tf 6.59 0 Td[(1 (k)]TJ /F3 11.955 Tf 11.96 0 Td[(1)!log(x)(2)holdsuniformlyink.Theessenceoftheirproofistheobservationthatkk+1(x)Xpp xkx pfromwhichthetheoremeasilyfollowsbyusingthePrimeNumberTheoremasthebasecaseofk=1andapplyinginductiononk.ForthecaseofNk(x)HardyandRamanujandemonstratedthatthereexistsaconstantDsuchthat: Nk(x)=Ox(loglog(x)+D)k)]TJ /F7 7.97 Tf 6.59 0 Td[(1 (k)]TJ /F3 11.955 Tf 11.95 0 Td[(1)!log(x)(2)holdsuniformlyfork(2)]TJ /F6 11.955 Tf 12.93 0 Td[()loglog(x),>0.Theadditionalrestrictiononkinequation(2-31)followsfromsomesubtleaspectsconcerningthefunction(n)whichwillbediscussedbelow.HardyandRamanujanmademanymoreobservationsintheirseminalpaper,oneofwhichwastointroducetheconceptofthenormalorderofanarithmeticfunction.Anaverageorderisaverynaturalestimatetoseek,forgivenanarithmeticfunctionf(n)theaverageorderissimply1 xXnxf(n).Furthermore,bysummingoverthefunctionf(n)fornxweareineffectsmoothingouttheirregularitiesofthefunctionwhichwillalmostcertainlyexist;hence,theaverageordermaybeinuencedbysporadicallyoccurringirregularvalues.Theseirregularvaluescansometimesgivemisleadingresults,andinthissectionwewillgiveafamousexampleofonesuchsituation.However,Hardyand 55

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Ramanujandevelopedadifferenttypeofstatisticcalledthenormalorder,whichisinmanywaysmorenaturalandinformativethantheaverageorder,andwhichwenowdene. Denition2.3.2. Wesaythatanarithmeticfunctionf(n)hasnormalorderg(n)ifg(n)isamonotonearithmeticfunctionsuchthatforany>0jf(n))]TJ /F4 11.955 Tf 11.95 0 Td[(g(n)jjg(n)jonasetofintegersnhavingnaturaldensity1.TheobservationsofHardyandRamanujanallowustoobtaininterestingprobabilisticinterpretationsofvariousarithmeticfunctions.RecallthatarandomvariableXisPoissonwithparameter>0ifP(X=j)=j j!e)]TJ /F8 7.97 Tf 6.59 0 Td[(forj=0,1,2,....TheexpectedvalueofaPoissonrandomvariableisgivenbyE[X]=,see[ 21 ].Equation(2-30)ofHardyandRamanujandemonstratesthatk(x)maybeinterpretedasbeingboundedbyaPoissonrandomvariablewithparameter=loglog(x);furthermore,thissameequationshowsthatk(x)issmallcomparedtoxwhenjk)]TJ /F3 11.955 Tf 11.96 0 Td[(loglog(x)jislarge.Thisisbecauseforevery>0 limt!1e)]TJ /F5 7.97 Tf 6.58 0 Td[(tXjk)]TJ /F5 7.97 Tf 6.58 0 Td[(tj>t1=2+tk k!!0(2)(see[ 21 ])sothat(2-30)demonstratesthatlimx!11 xXjk)]TJ /F7 7.97 Tf 6.59 0 Td[(loglog(x)j>(loglog(x))1=2+!(x)=0and,hence,thattheinequality 56

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j!(n))]TJ /F3 11.955 Tf 11.95 0 Td[(loglog(n)j<(loglog(n))1=2+holdsonasetofintegerswithdensity1.TheseobservationsledHardyandRamanujantodeducethat!(n)notonlyhasaverageorderloglog(n),butalsohasnormalorderloglog(n).In1934P.Turansucceededinprovingastrongerresultconcerning!(n)thanthatderivedbyHardyandRamanujan.Thisresult,knownasTuran'sinequality,hasfurtherprobabilisticimplicationsfortheHardy-Ramanujanfunctionsandisthesubjectofthenexttheorem. Theorem2.3.3. (Turan)Xnx(!(n))]TJ /F3 11.955 Tf 11.96 0 Td[(loglog(x))2=O(xloglog(x))Proof:Consider:Xnx(!(n))]TJ /F3 11.955 Tf 11.96 0 Td[(loglog(x))2=Xnx!2(n))]TJ /F3 11.955 Tf 10.17 0 Td[(2loglog(x)Xnx!(n)+x(loglog(x))2+O(xloglog(x))=Xnx!2(n))]TJ /F4 11.955 Tf 11.95 0 Td[(x(loglog(x))2+O(xloglog(x)).Now,asXnx!2(n)=Xnx0@Xpjn11A2=Xnx0@Xpqjn1+Xp2jn11A,whereitisunderstoodthatp6=qaredistinctprimes.ItiseasilyseenthatXnxXp2jn1=O(x)andthat 57

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Xp,qjnXn0(pq)1Xpqxx pq,whereitisimplicitthatonthelefthandsideoftheaboveequationtheoutersumissummedoverallpqxandtheinnersumissummedoverallnx.Now,asXpqxx pqx Xqx1 p! Xqx1 q!x(loglog(x))2+O(xloglog(x));hence,Xnx!2(n)x(loglog(x))2+O(loglog(x)).Wemaythereforeconcludethat:Xnx(!(n))]TJ /F3 11.955 Tf 11.96 0 Td[(loglog(x))2=O(xloglog(x)),whichisTuran'sresult.WemayfurthernotethatifT>0islargeandNT(x)denotesthesumoftheintegerslessthanx,suchthatj!(n))]TJ /F3 11.955 Tf 11.96 0 Td[(loglog(x)j>Tp loglog(x),thatis:NT(x)=Xj!(n))]TJ /F7 7.97 Tf 6.59 0 Td[(loglog(x)j>Tp loglog(x)1,wherethesumisoverallnx,thenTloglog(x)NT(x)Xj!(n))]TJ /F7 7.97 Tf 6.59 0 Td[(loglog(x)j>Tp loglog(x)(!(n))]TJ /F3 11.955 Tf 11.95 0 Td[(loglog(x))2O(xloglog(x)).Therefore,wemayconcludethatNT(x)=Ox T, 58

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andinparticular,thatlimT!1NT(x)=o(x).Wenowofferanalternativeproofofequations(2-27)and(2-28)viathefollowingcompanionfunction: Denition2.3.4. Letw(n)=Xdjn(d) log(d),thatisw(n)=Xpjn1 .Withthisdenitionitshouldbeclearthat!(n)w(n)(n)forallintegersn2.Withtheseinequalitieswecanimmediatelydeducethat:Xnxw(n)=xloglog(x)+O(x).However,wecanimprovetheO-termbynotingthatw(n)generatesthefollowingDirichletseries:(s)log(s)=1Xn=2w(n) ns.RecallthatinordertoprovethePrimeNumberTheoremitwasnecessarytoshowthat(s)6=0fors=1+it;hence,thefunctionlog(s)hasasolesingularityontheline<(s)=1,ats=1.ThiswasnecessaryasDelange'stheoremcouldnotbeappliedtofunctionswhichdonotsatisfyconditionsofholomorphyonsomeverticalline,andfromthepropertiesoflog(s)thefunction(s)log(s)mustbeholomorphicatallpointsontheline<(s)=1,savethesingularityats=1.Ofcourse,aswaspreviouslymentioned,thisresultfollowseasilyfromequation(2-28),andisthereforenotasdeepatheoremasthePrimeNumberTheorem.However,wewantedtoseehowtheHardy-Ramanujan 59

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resultstwithinthecontextofDelange'sTheoremaswewillprovideasimilartreatmentwhenaddressingtheAlladi-Erdosfunctions,whichwillbedevelopedinthenextchapter.WhiletheaboveobservationsaresufcienttoapplyDelange'sTheoremtoevaluatethesumXnxw(n),wecouldactuallyprovemuchstrongerresultsthroughmoreelaboratemethods.Thisisbecause(s)log(s)maybeanalyticallycontinuedforallcomplexs6=1inthesamezero-freeregionas(s).Although,forourpurposesthegainisminorfortheadditionalworkthatwouldbeinvolved. Theorem2.3.5. Thereexistsaconstantc3suchthatXnxw(n)=xloglog(x)+c3(x)+Ox log(x).Proof:Bytheabovediscussion(s)log(s)satisestheconditionsofDelange'stheorem;moreover,(s)log(s)hasasingularityofthetype:1 (s)]TJ /F3 11.955 Tf 11.95 0 Td[(1)log1 s)]TJ /F3 11.955 Tf 11.95 0 Td[(1+c3 (s)]TJ /F3 11.955 Tf 11.95 0 Td[(1)+c4log1 s)]TJ /F3 11.955 Tf 11.95 0 Td[(1+...forsnear1.Therefore,Delange'sTheoremimpliesthat:Xnxw(n)=xloglog(x)+c3x+Ox log(x),aswastobedemonstrated.IntheaboveproofitshouldbenotedthatwhileDelange'spowerfultheoremallowsustoeasilyderiveourresult,histheoremisinmanyrespectsinvokedunnecessarily.Inparticular,therstproofimpliesthat(aswasmentionedbefore)theonlyrealanalyticresultnecessaryisequation(2-29),whichisfarmoreeasilyderivedthanDelange'stheorem.Itisaratheramazingfactthatmanydeterministicarithmeticfunctionscanbeinterpretedinaprobabilisticmanner,muchlikehowtheprimenumbertheoremcanbeconsideredtobeastatementaboutthearithmeticmeanof(n).Furthermore,manyof 60

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theseprobabilisticinterpretationsshedmuchmorelightonvariousarithmeticfunctionsthanwhatcanbederivedanalyticallyorelementarily.ThemotivationforDelange'spowerfultheoremwassuppliedbytheworkofSatheandSelberg,whosetheoremsofferfurtherprobabilisticinterpretationsforthefunctionsk(x)andNk(x).Historically,EdmundLandauderivedtheestimates k(x)sx(loglog(x))k)]TJ /F7 7.97 Tf 6.59 0 Td[(1 (k)]TJ /F3 11.955 Tf 11.95 0 Td[(1)!log(x)(2)and Nk(x)sx(loglog(x))k)]TJ /F7 7.97 Tf 6.59 0 Td[(1 (k)]TJ /F3 11.955 Tf 11.96 0 Td[(1)!log(x)(2)in1909[ 16 ],butheonlyprovedtheseresultsforxedk.HardyandRamanujanderivedtheiruniformestimates(2-30)and(2-31)in1917[ 11 ],yetitwouldtakeafurther36yearsbeforeSatheproveduniformestimatesinkwhichwerecomparableinaccuracytothoseofLandau.SathenodoubtfoundhisinspirationinthepapersofHardyandRamanujanandproceededtoprovehisresultsinductively,althoughitshouldbenotedthatSathe'smethodsareverycomplicatedandhisresultsaredifculttoderive.However,byusingSelberg'sargumentonemayderiveinamoreclassicalandnaturalfashiontheresultsofSathe.Toarriveattheseresults,considerthefunctions: F(s,z)=1Xn=1z!(n) ns=Yp1+z ps)]TJ /F3 11.955 Tf 11.96 0 Td[(1=(s)zf(s,z)(2)and G(s,z)=1Xn=1z(n) ns=Yp1)]TJ /F4 11.955 Tf 15.42 8.09 Td[(z ps)]TJ /F7 7.97 Tf 6.59 0 Td[(1=(s)zg(s,z)(2)wherethefunctionin(2-35)representsananalyticfunctionofsandzfor<(s)>1,and(2-36)representsananalyticfunctioninsandzfor<(s)>1,jzj<2.Whenfactoredin 61

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termsof(s)z,f(s,z)becomesanalyticforallzwhere<(s)>1=2,andg(s,z)becomesanalyticwhen<(s)>1=2andjzj<2.Theadditionalrestrictionthatjzj<2arisesfromthepolewhichtheprimep=2contributesto(2-36).Inordertodeduceanasymptoticformulafork(x),SelbergrstappliesPerron'sformulato(2-35)S(z,x)=Xnxz!(n)=1 2iZk+i1k)]TJ /F5 7.97 Tf 6.58 0 Td[(i1(s)zf(s,z)xs sdswherek>1,inordertogetanestimateofthepartialsumsofthecoefcientsofthatseries;however,thisrequiresonetodeformthestraightlinecontourinPerron'sformulaintoazero-freeregionfor(s)inthestrip1=2<<(s)<1.Furthermore,ingeneral(s)zmaycontainbranchpointsingularities,whichwillnecessitatetakingaHankelcontouraroundthesingularityats=1.Afterdeformingthelineofintegration,onewillquicklynoticethat(asinTheorem3.1.8)themajorityofthecontributionfromtheintegralcomesfromthisbranchpointsingularity;therefore,afterSelbergdemonstratedthattheothercontourscontributelittletotheestimateofS(z,x),heobtainedtheresultthatS(z,x)=Xnxz!(n)=xlogz)]TJ /F7 7.97 Tf 6.59 0 Td[(1(x)f(1,z)1+O1 log(x)whichisuniformifzisbounded.In[ 26 ]Selbergthensolvesfork(x)byapplyingCauchy'stheoremtothesumS(z,x):k(x)=1 2iZjzj=rS(z,x) zk+1dzandthenoptimizingtheradiusrofthecontour.Theoptimalvalueturnsouttober=k loglog(x),andutilizingthisresultwemayobtainanimprovementonLandau'sasymptotick(x)sf(1,k loglog(x))x(loglog(x))k)]TJ /F7 7.97 Tf 6.58 0 Td[(1 (k)]TJ /F3 11.955 Tf 11.95 0 Td[(1)!log(x) 62

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whichisvaliduniformlyforkMloglog(x),Manarbitrarilylargepositiveconstant.WenowapplytheSathe-SelbergtechniquetoderiveanasymptoticforNk(x)whichholdsuniformly(insomerangeofk).ByapplyingPerron'sformulainamanneranalogoustothederivationofS(z,x)(i.e.bydeformingthepathofintegrationandevaluatingtheHankelcontouraroundthesingularityats=1)to(2-36)weobtainthesumT(z,x)=Xnxz(n)=xlogz)]TJ /F7 7.97 Tf 6.59 0 Td[(1(x)g(1,z)1+O1 log(x)which,byutilizingCauchy'stheoreminasimilarmannertothemethodusedtoderivek(x),givesNk(x)=1 2iZjzj=rT(z,x) zk+1dzwiththeoptimalvalueoftheradiusr=k loglog(x).However,asjzj<2thisforcesk(2)]TJ /F6 11.955 Tf 13.24 0 Td[()loglog(x),forany>0.Evaluatingthiscontourintegralyieldstheasymptotic:Nk(x)sx(loglog(x))k)]TJ /F7 7.97 Tf 6.59 0 Td[(1 (k)]TJ /F3 11.955 Tf 11.96 0 Td[(1)!log(x),whichholdsuniformlyink(2)]TJ /F6 11.955 Tf 11.95 0 Td[()loglog(x).Inordertostudyfunctionsofthistypeingeneral,Selbergin[ 26 ]thenderivesthefollowingdeeptheorem: Theorem2.3.6. (Selberg)LetB(s,z)=1Xn=1b(z,n) nsfor<(s)>1=2,andlet 63

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1Xn=1jb(z,n)j n(log(2n))B+3beuniformlyboundedforjzjB.Furthermore,letB(s,z)(s)z=1Xn=1a(z,n) nsfor<(s)>1.Then,wehaveA(z,x)=Xnxa(z,n)=B(1,z) \(z)xlogz)]TJ /F7 7.97 Tf 6.58 0 Td[(1(x)+O(xlogz)]TJ /F7 7.97 Tf 6.59 0 Td[(2(x)),uniformlyforjzjB,x2.Withthistheoreminhandwemaynotonlyobtaintheaboveasymptoticvalues,butalsothoseforamuchwiderclassoffunctions.Aswasmentionedpreviously,theaverageorderofafunctioncansometimesleadtomisleadingresults.Toillustratethisphenomenonconsider(n),thenumberdivisorsofanintegern.Itisawell-knownfact,rstdemonstratedbyDirichlet,thatXnx(n)=xlog(x)+(2)]TJ /F3 11.955 Tf 11.96 0 Td[(1)x+O(p x),andwemayconcludethatthearithmeticmeanof(n)islog(n).Itisthereforetemptingtoassumethat(n)willbeaboutthesizeoflog(n)quiteoften;however,thisiscompletelyfalse.Giventhecanonicaldecompositionofanintegern=!(n)Yi=1piiitiseasilyseenthat 2!(n)(n)=!(n)Yi=1(i+1)!(n)Yi=12i=2(n),(2)andfromtheHardy-Ramanujanresultsboth!(n)and(n)haveaverageandnormalorderloglog(n),hence(2-37)implies 64

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(n)=(log(n))log(2)+o(1)onsomesubsetoftheintegershavingnaturaldensity1.Thus(n)ismoreoftenthannotequalto(log(n))log(2)+o(1),andisthereforesignicantlylessthanitsarithmeticmeanonasetofdensity1(inparticular(n)cannothaveitsaverageorderequaltoitsnormalorder).Thisfactcannowbeexplained,butitrequiresthepoweroftheSathe-Selbergtechniquesandisthereforemuchdeeperthantheearlierresultsconcerning(n)describedabove.ItwillbeshownpresentlythatthesumXnx(n)isdominatedbyasmallnumberofintegerswithmanydivisors.AsXnx(n)sXnx2!(n)+o(1)sXnx2kk(n)s1Xk=02kx(loglog(x))k)]TJ /F7 7.97 Tf 6.59 0 Td[(1 (k)]TJ /F3 11.955 Tf 11.96 0 Td[(1)!log(x)andthisisanexponentialseriesin2loglog(x).Thus Xjk)]TJ /F7 7.97 Tf 6.59 0 Td[(2loglog(x)j<(loglog(x))1=2+2x(2loglog(x))k)]TJ /F7 7.97 Tf 6.59 0 Td[(1 (k)]TJ /F3 11.955 Tf 11.95 0 Td[(1)!log(x)xlog(x)(2)Themaincontributionin(2-38)comesfromthetermsforwhichjk)]TJ /F3 11.955 Tf 12.82 0 Td[(2loglog(x)j<(loglog(x))1=2+.Forsuchvaluesofk,(n)=2k(1+o(1))=22loglog(x)(1+o(1))=(log(x))2log(2)+o(1)andas2log(2)>1,thesevaluesarelargerthantheaverageorderoflog(n)for(n).Anotherwaytointerpretthisisthattheaverageorderof(n)isthrownoffbythesmallsetofintegerswiththepropertythat!(n)s2loglog(n),whichinlightofthefactthattheaverageandnormalordersof!(n)areloglog(n),shouldalsobequiterare.Thisexplainswhy(n)canhaveanaverageorderwhichislargerthanitsnormalorder.Toclosethissectionwegiveonemoreinterestingprobabilisticinterpretationofthefunction!(n).TheErdos-KacTheoremstatesthatforall2R 65

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limx!11 xnx:!(n)loglog(x)+p loglog(x)=1 p 2Ze)]TJ /F5 7.97 Tf 6.59 0 Td[(u2=2du.Itfollowsthatthefunction!(n)behavesasanormallydistributedrandomvariable,withmeanandvarianceloglog(n). 2.4RemarksItshouldbenotedthattheprimenumbertheoremisaratherdeeptheorem,andwhileelementaryproofsofthistheoremexist(thatis,theoremswhichdonotrequiretheuseofcomplexfunctiontheory),theyarehardlysimple.ThebrevityoftheaboveargumentsprovingthePrimeNumberTheorem(andinparticulartheveryshortproofoftheHardy-RamanujanTheorem)followsfromthefactthatmostoftheparticularsofthetheoremaresubsumedinDelange'sTheorem(Theorem2.2.1),whichisitselfahighlynontrivialresultrelyinguponthetheoryofanalyticfunctions.Theaboveproofsdonotuseanynewtechniquesfromthetheoryofcomplexvariables;however,IhaveneverseenanypublishedresultsontheasymptoticvalueofthesummationofthepowersofthevonMangoldtfunction: k(x)=Xnxk(n).Theaboveapproachfor 2(x)generalizeseasilyto k(x)fork2Z,k>2,andonlyrequiresonetodifferentiatelog(s)k-times,andthenapplyDelange'stheorem.Itiseasilydeducedthatanyasymptoticresultfor k(x)isequivalenttotheprimenumbertheorem;asasymptoticresultsconcerning(x)followbyapplyingpartialsummationtothefunction k(x)inthemanneroutlinedabove,yielding k(x)sxlogk)]TJ /F7 7.97 Tf 6.59 0 Td[(1(x).Furthermore,theformula 2(x)=xlog(x)+x+Ox log(x)allowsustoanalyzethesecondmoment(thevariance)ofthefunction(n).While(n)=0unlessnis 66

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aprimepower,weshouldexpectitsvariancetobequitelarge.ThisfactisinmanywayssurprisingasthePrimeNumberTheoremassuresusthatthemeanof(n)is (x) x=1+O1 log(x),whichisquitesmall.However,bydirectcalculationthisvarianceisgivenby1 xXnx(n))]TJ /F3 11.955 Tf 11.96 0 Td[(1+O1 log(n)2=1 xXnx2(n))]TJ /F3 11.955 Tf 11.96 0 Td[(2(n)+1+O(n) log(n)=1 x( 2(x))]TJ /F3 11.955 Tf 11.96 0 Td[(2 (x)+x+O((x)))=1 xxlog(x)+x)]TJ /F3 11.955 Tf 11.95 0 Td[(2x+x+Ox log(x)=log(x)+O1 log(x)=log(x)+o(1).Thisisfarinexcessoftheaverageorderof(n),whichisinmanywaystobeexpected.InclosingthissectionwemakeseveralremarksonthestillunprovenRiemannhypothesiswhichwouldhavemajorconsequencesfortheestimateof(x).Aswasmentionedearlier,theerrorterminthePrimeNumberTheoremdependsontheelusivezero-freeregionof(s),andRiemannHypothesizedthatif()=0and0<<()<1then<()=1=2;thisconjecture(iftrue)wouldallowustoimproveourasymptoticestimateto (x)=x+O(x1=2log2(x))and(x)=li(x)+O(x1=2log(x)) 67

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(forproofsoftheseresultsandfurtherconsequencesofRiemann'sHypothesissee[ 7 ]and[ 29 ]).ItshouldbenotedthatwhilethereisagreatdealofevidencetosupportRiemann'sconjecture,thisconjectureappearstobefarbeyondthescopeofourpresentunderstandingofmathematics.Despiteeffortsbysomeofthemosttalentedmathematiciansinhistory,theRiemannHypothesishaseludedallattemptstoeitherproveordisproveitsvalidity. 68

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CHAPTER3ARITHMETICFUNCTIONSINVOLVINGTHELARGESTPRIMEFACTOR 3.1TheAlladi-ErdosFunctionsInthissectionwewillintroducevariousarithmeticfunctionsandidentitieswhichwillbeofuselaterinapplicationstonumericalfactorization.Tobegin,wedenethefollowingfunctions: Denition3.1.1. LetA(n)=Xpjnp,thesumoftheprimefactorsofnandletA(n)=Xpjjnpbethesumoftheprimefactorsofn,weightedaccordingtomultiplicity(Notethatpjjnimpliesthatpjnbutp+1doesnotdividen).WewillcallthesetwofunctionstherstandsecondAlladi-Erdosfunctions,respectively.Furthermore,wenowaddacompanionfunctiontothetwofunctionsjustintroduced,namely,wedeneA0(n)=Xpjnp .Intheir1976paper[ 2 ],AlladiandErdosintroducedthefunctionsA(n)andA(n),aswellasdemonstratingsomeoftheirbasicproperties.WehaveintroducedthefunctionA0(n)asitgeneratesaDirichletSerieswhichisveryconvenienttoworkwithinlightofDelange'sTheorem.Firstly,thefunctionsA(n),A(n),andA0(n)arealladditiveandA(n)A0(n);furthermore,asA0(p2)=p+p2 2andA(p2)=pweseethatA(p2)A0(p2),anditfollowsbyasimpleinductiveargumentthatA(n)A0(n)forallvaluesofn2Z+.Otherinterestingpropertiesfollowbyconsideringtheasymptoticvalueofthefunctionsf(x)=XnxA(n),f(x)=XnxA(n),andf0(x)=XnxA0(n),allofwhichareequalto 2 12x2 log(x)+Ox2 log2(x).(3)ThisasymptoticwasalsoestablishedforthersttimebyAlladiandErdos;however,wewillimprovetheirasymptoticvalueto 69

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(2)Zx2t log(t)dt+O(x2e)]TJ /F5 7.97 Tf 6.59 0 Td[(Cp log(x)),(3)(where(2)=2=6)aswellasshowinghowRiemann'shypothesiswouldallowustofurtherimprovetheerrorterm.Wewillnowanalyticallyderivetheasymptoticoff0(x)whichcanbeapproachedmoreeasilythroughanalysisthanf(x)orf(x).Webeginwithasimplelemma: Lemma3.1.2. f0(x))]TJ /F4 11.955 Tf 11.96 0 Td[(f(x)=O(x3=2)Proof:Considerthedifferencef0(x))]TJ /F4 11.955 Tf 11.95 0 Td[(f(x)whichiseasilyseentobeequalto:Xnx(A0(n))]TJ /F4 11.955 Tf 11.96 0 Td[(A(n))=Xp2xp2 2x p2+Xp3xp3 3x p3+...+Xpxp x pwherepjjn.Clearlythisvalueismajorizedby:Xnx(A0(n))]TJ /F4 11.955 Tf 11.95 0 Td[(A(n))Xp2xx 2+...+Xpxx =x0@Xpx1=21 2+...+Xpx1=1 1A=xx1=2 log(x)+x1=3 log(x)+...+x1= log(x)+...xXnx1=2 log(x)+Oxx1=2 log2(x)=O(x3=2)wherewehaveusedthefactthat(x)=x log(x)+Ox log2(x),andthefactthatPn1=log(x) log(2)=O(log(x)).Thuswehaveobtainedtheresultofthelemma.Itisinterestingtocontrastthedifferencef0(x))]TJ /F4 11.955 Tf 12.37 0 Td[(f(x)fromtheabovelemmawiththedifferencef(x))]TJ /F4 11.955 Tf 11.02 0 Td[(f(x),whichismuchsmaller.ItwasrstprovedbyAlladiandErdosin[ 2 ],that Lemma3.1.3. (Alladi-Erdos)f(x))]TJ /F4 11.955 Tf 11.96 0 Td[(f(x)=xloglog(x)+O(x) 70

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Proof:Itiseasilyseenthatthedifferencef(x))]TJ /F4 11.955 Tf 11.95 0 Td[(f(x)isgivenbyXnx(A(n))]TJ /F4 11.955 Tf 11.95 0 Td[(A(n))=Xp2xpx p2+Xp3xpx p3+...,furthermoreXp2xpx p2=Xpp xx p+O0@Xpp xp1A=xloglog(x)+O(x)andXpixpx pi=xXpix1 pi)]TJ /F7 7.97 Tf 6.58 0 Td[(1=O0@Xpx1=ip1A.Therefore,asXi3Xpx pi=O(x)andwemayconcludethat Xnx(A(n))]TJ /F4 11.955 Tf 11.96 0 Td[(A(n))=xloglog(x)+O(x);(3)whichistheresultofAlladiandErdos.Giventheresultsoftheprecedinglemmas,thedifferencebetweenanythreeofthearithmeticfunctionsf(x),f0(x),andf(x)willnotexceedO(x3=2)=o(x2=log(x)).Itwillbeshownlaterinthischapterthatf0(x)isafunctionwhichisasymptoticallyequivalentto2 12x2 log(x),implyingthatthethreefunctionsmusthavethesameasymptoticvalues.Itisalsointerestingtonotehowmuchgreaterf0(x)growsasymptoticallythanthefunctionsf(x)andf(x).Whileforourpurposesthisdiscrepancywillnotbeimportant,equation(3.1.3)showsthatf(x))]TJ /F4 11.955 Tf 12.55 0 Td[(f(x)=xloglog(x)+O(x)whichisafarsmaller 71

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valuethenf0(x))]TJ /F4 11.955 Tf 12.35 0 Td[(f(x)=O(x3=2).Thisismosteasilyexplainedbythefactthatf0(x)essentiallycountsprimepowers,whereasthesummationoftheAlladi-Erdosfunctionsdonot.Therefore,itisinteresting(andperhapssomewhatcounterintuitive)thateachfunctionhasthesamedominanttermfortheirrespectiveasymptoticvalues.Thisisbestexplainedassayingthatthecontributionfromtheprimespwhichdivideanintegermorethanonce(i.e.pjn,where>1)isquitesmallcomparedtothecontributionfromthelargestprimedivisor.Tomakethisdiscussionmoreprecise,letP1(n)denotethelargestprimedivisorofanintegern.In[ 2 ]AlladiandErdosshowedthat XnxP1(n)=(2) 2x2 log(x)+Ox2 log2(x),(3)andhenceXnxP1(n)hasthesameasymptoticasf0(x).Therefore,themajorityofthecontributiontothesumXnxA(n)comesfromthelargestprimefactorsoftheintegersnx.ThisisconsistentwithHardyandRamanujan'stheoremthatXnx((n))]TJ /F6 11.955 Tf 11.95 0 Td[(!(n))=O(x),where!(n)isthenumberofdistinctprimedivisorsofanintegernand(n)isthenumberofdistinctprimepowersdividinganintegern;furthermore,fromequations(2-27)and(2-28)bothXnx!(n)andXnx(n)areasymptoticallyxloglog(x)+O(x).That!(n)and(n)areasymptoticallyverycloseallowsustosurmise(albeitheuristically)thattheprimefactorsofmostintegersoccursquare-free.Theorem3.1.4alsogivesaninterestingresultduetoErdos(seeTheorem3.1.4below)whichgivesfurtherevidenceofhowthelargestprimefactorofnisquitedominant,anddictatesthebehaviorofalargeclassofarithmeticfunctions.TheseobservationsfurthervalidatethefactthatXnxP1(n)isasymptoticallythesameorderofmagnitudeasf(x),asmostintegerswillonlybedivisiblebyaprimeponce,thehigherpowersofpwillberare;therefore,theyshouldcontributelittletothevalueoff(x),astheydo.Thislastobservationraisesaveryinterestingandnaturalquestion:letPm(n)tobethemthlargestprimefactorofanintegern,1m!(n)(thatis,Pm+1(n)
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1m!(n))]TJ /F3 11.955 Tf 12.51 0 Td[(1).WehavealreadynotedthatXnxP1(n)accountsforthedominantterminthefunctionsf(x),f(x),andf0(x);consequently,itisonlynaturalthatweaskwhetherthesumXnxP2(n)accountsforthedominantterminthemodiedAlladi-Erdosfunctions:Xnx(A(n))]TJ /F4 11.955 Tf 11.95 0 Td[(P1(n)),Xnx(A0(n))]TJ /F4 11.955 Tf 11.96 0 Td[(P1(n)),andXnx(A(n))]TJ /F4 11.955 Tf 11.95 0 Td[(P1(n))?ThisquestionwasposedtoPaulErdosbyKrishnaswamiAlladiduringtheirrstcollaborativeencounter,andwasprovednotlongafterinamuchmoregeneralformin[ 2 ].Theirsolutiontotheproblemissuppliedbyequation(1-13),however,wewillnowstateitformallyasatheorem: Theorem3.1.4. (Alladi-Erdos)Forallintegersm1,wehave:Xnx(A(n))]TJ /F4 11.955 Tf 11.96 0 Td[(P1(n))]TJ /F3 11.955 Tf 11.95 0 Td[(...)]TJ /F4 11.955 Tf 11.96 0 Td[(Pm)]TJ /F7 7.97 Tf 6.59 0 Td[(1(n))sXnxPm(n)skmx1+(1=m) logm(x),wherekm>0isaconstantdependingonlyonm,andisarationalmultipleof(1+1=m)where(s)istheRiemannZetafunction.Thus,form2thesumsXnxPm(n)=Ox1+(1=m) logm(x),whileasymptoticallyboundedbyf(x),f0(x),andf(x),dogrowappreciably.Notethatforthecasem=1,theabovetheoremimpliesthatXnxP1(n)=r2 6x2 log(x)+ox2 log(x),wherer2Q,r>0.Infact,form=1wemaytaker=1=2sothatXnxP1(n)=2 12x2 log(x)+ox2 log(x);however,atpresentwecannotprovethatr=1=2bysimplyappealingtotheabovetheorem.Nevertheless,theseobservationsfurthervalidatethepreviouscommentsconcerningthesumXnxP1(n). 73

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Tofurtheremphasizethedominanceofthelargestprimefactorofanintegern,wewillproveaninterestingresultrstderivedbyP.Erdos.Recallthatifpndenotesthenthprimenumber,thenTchebyschev'sestimatestatesthatpn=O(nlog(n));also,aweakformofthewell-knowntheoremduetoF.Mertens(1874)statesthat Ypx1)]TJ /F3 11.955 Tf 13.3 8.08 Td[(1 p)]TJ /F7 7.97 Tf 6.59 0 Td[(1=O(log(x)).(3)Alloftheseresultscanbemademoreprecise,andcanbefoundin[ 29 ].Ournexttheorem,duetoPaulErdos,givesaninteresting(andinmanywayssurprising)characterizationofanarithmeticfunctionintermsofitslargestprimefactor: Theorem3.1.5. (Erdos)Letf(n)>0beanon-decreasingarithmeticfunction.Thenthesum1Xn=11 f(n)nconvergesifandonlyifthesum1Xn=11 f(P1(n))nconverges.Proof:Asf(n)isnon-decreasingf(P1(n))f(n),thus1Xn=11 f(n)n1Xn=11 f(P1(n))n,sothatif1Xn=11 f(P1(n))nconverges,1Xn=11 f(n)nmustalsoconverge.Toprovetheconverse,considerthefollowing:Xnx1 f(P1(n))n=Xpx1 f(p)XP1(n)=p1 n 74

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=Xpx1 f(p)XP1(m)p1 mp=Xpx1 f(p)pYq
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=1Xn=11 ns+11Xn=1(n) log(n)n ns+1=(s+1)log(s).Usingthefactthat(s)isanalyticin<(s)>1,(1+it)6=0,8t2R,andhasasolesingularityats=1wemaydeducethat(s+1)log(s)isholomorphicatallpointslyingontheline<(s)=1,s6=1;therefore,wemayapplyDelange'stheoremtoconcludethefollowing: Theorem3.1.6. f0(x)=XnxA0(n)=2 12x2 log(x)+ox2 log(x)Proof:Asequation(3-6)demonstrates,1Xn=1A0(n) ns+1=(s+1)log(s).ApplyingDelange'stheorem(Theorem2.2.1)yields:XnxA0(n) n=(2)x log(x)+ox log(x)=2 6x log(x)+ox log(x),andastraightforwardapplicationofAbelsummation(equation(1-12))demonstratesthatf0(x)=XnxA0(n)=2 12x2 log(x)+ox2 log(x),whichisthedesiredresult.UsingMellin'sinversiontheoremwemayimprovethisasymptoticestimatebytakingintoaccountthezero-freeregionof(s)suppliedbyTheorem2.1.3.ToimprovetheestimatewewillnotneedthefullpowerofMellin'sinversiontheorem,infact,forourpurposesthefollowingtheoremwillsufce: Theorem3.1.7. Letk>1andx>0,thentheintegral1 2iZk+i1k)]TJ /F5 7.97 Tf 6.58 0 Td[(i1xs+1 s+1dsequals1ifx>1,1 2ifx=1,and0if0
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IfD(s)=P1n=1d(n) ns+1isanyDirichletseries,wherek>a1ischosensuchthatD(s)liesinadomainofabsoluteconvergence,then1 2iZk+i1k)]TJ /F5 7.97 Tf 6.59 0 Td[(i1D(s)xs sds=1 2iZk+i1k)]TJ /F5 7.97 Tf 6.59 0 Td[(i1 1Xn=1d(n) ns+1!xs+1 s+1ds=Xnx1 2iZk+i1k)]TJ /F5 7.97 Tf 6.59 0 Td[(i1d(n)x ns+1ds s+1+Xnx1 2iZk+i1k)]TJ /F5 7.97 Tf 6.59 0 Td[(i1d(n)x ns+1ds s+1=Xnxd(n)ifx2R)]TJ /F10 11.955 Tf 11.95 0 Td[(Z.Ifx2Zthentheaboveintegralequalsd(x) 2+Xn
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lineofintegrationtobeasuitablecontour.ForourpurposeswewillsimplytaketheclassicalcontourofintegrationchosenbydelaVallee-Poussin,whichrequiresustotake<(s)=>1)]TJ /F5 7.97 Tf 22.13 4.71 Td[(c log(t)forsomeconstantc>0.However,forsinthisregionthefunction(s+1)isanalytic,hencebounded,soweshouldsuspectthatitdoesnotmakeanymajorcontributiontotheintegral.Thus,themajorityofthecontributionfromthecontourwilloccuratthelogarithmicsingularityats=1of(s+1)log(s),whichhasacoefcientof(2);also,theerrorterminourestimatewillbedirectlyrelatedtohowfarwemaytakeourcontourintothecriticalstrip.Aswewillnotbetakingourcontourfurtherthan=1=2,and(s+1)(3=2)isboundedinthisregion,theerrortermwillbeboundedbyafunctiontimestheerrortermimpliedbythePrimeNumberTheorem.Hence,weshouldsuspectthattheintegraliscloselyapproximatedby:(2) 2iZk+i1k)]TJ /F5 7.97 Tf 6.59 0 Td[(i1log(s)xs+1 s+1ds=(2)Xnx(n) log(n)n=(2)Zx2t log(t)dt+O(x2e)]TJ /F5 7.97 Tf 6.59 0 Td[(cp log(x))whichfollowsfromthePrimeNumberTheorem.Thenexttheoremissimplythestatementthatalloftheseobservationsareinfactaccurate.Furthermore,weremindthereaderthatthefollowingtheoremismerelyasketchoftheproofasmanyofthedetailshavebeenomitted. Theorem3.1.8. Thereexistsaconstantc>0suchthatf0(x)=(2)Zx2t log(t)dt+O(x2e)]TJ /F5 7.97 Tf 6.59 0 Td[(cp log(x)).Proof:First,notethatfora>1f0(x)=1 2iZa+i1a)]TJ /F5 7.97 Tf 6.59 0 Td[(i1(s+1)log(s)xs+1 s+1dsfornon-integralx,whereastheintegralequalsf0(x)]TJ /F3 11.955 Tf 11.96 0 Td[(1)+A0(x) 2=f0(x)+o(x3=2)ifx2Z;thusthechoiceofanintegralornon-integralx>0willnotaffectourestimate.Inorder 78

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toevaluatethislineintegralwewilldeformthepathofintegrationasfarintothecriticalstrip,01,aspossiblewhileavoidingthesingularitiesoftheintegrand,alsotheintegrandhasabranchcutonthereallinewhichmustbehandledwithcare.Therefore,wechoosethecontourasfollows:rstletTbexed,b=1)]TJ /F5 7.97 Tf 23.38 4.71 Td[(c log(T),anda=1+c log(T)wherecistheconstantindelaVallee-Poussin'szero-freeregion.Theclassicpathofintegrationisgivenbytheverticallinefroma+i1toa+iT,followedbythehorizontallinefroma+iTtob+iT,thentheverticallinefromb+iTtob+i.Thecontourthenproceedsaroundthebranchcutavoidingthesingularityats=1,withasemicircleofradiusandcenter1,thenfollowsthehorizontalpathbelowthebranchcuttob)]TJ /F4 11.955 Tf 12.05 0 Td[(i.Theremainderofthepathismerelythecontourabovetherealaxisreectedabouttherealline,thatis,theverticallinefromb)]TJ /F4 11.955 Tf 12.21 0 Td[(itob)]TJ /F4 11.955 Tf 12.2 0 Td[(iT,thehorizontallinefromb)]TJ /F4 11.955 Tf 12.4 0 Td[(iTtoa)]TJ /F4 11.955 Tf 12.39 0 Td[(iT,andtheverticallinefroma)]TJ /F4 11.955 Tf 12.4 0 Td[(iTtoa)]TJ /F4 11.955 Tf 12.39 0 Td[(i1.NotethatbydelaVallee-Poussin'stheoremandourchoiceofaandbthiscontourliesinadomainofanalyticityof(s+1)log(s).Hence,byCauchy'stheorem,wemayconcludethatf0(x)=1 2iZa+i1a)]TJ /F5 7.97 Tf 6.59 0 Td[(i1(s+1)log(s)xs+1 s+1ds=1 2iZa)]TJ /F5 7.97 Tf 6.59 0 Td[(iTa)]TJ /F5 7.97 Tf 6.59 0 Td[(i1+Zb)]TJ /F5 7.97 Tf 6.58 0 Td[(iTa)]TJ /F5 7.97 Tf 6.59 0 Td[(iT+Zb)]TJ /F5 7.97 Tf 6.59 0 Td[(ib)]TJ /F5 7.97 Tf 6.58 0 Td[(iT+Zcut+Zb+iTb+i+Za+iTb+iT+Za+i1a+iT(s+1)log(s)xs+1 s+1ds=D(x)+E(x).whereD(x)denotesthedominantterminourasymptoticestimateandE(x)denotestheerrorterm,i.e.E(x)=o(D(x)).Itisawell-knownfactthattheerrorterminthePrimeNumberTheoremisdirectlyrelatedtohowfarwemaymovethepathofintegrationoflog(s)intothecriticalstrip,andasourintegrand(s+1)log(s)isverysimilartothefunctionevaluatedintheclassicalproofsoftheprimenumbertheoremitisnotdifculttojustifythattheerror 79

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termoff0(x)isalsorelatedtohowfarwemaydeformourcontourintothecriticalstrip.Infact,forourpurposestheerrortermE(x)willbesuppliedbythecontourswhicharenotaroundthebranchpoint(forajusticationofthisfactandfurtherdetailssee[ 4 ]).Specically,E(x)=1 2iZa)]TJ /F5 7.97 Tf 6.59 0 Td[(iTa)]TJ /F5 7.97 Tf 6.58 0 Td[(i1+Zb)]TJ /F5 7.97 Tf 6.59 0 Td[(iTa)]TJ /F5 7.97 Tf 6.58 0 Td[(iT+Zb)]TJ /F5 7.97 Tf 6.58 0 Td[(ib)]TJ /F5 7.97 Tf 6.59 0 Td[(iT+Zb+iTb+i+Za+iTb+iT+Za+i1a+iT(s+1)log(s)xs+1 s+1ds.For<(s)=>0thefunction(s+1)ismaximizedontherealline,andasthepathofintegrationdoesnotpenetrateasfaras=1 2intothecriticalstrip,wemayconcludethat,E(x)(3=2) 2iZa)]TJ /F5 7.97 Tf 6.58 0 Td[(iTa)]TJ /F5 7.97 Tf 6.59 0 Td[(i1+Zb)]TJ /F5 7.97 Tf 6.59 0 Td[(iTa)]TJ /F5 7.97 Tf 6.59 0 Td[(iT+Zb)]TJ /F5 7.97 Tf 6.59 0 Td[(ib)]TJ /F5 7.97 Tf 6.59 0 Td[(iT+Zb+iTb+i+Za+iTb+iT+Za+i1a+iTlog(s)xs+1 s+1ds.Fromhereitisastraightforward,thoughdetailed,processtoestimateE(x).However,wemayside-steptheissueofdirectlyevaluatingthesixcontoursbynotingthatXpxp =1 2iZa+i1a)]TJ /F5 7.97 Tf 6.59 0 Td[(i1log(s)xs+1 s+1ds,then,choosingthesamecontourasbefore,itfollowsfromthePrimeNumberTheoremthatXpxp =Zx2t log(t)dt+O(x2e)]TJ /F5 7.97 Tf 6.59 0 Td[(cp log(x)).Thus,theerrortermisgivenby1 2iZa)]TJ /F5 7.97 Tf 6.59 0 Td[(iTa)]TJ /F5 7.97 Tf 6.59 0 Td[(i1+Zb)]TJ /F5 7.97 Tf 6.58 0 Td[(iTa)]TJ /F5 7.97 Tf 6.59 0 Td[(iT+Zb)]TJ /F5 7.97 Tf 6.59 0 Td[(ib)]TJ /F5 7.97 Tf 6.59 0 Td[(iT+Zb+iTb+i+Za+iTb+iT+Za+i1a+iTlog(s)xs+1 s+1ds=O(x2e)]TJ /F5 7.97 Tf 6.58 0 Td[(cp log(x)),whichdiffersfromthecontourintegralwewishtoevaluatebyaconstant.ThereforeE(x)O((3=2)x2e)]TJ /F5 7.97 Tf 6.58 0 Td[(cp log(x))=O(x2e)]TJ /F5 7.97 Tf 6.58 0 Td[(cp log(x)).NotethattheimplicitconstantintheO-termisfarfromoptimal;however,forourpurposesweneedonlyshowthatsuchaconstantexists. 80

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ThenalstepnecessarytoestablishTheorem(3.1.7)istoevaluatetheintegralalongthebranchcut,whichwenowproceedtodo:1 2iZcut(s+1)log(s)xs+1 s+1ds=1 2iZcut(s+1)log((s)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(s))xs+1 s+1ds)]TJ /F3 11.955 Tf 18.63 8.09 Td[(1 2iZcut(s+1)log(s)]TJ /F3 11.955 Tf 11.96 0 Td[(1)xs+1 s+1dsTherstintegraliszerobecause(s+1)log((s)]TJ /F3 11.955 Tf 12.35 0 Td[(1)(s))isregularandsingle-valuedalongthecut;hence,theintegrandisregular,andtheintegralalongtheuppersidecancelstheintegralalongthelowerside.Toevaluatethesecondintegralwemakethesubstitutions)]TJ /F3 11.955 Tf 12.23 0 Td[(1=eifor)]TJ /F6 11.955 Tf 9.3 0 Td[(<<.Withthissubstitutionlog(s)]TJ /F3 11.955 Tf 12.23 0 Td[(1)=log()+i,andthereforethevalueoflog(s)]TJ /F3 11.955 Tf 10.98 0 Td[(1)alongthelowerportionofthebranchcutdiffersfromthevalueoflog(s)]TJ /F3 11.955 Tf 12.4 0 Td[(1)alongtheupperportionofthebranchcutby2i.Lettingbeasemicircleofradiuscenteredats=1,thentheaboveintegralreducestoevaluating: 1 2iZcut(s+1)log(s)]TJ /F3 11.955 Tf 11.96 0 Td[(1)xs+1 s+1ds(3)=1 2iZ1b(2+)(log())]TJ /F4 11.955 Tf 11.95 0 Td[(i)xs+1 s+1ds+1 2iZb1(2+)(log()+i)xs+1 s+1ds+1 2iZ(2+)(log()+i)xs+1 s+1ds.Now,weneedonlyevaluatethethreeintegralsin(3-7);letting!0,thethirdintegraliseasilyseentobe1 2iZ(2+)(log()+i)xs+1 s+1ds=O)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(log(+)x2+2=o(1).ToobtainthedesiredvalueforD(x)wenotethat, 81

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1 2iZ1b(2+)(log())]TJ /F4 11.955 Tf 11.96 0 Td[(i)xs+1 s+1ds)]TJ /F3 11.955 Tf 18.63 8.09 Td[(1 2iZ1b(2+)(log()+i)xs+1 s+1ds=)]TJ /F12 11.955 Tf 11.29 16.27 Td[(Z1b(2+)xs+1 s+1ds=)]TJ /F12 11.955 Tf 11.29 16.27 Td[(Z1b(2)xs+1 s+1ds,as!0.Thusweareleftwithevaluatingtheintegral, (2)Z1bxs+1 s+1ds.(3)Lettingu2=xs+1in(3-8)gives,(2)Z1bxs+1 s+1ds=(2)Zxxb 2+1 2u log(u)du=(2)Zx2u log(u)du+(2)Z2xb 2+1 2u log(u)du;furthermore,aslog(u)>log(2)for2uxb 2+1 2weseethat(2)Zxb 2+1 22u log(u)du=O(xb 2+1 2)whichwillbeabsorbedintotheerrortermofE(x)=O(x2e)]TJ /F5 7.97 Tf 6.59 0 Td[(cp log(x)).ThereforeD(x)=(2)Zx2t log(t)dt,andasf0(x)=D(x)+E(x)wehaveourdesiredestimate: f0(x)=(2)Zx2t log(t)dt+O(x2e)]TJ /F5 7.97 Tf 6.59 0 Td[(cp log(x))(3)whichcompletestheproof.Wereiteratethattheaboveproofhasomittedseveralimportantdetails,specically,theexplicitevaluationofthecontourswhichareawayfromthebranchpoint.However,thereaderneednotworryabouttherigoroftheaboveproofasintegralsofthisformarenowclassicalinanalyticnumbertheory,andcanbefoundinmosttextsonthe 82

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topic.Infact,in1903,usingavariantoftheaboveintegralrepresentation,EdmundLandausucceededingivinghisownproofofthePrimeNumberTheorembyusingtheverysamecontourchosenabove,anditisthisproofwhichLandauincludedinhisclassictext[ 16 ](atextwhichachievedsomuchfamethatmathematicianssuchasG.H.HardysimplyreferredtoitastheHandbuch).Theevaluationofeachofthesecontoursisnotterriblydifcult,althoughthereareseveralparticularswhichmustbetakenintoaccount.Forexample,log(s)isamultiple-valuedfunctionwithasingularityats=1,andthisnecessitatestakingabranchofthelogarithmwhichinturnmakesthecontourintegrationmoredifcult.Furthermore,someofthecontoursarenotabsolutelyconvergent,whichfurthercomplicatestheirevaluation.IntheaboveproofweavoidedthesedifcultiesbynotingthatthecontoursawayfromthebranchpointcorrespondtotheerrorterminthePrimeNumberTheorem,andhencecanbederivedinasimplefashionfromthisobservation.However,itshouldbementionedthattheerrorterminthePrimeNumberTheoremarisesfromtheevaluationthesecontours.Inessence,wehavenotavoidedthetaskofevaluatingthesecontours,butrather,wehaveinvokedatheoremwhichallowsustoavoidtheirexplicitcomputation.Furthermore,notethatbyapplyingintegrationbypartstotheintegralin(3-9)that:f0(x)=(2)Zx2t log(t)dt=(2) 2x2 log(x)+Ox2 log2(x)wemayre-verifyourpreviousestimateoff0(x).AssumingtheRiemannHypothesis,allofthecomplex-valuedsingularitiesoflog(s)willhave<(s)=1 2.Ifonemakesthisassumptionwhenevaluatingtheabovecontourintegral,thenwemayimproveourestimateto:f0(x)=(2)Zx2t log(t)dt+O(x3=2log(x)),whichisthebestpossibleestimateusingthesemethods.Asf0(x))]TJ /F4 11.955 Tf 12.58 0 Td[(f(x)=O(x3=2)andf0(x))]TJ /F4 11.955 Tf 12.97 0 Td[(f(x)=O(x3=2)thisalsoimprovestheerrortermsoff(x)andf(x). 83

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Furthermore,itshouldbenotedthatimprovingtheerrortermintheasymptoticestimateoff0(x)(orintheestimateoff(x)orf(x))toO(x3=2log(x))isequivalenttotheRiemannhypothesis.RecallthatTheorem3.1.3above,duetoAlladiandErdos,demonstratesthatthesumXnxPm(n)=o(x3=2)forallm2.Thisshowsthatsumsofthisformarefarsmallerthaneventhebesterrortermsforf(x),f0(x),andf(x),whichistobeexpected,astheircombinedsum(inmanyways)willdeterminethiserrorterm.Aswasalreadystated,AlladiandErdosderivedtheasymptoticresultsf(x)=2 12x2 log(x)+Ox2 log2(x)andXnxP1(n)=2 12x2 log(x)+Ox2 log2(x)in[ 2 ]usingentirelyelementarymethods.FurtherresultsofthisnaturewerealsoderivedbyKnuthandPardowho,usingonlyelementarymethods,derivedtheasymptoticvaluesforthemeanandstandarddeviationofthelargestprimefactorofn,P1(n).Thefollowingtheoremisarestatementofthisresult,whichwehaveincludedtoprovideaclearerpictureofthefunctionsunderdiscussion.TheemphasisofKnuthandPardoin[ 14 ]istheiralgorithmicprocess,andforthisreasontheyonlysketchhowonemayderivetheirtheorem.Ofcourse,wewishtoemphasizethemathematicalaspectsoftheirpaper,sowhiletheproofbelowisessentiallyduetoKnuthandPardo,weincludecertaindetailstomaketheirproofmorerigorous.UsingKnuthandPardo'snotation,let(t)betheprobabilitythatP1(n)twhennisintherange1nN.ThisfunctionscanbeidentiedwiththeDickmanfunction(u)(seeDenition:3.2.5)discussedintherstsection,andwhichwillbediscussedingreaterlengthinsection3.2.Infactifonesetsu=log(N) log(t)thenfor1u2wehave(u)=(t).In[ 14 ]theauthorsdemonstratethat: 84

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(t)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(loglog(N) log(t)+1 log(N)ZN=t1fugdu u2+O1 log(N)2forp NtN.Now,letEk(P1(n))=ZN1tkd(t)=(N)Nk)]TJ /F3 11.955 Tf 11.96 0 Td[((1))]TJ /F4 11.955 Tf 11.95 0 Td[(kZN1(t)tk)]TJ /F7 7.97 Tf 6.58 0 Td[(1dt,thatis,Ek(P1(n))isthekthmomentofP1(n).Notethatastheaboveintegralfrom1top NisO Nk=2Zp N1d(t)!=O(Nk=2),itwillbeabsorbedintotheerrortermbelow. Theorem3.1.9. Ek(P1(n))=(k+1) k+1Nk log(N)+ONk log2(N).Proof:Ignoringtheintegralfrom1top NweareleftwithZNp Ntkd(t)=(N)Nk)]TJ /F3 11.955 Tf 11.95 0 Td[((p N)Nk=2)]TJ /F4 11.955 Tf 11.95 0 Td[(kZN1(t)tk)]TJ /F7 7.97 Tf 6.59 0 Td[(1dt,=ZNp Ntkd 1+loglog(t))]TJ /F3 11.955 Tf 11.95 .01 Td[(loglog(N)+1 log(N)ZN=t1fugdu u2+O1 log2(N)!=ZNp Ntkd(loglog(t))+1 log(N)Z1p NN vkdZv1fugdu u2+ONk log2(N)byreplacingtbyN=vinthesecondintegral.TheO-estimateisjustiedbythesimpleobservationthatifRbaf(t)dg(t)andRbaf(t)dh(t)exist,whereh(t)=O(g(t)),andwherebothfandgarepositivemonotonefunctionson[a,b],thenZNp Ntk)]TJ /F7 7.97 Tf 6.59 0 Td[(1dt log(t)=NkZp N1dv vk+1(log(N))]TJ /F3 11.955 Tf 11.95 0 Td[(log(v))=Nk log(N) Zp N1dv vk+1+Zp N1log(v)dv vk+1(log(N))]TJ /F3 11.955 Tf 11.95 0 Td[(log(v))! 85

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=Nk klog(N)+ONk log2(N).Notethatthesecondintegralis)]TJ /F4 11.955 Tf 9.3 0 Td[(Nk=log(N)timestheintegralZp N1fvgdv=vk+2,whichiswithinO(N)]TJ /F7 7.97 Tf 6.59 0 Td[((k+1)=2)ofZ11fvgdv vk+2=Xj1Zj+1j(v)]TJ /F4 11.955 Tf 11.95 0 Td[(j)dv vk+2=Xj11 k1 jk)]TJ /F3 11.955 Tf 30.09 8.09 Td[(1 (j+1)k)]TJ /F4 11.955 Tf 24.94 8.09 Td[(j k+11 jk+1)]TJ /F3 11.955 Tf 35.6 8.09 Td[(1 (j+1)k+1=Xj11 k(k+1)1 jk)]TJ /F3 11.955 Tf 30.1 8.09 Td[(1 (j+1)k)]TJ /F3 11.955 Tf 23.9 8.09 Td[(1 k+11 (j+1)k+1=1 k(k+1))]TJ /F3 11.955 Tf 23.9 8.09 Td[(1 k+1((k+1))]TJ /F3 11.955 Tf 11.96 0 Td[(1)=1 k)]TJ /F6 11.955 Tf 13.15 8.09 Td[((k+1) k+1.Hence,wehaveshownthatEk(P1(n))=(k+1) k+1Nk log(N)+ONk log2(N),whichisthedesiredresult.ThisveriestheresultofAlladiandErdosthattherstmoment(themean)ofP1(n)isasymptotically2 12N log(N).Furthermore,theabovetheoremdemonstratesthatP1(n)hastheasymptoticstandarddeviationofr (3) 3N p log(N)(withinafactorof1+O(1=log(N))).In[ 14 ]KnuthandPardoalsomaketheimportantobservationthattheratioofthestandarddeviationtothemeandeviatesasN!1.Thiswillbeimportantfortheanalysisoftheirfactorizationalgorithm(inChapter4)asitshows 86

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thatatraditionalmeanandvarianceapproachisunsuitablewhendealingwithsuchfactorizationalgorithms. 3.2(x,y)ItwillbeveryinformativetoconsiderthemostfrequentlyoccurringvalueforthelargestprimefactorP1(n)ofagivenintegern,nx.Thisvaluecannotbeobtainedfromtheaboveestimateswhichgivetheaverageofthelargestprimefactorsofintegersnx.Itturnsoutthatthemostfrequentlyoccurringvalueforthelargestprimefactorofanintegernxisfarsmallerthanf(x)=x,owingtoasmallnumberofintegerswithverylargeprimefactorswhichinuencetheestimateforf(x)=x(andmakeitmuchlarger).Inordertodothisanalysiswemustintroducethefunction(x,y): Denition3.2.1. TheBuchstab-deBruijnfunction(x,y)isdenedtobethenumberofpositiveintegersnxsuchthatP1(n)y.Thatis,(x,y)isthenumberofintegerslessthanxwhoseprimefactorsarelessthany.Asthevaluelog(x) log(y)playsanimportantroleinthebehaviorofthisfunction,itiscustomaryandconvenienttodeneu:=log(x) log(y),providedxy2;fortheremainderofthissectionanyreferencetouwillcorrespondtothisdenition.Aswasmentionedintheintroduction,(x,y)wasrststudied(inisolation)byS.Ramanujanand(publicly)byR.Rankinin[ 23 ],whousedittoinvestigatethedifferencesbetweenprimenumbers.However,Rankin'sresultsonlypertainedtothefunction(x,y)inaratherlimitedrangeforthevalueofy.OneyearpriortoRankin'spublicationA.A.Buchstabderivedaverygeneralrecurrenceformulain[ 6 ]toevaluaterecurrenceswhichcommonlyariseinsievetheory.Initsmostgeneralformthisformulacanquicklybecomecomplicatedtothepointoflosingmuchofitsusefulness,butasitpertainsto(x,y)theformulaisfarlessdaunting.Thefollowingidentitywillbereferred 87

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toasBuchstab'srecurrencerelation(theproofisfairlyeasyandcanbefoundin[ 29 ]):forx1,y>0,wehave (x,y)=1+Xpyx p,p,(3)wherepnis,asusual,theprimenumberslessthanorequalton.Itwasarecurrenceofthissort(thoughinadifferentguise)whichwasdiscoveredbyRamanujan,andwhoserediscoveryisrecountedintheinterestingarticle[ 27 ].Iteratingthisrecurrencegivesusthefollowingtheorem,knownasBuchstab'sidentity,whoseproofcanalsobefoundin[ 29 ].Thus(2-26)implies: Theorem3.2.2. Forx1,zy>0,wehave(x,y)=(x,z))]TJ /F12 11.955 Tf 16.93 11.36 Td[(Xy
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(x,y)=O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(xe)]TJ /F5 7.97 Tf 6.58 0 Td[(u=2holdsuniformly.Itisworthnotingthatthevalueucanvarywithxintheabovetheorem,whichisoneofthemajorreasonsforthesuperiorityofdeBruijn'sresultoverthoseofhispredecessor's.However,thisresultcanbestrengthened,aswasdemonstratedbydeBruijnin[ 9 ]thatifx>0,y2,wherelog2(x)yxandudenedasbefore,then (x,y)0wemayapplyPerron'sformulatoobtain:(x,y)=1 2iZ+i1)]TJ /F5 7.97 Tf 6.59 0 Td[(i1Ypy1)]TJ /F3 11.955 Tf 15.57 8.08 Td[(1 ps)]TJ /F7 7.97 Tf 6.59 0 Td[(1xs sds,whichdeBruijnthenevaluates,fromwhichheobtains(3-11).Forourpurposesweonlyrequiretheuniformestimatefor(x,y)givenbyTheorem3.2.3,whichholdsforlargevaluesofy.WewillnowstudytheDickmanfunction,afunctionwhichfrequentlyarisesinthestudyof(x,y)andwhichitselfsatisesmanyinterestingandusefulequations.AstudyoftheDickmanfunction,andinparticulartherateofdecayofthisfunction,willhelpustobetterunderstandtheasymptoticrelationinTheorem3.2.3 Denition3.2.4. Letu=log(x) log(y)suchthat2u3,theDickmanfunctionisdenedas(u):=1)]TJ /F3 11.955 Tf 11.95 0 Td[(log(u)+Zu2log(v)]TJ /F3 11.955 Tf 11.95 0 Td[(1)dv v. 89

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ItmaynotbeimmediatelyclearhowtheDickmanfunctionisrelatedtotheBuchstab-deBruijnfunction,however,intheproofoftheorem(3.1.8)whatKnuthandPardorefertoas(t)isessentially(u)withu=log(N) log(t).Thefollowingtheoremmakestheconnectionbetween(u)and(x,y)moreexplicit(see[ 29 ],[ 9 ],and[ 27 ]) Theorem3.2.5. Forxy2,andxedu=log(x) log(y),wehavelimx!1(x,x1=u) x=(u).Hence,theabovetheoremdemonstratesthatforxedu(x,x1=u)s(u)x.Itisafactthat(u)isuniquelydenedbytheinitialcondition(u)=1for0u1andtherecurrence(u)=(k))]TJ /F12 11.955 Tf 11.96 16.27 Td[(Zuk(v)]TJ /F3 11.955 Tf 11.96 0 Td[(1)dv vfork1.Furthermore,theabovepropertiesof(u)demonstratethattheDickmanfunctionisdifferentiableforu>1,andusingTheorems3.2.3and3.2.5DeBruijnshowedthatforu>3 (u)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(ulog(u))]TJ /F5 7.97 Tf 6.59 0 Td[(uloglog(u)+O(u).(3)Wearenowinapositiontoanswerthemotivatingquestionofthissection. 90

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Theorem3.2.6. Forall>0,themostfrequentlyoccurringvalueforP1(n)forallnxliesbetweene(1)]TJ /F8 7.97 Tf 6.58 0 Td[()p log(x)loglog(x)ande(1+)p log(x)loglog(x).Proof:ThenumberofintegersmxwiththepropertythatP1(m)=pisgivenbyx p,p,soweneedtomaximizethisfunction.Fromequation(3-13)wemayconcludethatx p,p=x pe)]TJ /F15 5.978 Tf 7.78 4.02 Td[(log(x=p) log(p)+O(u)=x pe)]TJ /F15 5.978 Tf 7.78 3.86 Td[(log(x) log(p)+1+o(1);hence,maximizingthefunctionx te)]TJ /F15 5.978 Tf 7.79 3.86 Td[(log(x) log(t)+1fortwillgivearoughestimateoftheaveragesizeofthelargestprimefactorofn,fornx.Thisisasimpleoptimizationproblem,thegreatestvalueoftheaboveequationoccurswhent=eq log(x) loglog(x)andinsertingthisintotheequationyieldse(1+o(1))p log(x)loglog(x),fromwhichwemayconcludethedesiredresult.Thefunction(x,y)wasgeneralizedbyKnuthandPardoin[ 14 ],tok(x,y)=jfnx:Pk(n)ygj.Clearly1(x,y)=(x,y),butthereareseveralsimilaritiesbetween(x,y)andk(x,y),namely,onemaystudythesefunctionsinductively.If>0then k(x,x)=k()x+Ox log(x)(3)wherek()arefunctionsanalogoustotheDickmanfunctioninthecasewhenk=1.KnuthandPardodemonstratethatthek()functionssatisfysimilarrecurrencerelationsto(u).If>1andk1then,k()=1)]TJ /F12 11.955 Tf 11.96 16.27 Td[(Z1(k(t)]TJ /F3 11.955 Tf 11.96 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(1))dt t,for0<1andk1,k()=1, 91

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andif0ork=0k()=0.Furthermore,KnuthandPardodemonstratethefollowingimportantasymptoticresults,fork=2thereexistconstantsc0,c1,...,crsuchthat 2()=ec0 +c1 2+...+cr)]TJ /F7 7.97 Tf 6.59 0 Td[(1 r+O()]TJ /F5 7.97 Tf 6.59 0 Td[(r)]TJ /F7 7.97 Tf 6.58 0 Td[(1)(3)andfork3wehave k()=elogk)]TJ /F7 7.97 Tf 6.59 0 Td[(2() (k)]TJ /F3 11.955 Tf 11.96 0 Td[(2)!+Ologk)]TJ /F7 7.97 Tf 6.58 0 Td[(3() .(3)Equations(3-14),(3-15),and(3-16)showthatthereisadramaticdifferencebetweenthefunctionsk(x,y)fork2and1(x,y),inparticular,k(x,y)decreasesmuchmorerapidlyfork=1thanitdoesfork2.Asasimpleexampleofthisdifference,considerthefunctions1(x,2)and2(x,2).Astherearenoprimesp2exceptp=2,itfollowsthat1(x,2)willsimplybeacountofthenumberofpowersof2lessthanorequaltox.Hence,1(x,2)=log(x) log(2);however,noticethateverynumberoftheform2pwherepx 2willbecountedbythefunction2(x,2).Bytheprimenumbertheoremthisimpliesthatthereareaboutx 2log(x)suchnumbersoftheform2px,thus,x 2log(x)2(x,2).Althoughasimpleexample,itisclearthat1(x,2)growsfarmoreslowlyasafunctionofxthandoes2(x,2). 3.3GeneralizedAlladi-DualityThereisaninterestingdualitybetweenthekthlargestandthekthsmallestprimefactorsofanintegern,rstnotedbyK.Alladiin[ 1 ],whichwillbeofuseinlaterobservationsconcerningnumericalfactorization.WhereasAlladi'streatmentisentirelyelementary,andholdsforallarithmeticfunctions,thefollowingproofwilldemonstrateAlladi'sdualityanalytically.WealsonotethatalthoughAlladi'sproofofthedualityin 92

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[ 1 ]generalizestogivethefollowingresult,heonlysuppliesaproofoftheprincipleforthespecialcaseofk=1(whichisthedualityamongstthelargestandsmallestprimefactorsofn).TheonlydetrimenttotheanalyticapproachisthatwemustplaceboundsonthearithmeticfunctionsbeingdiscussedtoensuretheconvergenceoftheDirichletserieswhichtheygenerate;however,thisisarelativelyminorrestriction,andisinfactequivalenttothestatementthattheDirichletseriesgeneratedbythearithmeticfunctionhasanabscissaofconvergencea6=.Furthermore,asabonus,wewillderiveanewrepresentationforthefunction(s).Letg(n)beanarithmeticfunctionsuchthatg(1)=0andletpk(n)andPk(n)denotethekthsmallestandkthlargestprimefactorsofn,respectively(whilethesetwovaluesmaycoincide,thehopeisthatwiththisdenitionthenotationwillnotcauseanyconfusion).Furthermore,let(n)beMobius'snumbertheoreticfunction.Inthissection,unlessotherwiseindicated,sumsaretobetakenforalln2. Lemma3.3.1. (Alladi)Xdjn(d)g(Pk(d))=()]TJ /F3 11.955 Tf 9.29 0 Td[(1)k!(n))]TJ /F3 11.955 Tf 11.95 0 Td[(1k)]TJ /F3 11.955 Tf 11.96 0 Td[(1g(p1(n)),Xdjn(d)g(pk(d))=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)k!(n))]TJ /F3 11.955 Tf 11.96 0 Td[(1k)]TJ /F3 11.955 Tf 11.95 0 Td[(1g(P1(n)),Xdjn(d)!(d))]TJ /F3 11.955 Tf 11.96 0 Td[(1k)]TJ /F3 11.955 Tf 11.96 0 Td[(1g(P1(d))=()]TJ /F3 11.955 Tf 9.29 0 Td[(1)kg(pk(n)),Xdjn(d)!(d))]TJ /F3 11.955 Tf 11.96 0 Td[(1k)]TJ /F3 11.955 Tf 11.96 0 Td[(1g(p1(d))=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)kg(Pk(n)),inparticular,Xdjn(d)g(P1(d))=)]TJ /F4 11.955 Tf 9.29 0 Td[(g(p1(n)), 93

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andXdjn(d)g(p1(d))=)]TJ /F4 11.955 Tf 9.3 0 Td[(g(P1(n)).Proof:ConsidertheDirichletseriesgeneratedby(n)z!(n)g(p1(n)),forjzj<1,andtheeasyidentity: E(z;s)=1Xn=1(n)z!(n)g(p1(n)) ns=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xpg(p) psYq>p1)]TJ /F4 11.955 Tf 15.52 8.09 Td[(z qs,(3)subjectonlytotherestrictionthatg(n)growsataratewheretheDirichletseriesunderconsiderationconvergesforallssuchthat<(s)>a.Thisidentityisvalidforalls2C,<(s)>a,whereaistheabscissaofabsoluteconvergenceoftheDirichletseries1Xn=1(n)g(p1(n)) ns,providedjzj1.Therefore, 1 (k)]TJ /F3 11.955 Tf 11.96 0 Td[(1)!@k @zk(s)E(z;s)=1Xn=10@Xdjn(d)!(d))]TJ /F3 11.955 Tf 11.96 0 Td[(1k)]TJ /F3 11.955 Tf 11.96 0 Td[(1z!(d))]TJ /F5 7.97 Tf 6.59 0 Td[(kg(p1)1A1 ns.(3)However, (s)E(z;s)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xpg(p) ps"Yq>p1)]TJ /F4 11.955 Tf 15.52 8.09 Td[(z qs1)]TJ /F3 11.955 Tf 15.67 8.09 Td[(1 qs)]TJ /F7 7.97 Tf 6.59 0 Td[(1#Yrp1)]TJ /F3 11.955 Tf 15.04 8.09 Td[(1 rs)]TJ /F7 7.97 Tf 6.59 0 Td[(1(3)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xpg(p) psYrp1)]TJ /F3 11.955 Tf 15.05 8.09 Td[(1 rs)]TJ /F7 7.97 Tf 6.59 0 Td[(1Yq>p1+1 qs+1 q2s+...)]TJ /F4 11.955 Tf 15.52 8.09 Td[(z qs)]TJ /F4 11.955 Tf 17.74 8.09 Td[(z q2s)]TJ /F3 11.955 Tf 11.95 0 Td[(...=)]TJ /F12 11.955 Tf 11.29 11.35 Td[(Xpg(p) psYrp1)]TJ /F3 11.955 Tf 15.04 8.09 Td[(1 rs)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xp1(m)>p()]TJ /F4 11.955 Tf 9.3 0 Td[(z)!(m))]TJ /F7 7.97 Tf 6.58 0 Td[(1 ms;hence, limz!1)]TJ /F3 11.955 Tf 27.02 13.9 Td[(1 (k)]TJ /F3 11.955 Tf 11.96 0 Td[(1)!@k @zk(s)E(z;s)=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)kXpg(p) psYrp1)]TJ /F3 11.955 Tf 15.04 8.08 Td[(1 rs)]TJ /F7 7.97 Tf 6.58 0 Td[(1X!(m)k)]TJ /F7 7.97 Tf 6.59 0 Td[(1,p1(m)>p1 ms(3) 94

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=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)k1Xn=1g(Pk(n)) ns.Byequating(3-18)and(3-20)weseethat()]TJ /F3 11.955 Tf 9.29 0 Td[(1)k1Xn=1g(Pk(n)) ns=1Xn=10@Xdjn(d)!(d))]TJ /F3 11.955 Tf 11.95 0 Td[(1k)]TJ /F3 11.955 Tf 11.95 0 Td[(1g(p1(d))1A1 ns.Now,fromtheuniquenessofDirichletserieswemayequatethecorrespondingcoefcientsof(3-18)and(3-20)intheaboveequalitytoconcludethat()]TJ /F3 11.955 Tf 9.3 0 Td[(1)kg(Pk(n))=Xdjn(d)!(d))]TJ /F3 11.955 Tf 11.96 0 Td[(1k)]TJ /F3 11.955 Tf 11.96 0 Td[(1g(p1(d)).Provingthefourthidentity.Similarly,considertheDirichletseriesgeneratedby(n)z!(n)g(P1(n)) D(z;s):=1Xn=1(n)z!(n)g(P1(n)) ns=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xpg(p) psYqp1)]TJ /F4 11.955 Tf 15.52 8.09 Td[(z qs;(3)then, 1 (k)]TJ /F3 11.955 Tf 11.95 0 Td[(1)!@k @zk(s)D(z;s)=1Xn=10@Xdjn(d)z!(d))]TJ /F5 7.97 Tf 6.59 0 Td[(k!(d))]TJ /F3 11.955 Tf 11.96 0 Td[(1k)]TJ /F3 11.955 Tf 11.96 0 Td[(1g(P1(n))1A1 ns.(3)However, (s)D(z;s)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xpg(p) psYr>p1)]TJ /F3 11.955 Tf 15.04 8.09 Td[(1 rs)]TJ /F7 7.97 Tf 6.59 0 Td[(1Yqp1)]TJ /F4 11.955 Tf 15.52 8.09 Td[(z qs1)]TJ /F3 11.955 Tf 15.67 8.09 Td[(1 qs)]TJ /F7 7.97 Tf 6.58 0 Td[(1(3)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xpg(p) psYr>p1)]TJ /F3 11.955 Tf 15.04 8.09 Td[(1 rs)]TJ /F7 7.97 Tf 6.59 0 Td[(1XP1(m)p()]TJ /F4 11.955 Tf 9.3 0 Td[(z)!(m))]TJ /F7 7.97 Tf 6.59 0 Td[(1 ms,andtakingthelimitasz!1)]TJ /F1 11.955 Tf 10.41 -4.33 Td[(by(3-21),(3-22),and(3-23)wehave 95

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limz!1)]TJ /F3 11.955 Tf 27.02 13.9 Td[(1 (k)]TJ /F3 11.955 Tf 11.96 0 Td[(1)!@k @zk(s)D(z;s)=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)kXpg(p) psYr>p1)]TJ /F3 11.955 Tf 15.04 8.09 Td[(1 rs)]TJ /F7 7.97 Tf 6.59 0 Td[(1X!(m)k)]TJ /F7 7.97 Tf 6.59 0 Td[(1,P1(m)
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=(n)()]TJ /F3 11.955 Tf 9.29 0 Td[(1)kXdjn(d)g(pk(d));thus,()]TJ /F3 11.955 Tf 9.3 0 Td[(1k)!(d))]TJ /F3 11.955 Tf 11.95 0 Td[(1k)]TJ /F3 11.955 Tf 11.95 0 Td[(1g(P1(n))=Xdjn(d)g(pk(d))whichisthesecondidentity.Therstidentityfollowssimilarly,provingthelemma.ThenextlemmacanalsobeviewedasageneralizationofAlladi'sdualityprincipleforsubsetsPwherePdenotesthesetofallprimenumbers.NotethataswehavealreadyprovedtheAlladidualityidentitiesforallnthefollowinglemmafollowstriviallyifweconsideronlysumswhicharetakenovertheintegersn2suchthatifaprimepjnthenp2. Lemma3.3.2. )]TJ /F4 11.955 Tf 9.29 0 Td[(g(p1(n))=Xdjn(d)g(P1(d))and)]TJ /F4 11.955 Tf 9.29 0 Td[(g(P1(n))=Xdjn(d)g(p1(n)),whereitisunderstoodthatg(n)isaboundedfunction(inthesenseofLemma3.3.1)andthesumsaretakenovertheintegersn2suchthatifpjnthenp2.FromLemma3.3.2andthepropertiesof(s)wemayestimatesumsoftheformXnx,P12g(p1(n))usingTheorem2.2.1(Delange'stheorem).Thus,Xnx,P1(n)21=Cx+o(x)ifandonlyifXp1(n)2(n)g(p1(n)) ns=)]TJ /F4 11.955 Tf 9.3 0 Td[(C<+1;Nowconsider 97

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Xp1(n)2(n) ns=)]TJ /F3 11.955 Tf 18.15 8.08 Td[(1 (s)Xp21 psYqp1)]TJ /F3 11.955 Tf 15.57 8.08 Td[(1 ps)]TJ /F7 7.97 Tf 6.59 0 Td[(1=)]TJ /F3 11.955 Tf 18.14 8.08 Td[(1 (s)XP1(n)21 ns(3)fors2C,<(s)>1.FromTheorem2.2.1itfollowsthatXnx,p1(n)21=e1x+o(x)ifandonlyif)]TJ /F4 11.955 Tf 9.3 0 Td[(e1=XP1(n)2(n) n,andXnx,p1(n)21=d1x+o(x)ifandonlyif)]TJ /F4 11.955 Tf 9.3 0 Td[(d1=Xp1(n)2(n) n.Inparticular,if=Pthentheaboveidentitiesbecome:1 (s))]TJ /F3 11.955 Tf 11.95 0 Td[(1=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xp1 psYq
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andasaresultof(3-21),Yq>nloglog(n);equivalently,(x)<
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l,m(x)=1 (m)x log(x)+Ox log2(x).ItisspeculatedbyAlladiin[ 1 ],andthisauthor,thatequation(3-22)iselementarilyequivalenttotheprimenumbertheoremforarithmeticprogressions.However,atpresentthisisonlyaconjecture.ThiscompletesChapter3andthetheoreticalresultsnecessarytoanalyzethefactorizationalgorithmofKnuthandPardo. 100

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CHAPTER4THEKNUTH-PARDOALGORITHMThealgorithmintroducedbyKnuthandPardointheir1976paper[ 14 ]isperhapsthesimplestwaytofactoranintegeralgorithmically.Theessentialidearestsinattemptingtodivideanintegernbyvarioustrialprimedivisorssuchas2,3,5,...andthencastingoutallfactorswhicharediscovered,andthenrepeatingtheprocess.Asthisprocesswilldiscoverallprimedivisorslessthanp n,thealgorithmwillterminatewhenthetrialdivisorsexceedthesquarerootoftheremainingun-factoredpart.Thereasonwhywemayrestrictourattentiontofactorslessthenp nisthatifniscomposite,thenitmusthaveaprimedivisorp n;or,statedanotherway,halfofthedivisorsofanintegernmustbep n,withtheotherhalfofthedivisorsbeingobtainedbyevaluatingn=d,fordjn,dp n.Also,ifp>p n,andpjn,thenp2willnotdividen.Thespeedofthisalgorithmisintimatelyrelatedtothesizeoftheprimefactorsofn.Ifnisitselfaprimenumberthentherewillbeapproximatelyp ntrialdivisions(ofcourse,wedonotknowaprioriwhetherornotnisprime);whereas,ifnisapowerof2(i.e.n=2aforsomepositiveintegera)thenumberoftrialdivisionswillbeO(log(n)).KnuthandPardoconsiderarandomintegernanddeterminetheapproximateprobabilitythatthenumberoftrialdivisionsisnxwith0x1=2.Theythendemonstratethatthenumberoftrialdivisionswillben0.35abouthalfofthetime(fortheseresultssee[ 14 ]).KnuthandPardoreachtheirconclusionbyanalyzingthekthlargestprimefactorofaninteger,andthendeterminetherunningtimeoftheiralgorithmbythesizeofthelargesttwoprimefactors.Wewillnowintroducetheirstandarddivideandfactoralgorithm.Forn2letn=p1(n)p2(n)...pt(n)m,wherep1(n),...,pt(n)areprimenumberslistedinnon-decreasingorderandmdwithallprimefactorsofmgreaterthanorequaltod,i.e.pi(m)d.Itisunderstoodthatwehavesuchalistoftheprimenumberswhicharerelevanttothealgorithm,andhencewedonotneedtoaddmoretimetothe 101

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algorithmbydeterminingthetrialprimedivisors.WhatfollowsisaninformalALGOL-likedescriptionofthealgorithm,suppliedin[ 14 ]bytheauthors:t:=0,m:=n,d:=2While:d2mdoBegin:increasedordecreasem;IfdjmthenBegin:t:=t+1pt(n):=dm:=m=dendElse:d:=d+1endt:=t+1;pt(n)=m;m:=1;d:=1IfwedenoteDasthenumberoftrialdivisionsperformedandTasthenumberofprimefactorsofn(countingmultiplicity)thentheabovealgorithm'swhile-looprequiresapproximatelyD+1operations,theif-looprequiresDoperations,thebegin-looprequiresT)]TJ /F3 11.955 Tf 11.95 0 Td[(1operations,andtheElse-looprequiresD)]TJ /F4 11.955 Tf 11.96 0 Td[(T+1operations.KnuthandPardoremarkin[ 14 ]thattheiralgorithmcanberenedinseveralsimplewaysbyavoidinglargenumbersofnon-primedivisors,forinstance,ifd>3wemayconsiderprimedivisorsoftheform6k1.Thishastheeffectofdividingthenumberoftrialdivisionsperformed,D,byaconstant.Theyfurthercommentthattheanalysisofthesimplecaseappliestomorecomplicatedsettingswithonlyminorvariations.LetPk(n)bethekthlargestprimefactorofn;therefore,Pk(n)=pT+1)]TJ /F5 7.97 Tf 6.59 0 Td[(k(n)afterthealgorithmterminates,with1kT.Ifnhaslessthank-primefactorsthenletPk(n)=1,andforconvenienceletP0(n)=1.KnuthandPardoobservein[ 14 ]that 102

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thewhile-loopinthealgorithmcanterminateinthreedifferentwaysdependinguponthenalinputsintotheloop:Case1:Ifn<4thenD=0,asd=2impliesthatd2>n;hence,thealgorithmwillterminate.Case2:Ifn4andtheDthtrialdivisionsucceeds.Inthiscasethenaltrialdivisionwasbyd=P2(n),whered2>P1(n).Asdisinitially2theoperationd:=d+1isperformedD)]TJ /F4 11.955 Tf 12.06 0 Td[(T+1times,andhenceD)]TJ /F4 11.955 Tf 11.95 0 Td[(T+1=P2(n))]TJ /F3 11.955 Tf 11.96 0 Td[(2orD=P2(n)+T)]TJ /F3 11.955 Tf 11.96 0 Td[(3forP2(n)2>P1(n).Case3:Ifn4andtheDthtrialdivisionfails,thenthenaltrialdivisionwasbyd,whereP2(n)d,d2P1(n).Thus: D=lp P1(n)m+T)]TJ /F3 11.955 Tf 11.95 0 Td[(3(4)whereP2(n)2
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Theorem4.0.3. ThelimitlimN!1Probk(x,N) N=Fk(x)exists.Proof:ConsiderProbk(t+dt,N))]TJ /F4 11.955 Tf 12.62 0 Td[(Probk(t,N),thenumberofnNwhereNtPk(n)Nt+dt,wheredtissmall.TocountthenumberofsuchnwetakeallprimessuchthatNtpNt+dtandmultiplybyallnumbersmN1)]TJ /F5 7.97 Tf 6.59 0 Td[(tsuchthatPk(m)pandPk)]TJ /F7 7.97 Tf 6.58 0 Td[(1(m)p.Ifn=mpthennNt+dtandPk(n)=p;conversely,everynNwithNtPk(n)Nt+dtwillhavetheformn=mpfortheabovestatedpandm.NotethatthenumberofsuchmN1)]TJ /F5 7.97 Tf 6.59 0 Td[(tsuchthatPk(m)pisapproximatelyProbk(t 1)]TJ /F5 7.97 Tf 6.59 0 Td[(t,N1)]TJ /F5 7.97 Tf 6.58 0 Td[(t),theunwantedsubsetconsistingofthosemsuchthatPk)]TJ /F7 7.97 Tf 6.59 0 Td[(1(m)
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AsFk(0)=0wemayintegrateequation(4-6)toobtain: Fk(x)=Z10Fkt 1)]TJ /F4 11.955 Tf 11.96 0 Td[(t)]TJ /F4 11.955 Tf 11.95 0 Td[(Fk)]TJ /F7 7.97 Tf 6.59 0 Td[(1t 1)]TJ /F4 11.955 Tf 11.96 0 Td[(tdt t.(4)AccordingtotheconventionP0(n)=1wedeneF0(x)=0,forallx.WemustalsohaveFk(x)=1forx1,k1.Notethatequation(4-7),togetherwiththeseinitialconditions,uniquelydetermineFk(x)for0x1,andaswealsohavetheequation: Fk(x)=1)]TJ /F12 11.955 Tf 11.95 16.27 Td[(Z1xdt tFkt 1)]TJ /F4 11.955 Tf 11.95 0 Td[(t)]TJ /F4 11.955 Tf 11.96 0 Td[(Fk)]TJ /F7 7.97 Tf 6.58 0 Td[(1t 1)]TJ /F4 11.955 Tf 11.95 0 Td[(t(4)for0x1,Fk(x)isalsouniquelydenedintermsofitsvaluesatpoints>x.Hencethelimitiswell-denedandthereforeexists.Wenowreturntothegeneralizedfunctionsk(x,y)tobetterenableustounderstandtheKnuth-Pardoalgorithm.Intheirpaper,KnuthandPardousethekthmomentcalculatedinTheorem3.1.9todeducemanyusefulpropertiesaboutk(x,y)and,consequently,toderivebetterresultsonPk(n).ItshouldbeclearthatProbk(x,N)=k(N,Nx)sothatk(N,Nx)sFk(x)N.ByanalyzingthevaluesEk(P1(n))(asinTheorem3.1.9)KnuthandPardogoontoshowthatk(x,x)=k()x+Ox log(x)forallxed>1. 105

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Toclosethediscussionconcerningthisfactorizationalgorithm,wewillmakeseveralremarksaboutthemodel.Foronething,themodelislargelyprobabilisticinthatitconsidersarandomnbetween1andN,andforrelationssuchasnkNx;however,theauthorsnotein[ 14 ]thatfromanintuitivestandpointitmaybemorenaturaltoaskfortheprobabilityofarelationsuchasnknxwithoutconsideringN.Furthermore,theycommentthatitisquiteeasytoconvertfromtheonemodeltotheotherasmostnumbersbetween1andNarelarge.Tomakethisdiscussionmoreprecise,theauthorsof[ 14 ]considerthenumberofintegersn,1 2NnN,suchthatPk(n)Nx.Thisis:Probk(x,N))]TJ /F4 11.955 Tf 12.38 0 Td[(Probk)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(x,N 2=1 2NFk(x)+O(N=log(N));whicheasilyfollowsfrom:Probk(x,N)=NFk(x)+O(N=log(N)).Furthermore,considerhowmanyofthesenhavenx(1 2N)x=Nx)]TJ /F15 5.978 Tf 8.59 3.86 Td[(log(2) log(N),andFkx)]TJ /F7 7.97 Tf 14.22 5.48 Td[(log(2) log(N)=Fk(x)+O(1=log(N)),asFk(x)isdifferentiable.Hence,thenumberofsuchnisatmost Probk(x,N))]TJ /F4 11.955 Tf 11.95 0 Td[(Probkx)]TJ /F3 11.955 Tf 14.71 8.08 Td[(log(2) log(N),N=ON log(N),(4)wheretheconstantimpliedbytheO-termin(4-9)isindependentofxinaboundedregionaboutx.Hence,wehaveshownthatFk(x)+ON log(N)forallnsuchthat1 2NnNsatisfyPk(n)nx.Therefore,ifQk(x,N)denotesthetotalnumberofnNsuchthatPk(n)nx,wehave:Qk(x,N)=X1jlog2log(N)N 2jFk(x)+Olog)]TJ /F7 7.97 Tf 6.59 0 Td[(1N 2j+ON log(N) =NFk(x)+ON log(N),(4)bydividingtherangeN log(N)nNintolog2log(N)parts.DenetheprobabilityofastatementS(n)aboutthepositiveintegernbytheformula: 106

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Pr(S(n))=limN!11 Njfn:nNs.t.S(n)istruegj(4)whenthelimitexists.HencewemayconcludethatPr(Pk(n)nx)=Fk(x),forallxedx.AnotherimportantobservationconcerningthetheoreticalmodelusedinthispaperisthatresultswerestatedintermsoftheprobabilitythatthenumberofoperationsperformedisNx(ornx).Typicallyitiscustomarytoapproachtheaveragenumberofoperationsofanalgorithmintermsofmeanvaluesandstandarddeviations;however,thisapproachappearstobeparticularlyuninformativeforthisalgorithm.KnuthandPardocommentin[ 14 ]thatthisphenomenonisapparentwhenconsideringtheaveragenumberofoperationsperformedoverallnN,whichwillberelativelyneartheworstcasen1=2;however,inmorethan70percentofthecasestheactualnumberofoperationsperformedwillbelessthann0.4.Anotherreasonwhythetypicalmean-varianceapproachisuninformativeisbecause,aswasnotedinTheorem3.1.4ofChapterIII,theratioofthestandarddeviationofthekthprimefactortoitsstandarddeviationisadivergentquantity.Anotherpoint,whichisworthnoting,isthat(asthenamesuggests)theKnuth-Pardoalgorithmisaverysimplealgorithm,andinrecentyearsmoreefcientalgorithmshavebeenintroduced(suchastheellipticcurve[ 17 ]andquadraticsievemethod[ 20 ])whichcandeterministicallyfactorintegersnwithrunningtimee(1+o(1)p log(n)loglog(n)).Thesealgorithmsrendermuchofouranalysissuperuous,asitwilltakefarfewerstepstofactoranintegercompletelyusingthesemethodsthanbyutilizingtheKnuth-Pardoalgorithm.Inclosing,wenotethattherearesomeinterestingavenuesforfutureresearchusingtheresultsofthisthesis.Alladiin[ 1 ]showedthatforlandmrelativelyprime,wehaveequation(3-27),whichistheidentity 107

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Xp1(n)l(m)(n) n=)]TJ /F3 11.955 Tf 20.69 8.08 Td[(1 (m)where,asinsection3.3,thesumistobetakenoveralln2.ThisidentityfollowsasaconsequenceofthePrimeNumberTheoremforArithmeticProgressions,thecasek=1inLemma3.3.1,andsomeresultson(x,y).NowthatKnuthandPardohavesupplieduswithresultsonthemoregeneralfunctionsk(x,y),itwouldbeinterestingtoseewhetherfurtherresultswouldfollowifwecouldtakesimilarsumsoverthoseintegersn2suchthatpk(n)l(m).However,aswasnotedinsection3.2,thereisasignicantdifferenceinthebehaviorof1(x,y)andk(x,y)whenk2.Inparticular,1(x,y)=O(xe)]TJ /F5 7.97 Tf 6.59 0 Td[(u)decaysexponentially,whereask(x,y)sk(u)xandfork2equations(3-14)and(3-15)showthatthefunctionsk(u)donotdecaynearlyasrapidly.Itwouldbeinterestingtoseewhatconsequencesthiswouldhaveforsumssimilarto(3-27). 108

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REFERENCES [1] KrishnaswamiAlladi,DualitybetweenPrimeFactorsandanApplicationtothePrimeNumberTheoremforArithmeticProgressions,JournalofNumberTheory,vol.9(1977),p.436. [2] K.Alladi,P.Erdos,OnanAdditiveArithmeticFunction,PacicJournalofMathematics,vol.71(1977),no.2,p.275. [3] K.Alladi,P.Erdos,OntheAymptoticBehaviorofLargePrimeFactorsofIntegers,PacicJournalofMathematics,vol.82(1979),no.2,p.295-315. [4] RaymondAyoub,AnIntroductiontotheAnalyticTheoryofNumbers,MathematicalSurveys,no.10,1963. [5] CarlB.Boyer,AHistoryofMathematicsJohnWileyandSons,1968. [6] A.A.Buchstab,Anasymptoticestimationofageneralnumber-theoreticfunction,Mat.Sbornik,vol.2(1937),no.44,p.1239-1246. [7] H.M.Edwards,Riemann'sZetaFunction,DoverPublications,Inc.,2001. [8] P.Erdos,UberdieReiheP1 p,Mathematica,ZutphenB7(1938),p.1-2. [9] N.G.deBruijn,Onthenumberofpositiveintegersxandfreeofprimefactors>y,Indag.Math.,vol.13(1951),p.50-60. [10] KarlDickman,Onthefrequencyofnumberscontainingprimefactorsofacertainrelativemagnitude,Ark.Mat.,AstronomiochFysic22A(1930),10,p.1-14. [11] G.H.Hardy,S.Ramanujan,Onthenormalnumberofprimefactorsofanumbern,QuarterlyJournalofMathematics,Oxford,vol.48(1917),p.76-92. [12] KennethIreland,MichaelRosen,AClassicalIntroductiontoModernNumberTheory,Springer-Verlag,1990. [13] I.MartinIsaacs,Algebra,aGraduateCourse,Brooks/ColePublishingCompany,1994. [14] DonaldE.Knuth,LuisTrabbPardo,AnalysisofaSimpleFactorizationAlgorithm,TheoreticalComputerScience,vol.3(1976),no.3,p.321. [15] Jean-MarieDeKoninck,AndrewGranville,andFlorianLuca(editors),AnatomyofIntegers,AmericanMathematicalSociety,2008. [16] EdmundLandau,HandbuchderLehrevonderVerteilungderPrimzahlen,Leipzig:Teubner,1909.ReprintedbyChelsea,1953. [17] H.W.LenstraJr.,Factoringintegerswithellipticcurves,AnnalsofMathematics,vol.123(1987),issue3,p.649-673. 109

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[18] HansVonMangoldtZuRiemann'sAbhandlung'UeberdieAnzahlderPrimzahlenuntereinergegebenenGrosse,JournalfurdiereineundangewandtMathematik,BD.114(1895),225-305. [19] OskarPerron,ZurTheoriederDirichletschenReihen,JournalfurdiereineundangewandteMathematik,BD.134(1908),95-143. [20] CarlPomerance,Thequadraticsievefactoringalgorithm,AdvancesinCryptology,Proc.Eurocrypt'84.LectureNotesinComputerScience.Springer-Verlag,1985,p.169-182. [21] JeanJacodandPhilipProtter,ProbabilityEssentials,secondedition,Springer-Verlag,2004. [22] V.Ramaswami,Thenumberofpositiveintegersxc,andaproblemofS.S.Pillai,DukeMath.J.,vol.16(1949),p.99-109. [23] R.Rankin,Thedifferencebetweenconsecutiveprimenumbers,J.LondonMath.Soc,vol.13(1938),p.242-247. [24] L.G.Sathe,OnaproblemofHardyonthedistributionofintegershavingagivennumberofprimefactorsI-IV,J.IndianMath.Soc.,vol.17(1953),p.63-82,83-141;vol.18(1954),p.27-42,43-81. [25] BruceSchechter,Mybrainisopen:themathematicaljourneysofPaulErdos,SimonandSchuster,Inc.,1998. [26] A.Selberg,NoteonapaperbyL.G.Sathe,J.IndianMath.Soc.,vol.18(1954),p.83-87. [27] KannanSoundararajanAnasymptoticexpansionrelatedtotheDickmanfunction,RamanujanJournal,vol.29(2012)(toappear). [28] J.J.SylvesterOnTchebycheff'stheoremofthetotalityofprimenumberscom-prisedwithingivenlimits,Amer.J.Math.,vol.4(1881),p.230-247. [29] GeraldTenenbaum,Introductiontoanalyticandprobabilisticnumbertheory,CambridgeUniversityPress,1995. [30] E.C.Titchmarsh,TheTheoryoftheRiemannZeta-function,Oxford,ClarendonPress,1951. 110

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BIOGRAPHICALSKETCH ToddMolnargraduatedfromtheUniversityofDelawarein2007withaB.S.inmathematicsandeconomics.HehasbeenagraduatestudentattheUniversityofFloridasince2008. 111