Landau ' S Problems on Primes

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Journal de Theacuteorie des Nombres \noindentde BordeauxJournal 2 1 open de parenthesis Th\ ’{ e} 2009orie closing des Nombresparenthesis comma 357 hyphen 404 deLandau Bordeaux quoteright 2 1 s ( .. 2009 problems ) , .. 357on ..− primes404 par Jaacutenos PINTZ \ centerlineReacutesumeacute{Landau period ’ s ..\ Auquad congrproblems egrave s international\quad on de\quad Cambridgeprimes en 1} 9 1 2 comma Lau hyphen dau dressa la liste de quatre probl egrave mes de base sur les nombres pre hyphen \ centerline { parJournal J \ ’{ a de} nos Théoriedes PINTZ } Nombres de Bordeaux 2 1 ( 2009 ) , 357 - 404 miers period Ces probl egraveLandau mes furent ’ s caracteacuteriseacutes problems dans on son discours primes comme quotedblleft inaccessibles en l quoteright eacutetatpar Janos´ actuel dePINTZ la scienceRow 1 quotedblright Row 2 period science\ centerline probl egrave{R\ ’{ mese}sum sont\ ’{ e} . \quad Au congr $ \grave{e} $ s international de Cambridge en 1 9 1 2 , Lau − } Resum´ e´ . Au congre ` s international de Cambridge en 1 9 1 2 , Lau - les suivants : dau dressa la liste de quatre proble ` mes de base sur les nombres pre - \ centerlineopen parenthesis{dau 1 dressa closing parenthesis la liste .. de Existe quatre hyphen probl t hyphen $ \ igrave l une infiniteacute{e} $ mes de de nombres base premiers sur les nombres pre − } miers . Ces proble ` mes furent caractérisésdans son discours comme de la forme 00 \ centerlinen to the power{ miers of 2 plus .“ Ces inaccessibles1 ? probl en $ l\ ’grave étatactuel{e} de$ la messcience furentCes caractproble `\mes’{ e} sontr i s \ ’{ e} s dans son discours comme } . open parenthesis 2 closing parenthesis .. La conjecture open parenthesis binaire closing parenthesis de Gold- \ centerline { ‘‘ inaccessibles en l ’ \ ’{ eles} tat suivants actuel : de la $\ l e f t . s c i e n c e \ begin { array }{ c} ’’ \\ bach comma que chaque nombre( 1 ) Existe - t - i l une infinitéde nombres premiers de la forme . \pairend supeacuterieur{ array }Ces\ agraveright . 2 $ est problsomme de $ deux\grave nombres{e} premiers$ mes period sont } open parenthesis 3 closing parenthesis .. La conjecturen des2 + nombres 1? premiers jumeaux period \ centerlineopen parenthesis{ les 4 suivants closing parenthesis : } .. Existe hyphen t hyphen i l toujours un nombre premier entre deux carreacutes ( 2 ) La conjecture ( binaire ) de Goldbach , que chaque nombre pair supérieurà2 \ centerlineconseacutecutifs{(est 1 ? ) somme\quad deExiste deux nombres− t − premiersi l une . (3 infinit ) La\ conjecture’{ e} de des nombres nombres premiers premiers de la forme } Tous .. ces probljumeaux egrave mes. ( 4 sont ) Existeencore - ouverts t - i l toujours period .. un Le nombre travail premierpreacutesenteacute entre deux carrés \ [i n ci ˆ est{ 2 un} exposeacute+ 1 des ? \ reacutesultats] partiels .. aux probl egrave mes .. open parenthesis 2 closing parenthesis endash open parenthesis 4 closing parenthesiscons commaecutifs´ ? avec une attention particuliere concernant les reacutesultats reacutecents de \noindentD period Goldston( 2 )Tous\ commaquad cesLa C probl period conjecturee ` mes Y i sont ld i rencore ( i m binaire et ouverts de l quoteright ). de Le Goldbach travail auteur présentéi sur , les que petits ci estchaque eacutecarts un exposé nombre entre p anombres i r sup premiers\ ’{ e} r i eperiod u r $ \gravedes résultatspartiels{a} 2 $ est aux somme proble ` demes deux ( 2 nombres ) – ( 4 ) , premiers . (Abstract 3 ) \quad periodLa .. conjectureAt the 1 9 1 2 Cambridgedes nombres International premiers Congress jumeaux Lan hyphen . avec une attention particuliere concernant les résultatsrécents de D . Goldston , C . (dau 4 ) listed\quad fourExiste basic problems− t − abouti l primes toujours period un These nombre problems premier were entre deux carr \ ’{ e} s Y ı ld ı r ı m et de l ’ auteur sur les petits écartsentre nombres premiers . characterised in his speech as .. quotedblleft unattackable at the present state \ [of cons scienceRow\acute 1{ quotedblrighte} c u t i f s Row ? 2\ period] science problems were the following : Abstract . At the 1 9 1 2 Cambridge International Congress Lan - open parenthesis 1 closingdau parenthesis listed four .. basic Are problems there infinitely about primes many primes . These of problems the form were n to the power of 2 plus 1 ? \ beginopen{ parenthesisc e n t e r } 2 closing parenthesis .. The open parenthesis Binary closing parenthesis Goldbach00 Conjecture Tous \quad cescharacterised probl $ in\grave his speech{e} as$ mes “ unattackable sont encore at the ouverts present state . \ ofquadscienceLe travailT he pr \ ’{ e} sent \ ’{ e} comma that every even number . i ci est un expos \ ’{ e} des r \ ’{ e} sultats partiels \quad aux probl $ \grave{e} $ exceeding 2 can be written as the sum of twoproblems primes periodwere the following : mesopen\quad parenthesis( 2 ) 3−− closing( 4 parenthesis ) , .. The Twin Prime Conjecture period \endopen{ c e parenthesis n t e r } 4 closing parenthesis( 1 ) Are .. there Does infinitely there exist many always primes at least of the one form primen2 + between 1? neigh hyphen bouring squares ? ( 2 ) The ( Binary ) Goldbach Conjecture , that every even number \ beginAll these{ c e n problems t e r } are still openexceeding period In 2 canthe presentbe written work as a the survey sum of two primes . avecwill une be given attention about partial particuliere results in Problems(concernant 3 ) open The Twinparenthesis les Prime r \ 2 Conjecture’ closing{ e} s u parenthesisl t a . t s r \ ’ endash{ e} cents open de parenthesis 4D closing . Goldston parenthesis ,C comma . Y( with 4$ ) special\imath Does there$ exist ld always $ \imath at least$ one r prime $ \imath between$ neigh met - de l ’ auteur sur les petits \ ’{ e} c a r t s ent re nombresemphasis premiers on the recent . results of D period Goldstonbouring comma squares C period ? Y i ld i r i m and the \endauthor{ c e n on t e rsmall} gaps between primes period hline All these problems are still open . In the present work a survey will be given about \ centerlineThe author{ wasAbstractpartial supported results . by\quad in OTKA ProblemsAt grants the ( 2 No 1) – 9 period ( 4 1 ) 2 , with67676 Cambridge special comma emphasis 7273International 1 comma on the recent ERC Congress hyphen results of AdG Lan No− } period 228005 and the D . Goldston , C . Y ı ld ı r ı m and the \ centerlineBalaton program{dau period listed four basic problems about primes . These problems were } author on small gaps between primes . \ begin { c e n t e r } characterised in his speech as \quad ‘‘ unattackable at the present state o f $\ l e f t . s c i e n c e \ begin { array }{ c} ’’ \\ . \end{ array }The\ right .$ problems were the following : \end{ c e n t e r } The author was supported by OTKA grants No . 67676 , 7273 1 , ERC - AdG No . 228005 and \ centerline {(the 1 ) Balaton\quad programAre . there infinitely many primes of the form $ n ˆ{ 2 } + 1 ? $ }

\ centerline {( 2 ) \quad The ( Binary ) Goldbach Conjecture , that every even number }

\ centerline { exceeding 2 can be written as the sum of two primes . }

\ centerline {( 3 ) \quad The Twin Prime Conjecture . }

\ centerline {( 4 ) \quad Does there exist always at least one prime between neigh − }

\ centerline { bouring squares $ ? $ }

\ begin { c e n t e r } All these problems are still open . In the present work a survey will be given about partial results in Problems ( 2 ) −− ( 4 ) , with special emphasis on the recent results of D . Goldston , C . Y $ \imath $ ld $ \imath $ r $ \imath $ m and the \end{ c e n t e r }

\ centerline { author on small gaps between primes . }

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

The author was supported by OTKA grants No . 67676 , 7273 1 , ERC − AdG No . 228005 and the Balaton program . 3 58 .. Jaacutenos Pintz \noindent1 period ..3 Introduction 58 \quad J \ ’{ a} nos Pintz In his invited address at the 1 91 2 International Congress of Mathemati hyphen \ centerlinecians comma{1 held . in\quad CambridgeIntroduction comma Edmund} Landau open parenthesis 1 9 1 2 closing parenthesis gave a survey about de hyphen Invelopments his invited in the address theory of primeat the numbers 1 91 and 2 International the Riemann zeta Congresshyphen function of Mathemati period − cians , held3 in 58 CambridgeJános Pintz , Edmund Landau ( 1 9 1 2 ) gave a survey about de − Besides this he mentioned .. open parenthesis1 . without Introduction .. any further discussion closing parenthesis .. four specificvelopments in the theory of prime numbers and the Riemann zeta − f u n c t i o n . Besides this heIn mentioned his invited\ addressquad ( at without the 1 91\ 2quad Internationalany further Congress discussion of Mathemati ) \quad four specific problems about- primes cians which , held he in considered Cambridge as .. ,quotedblleft Edmund Landauunattackable ( 1 at 9 the1 2 present ) gave a survey problemsstate of science about quotedblright primes which period he The considered four problems as open\quad parenthesis‘‘ unattackable in the original order at the closing present paren- state of scienceabout ’’ de . - The velopments four problems in the theory ( in of the prime original numbers order and ) the were Riemann the follow − thesis were the followzeta hyphen - function . Besides this he mentioned ( without any further ing \noindent ing discussion ) four specific problems about primes which he considered open parenthesisas 1 closing “ unattackable parenthesis Does at the the present function u state to the of power science of 2 ” plus . The 1 represent four problems infinitely many primes for integer ( in the original order ) were the follow - ( 1values ) Does of u ? the function $ u ˆ{ 2 } + 1 $ represent infinitely many primes for integer values of $uing ?$ open parenthesis 2( closing 1 ) Does parenthesis the function Does the equalityu2 + 1 represent m = p plus infinitely p to the power many of primesprime have for for any even m greater 2 a solution ? \ centerline {(2)Doestheinteger values of equalityu? $m = p + pˆ{\prime }$ have for any even open parenthesis 3 closing parenthesis Does the equality0 2 = p minus p to the power of prime have infinitely $ m > 2$( a 2 solution ) Does the equality$?$ }m = p + p have for any even m > 2 a solution ? many solutions in primes ? ( 3 ) Does the equality 2 = p − p0 have infinitely many solutions in open parenthesis 4 closing parenthesis Does there exist at least one prime between n to the power of 2 and \ hspace ∗{\ f i l lprimes}( 3 )? Does the equality $2 = p − p ˆ{\prime }$ have infinitely many solutions in primes open parenthesis n plus 1 closing parenthesis to the power of 2 for any 2 2 $ ? $ ( 4 ) Does there exist at least one prime between n and (n + 1) for positive integerany n ? positive integer n? In the present work we will begin with some historical remarks referring ( 4 ) Does thereIn exist the present at least work one we will prime begin between with some $ n historical ˆ{ 2 }$ remarks and $referring ( n + to these problemsto including these problems the few results including known the in 1few 91 2results about those known prob in hyphen 1 91 2 about those 1lems ) ˆ and{ 2 analyse}$ f the o r connections any between the four problems period After this we positive integerprob - $ lems n and ? $ analyse the connections between the four problems . After will give a surveythis of wethe most will give important a survey results of of the the most past nearlyimportant 100 years results period of the past nearly We will discuss the results in connection with Problems open parenthesis 2 closing parenthesis endash open In the present100 work years we . will We will begin discuss with the some results historical in connection remarks with referring Problems ( 2 ) parenthesis 4 closing– ( parenthesis 4 ) in more in more de - de hyphen totail these and briefly problems those connected including with the Problem few 1 results open parenthesis known Section in 1 9119 closing 2 about parenthesis those comma prob − with lems and analysetail and the briefly connections those connected between with the Problem four problems 1 ( Section . 19 After ) , with this special we special emphasis on recent developments concerning various approximations of the willemphasis give on a recent survey developments of themost concerning important various approximations results of ofthe the past nearly 100 years . We will discussGoldbach the results and Twin in Prime connection Problems with . Problems ( 2 ) −− ( 4 ) in more de − Goldbach and2 Twin . Prime History Problems of periodthe problems and related results before 19 1 2 2 period .. History of the problems and related results before 19 1 2 \noindent tail andWhereas briefly the those conjecture connected that there with are Problem infinitely 1 many ( Section twin primes 19 ) may , with special Whereas the conjectureoriginate that from there the are time infinitely of Euclid many twin and primes Eratosthenes may , it seems that it ap emphasisoriginate from on recent the time ofdevelopments Euclid and Eratosthenes concerning comma various it seems approximations that it ap hyphen of the Goldbach and- Twin peared Prime first Problems in print in . the work of de Polignac ( 1 849 ) , although in peared first in printa more in the general work of form de Polignac already open . We parenthesis know much 1 849 closingmore about parenthesis the origincomma of although in a more \ centerline {2Goldbach . \quad History ’ s Conjecture of the , however problems there and are related some interesting results (before and partly 19 1 2 } general form alreadynot well period known We know ) facts much to more mention about the concerning origin of Goldbach it s origin quoteright . In a s letter to Conjecture comma however there are some interesting open parenthesis and partly not well known closing Whereas the conjectureEuler , written that June there 7 ,are 1 742 infinitely , Goldbach many formulated twin primes his conjecture may in parenthesis two different forms . The first one asserted that originatefacts to mention from concerning the time it s of origin Euclid period and In a Eratosthenes letter to Euler comma , it written seems June that 7 comma it ap − peared1 742 comma first Goldbach in print formulated in the his work conjecture of de in twoPolignac different ( forms 1 849 period ) , The although first in a more general form alreadya . We know much moreX about the origin of Goldbach ’ s one asserted that if number Ni sthe P ofof 2primes , thenitprimes Conjecture , howevercan therebewritten are someas interestinga arbitrarily ( and partlymany not well known. ) Equation: open parenthesis 2 period 1 closing parenthesis .. if can to the power of a number sub(2 be.1) written Nfacts i as s the to a mention sum sum of concerning of sub arbitrarily it s2 primes origin many . In comma a letter then it primes to Euler sub period , written June 7 , 1In 742 these , formulations Goldbach formulated we have to keep his in mind conjecture that in his in t ime two the different number forms . The first one assertedIn that these formulations we have to keep in mind that in his t ime the num- one was consideredber toone be a primewas considered period The second to be formulation a prime . was The interestingly second formulation was found on the margin of the same letter period This states that \ begin { a l i g n ∗}interestingly found on the margin of the same letter . This states that open parenthesis( 2 period . 2 ) 2 closing parenthesis \ tagevery∗{$ number ( 2 greater . than 1 2 can ) $ be} writteni f { ascan the} sumˆ{ a of} threenumber primes period{ be written } N i { as } s the { a every\sum number}\sum greatero f than{ o f 2} can{ bearbitrarily written as the} 2 sum primesof three primes{ many . } , Euler pointed outEuler in his pointed answer of out June in 30 his that answer the first of formulation June 30 of that the first formulation thenGoldbach i t quoteright{ primes s} Conjecture{ . } follows from the conjecture that every even number \end{ a l i g n ∗} of Goldbach ’ s Conjecture follows from the conjecture that every even number \noindent In these formulations we have to keep in mind that in his t ime the number one was considered to be a prime . The second formulation was interestingly found on the margin of the same letter . This states that ( 2 . 2 )

\ centerline { every number greater than 2 can be written as the sum of three primes . }

\noindent Euler pointed out in his answer of June 30 that the first formulation of Goldbach ’ s Conjecture follows from the conjecture that every even number Landau quoteright s problems on primes .. 359 \ hspacecan be∗{\ writtenf i l l as}Landau the sum of ’ twos problems primes period on As primes Euler remarks\quad in359 his letter this latter conjecture was communicated to him earlier orally by Goldbach \noindenthimself periodcan So be it writteni s really justified as the to sum attribute of two the Binary primes Goldbach . As Euler Conjecture remarks in his letter thisto Goldbach latter period conjecture The correspondence was communicated of Euler and to Goldbach him earlier appeared orally already by Goldbach himself . So it i s really justified to attribute the Binary Goldbach Conjecture in 1 843 open parenthesis cf period Fuss open parenthesis 1 843 closingLandau parenthesis ’ s problems closing on primes parenthesis359 period Whileto Goldbach the secondformulation . The correspondence open parenthesis of 2 period Euler 2 closing and Goldbach parenthesis appeared of Goldbach already in 1 843 ( cfcan . Fuss be written ( 1 843 as the ) ) sum . While of two the primes second . As formulation Euler remarks ( in 2 his . 2 letter ) of Goldbach i s clearly equivalentthis latter to the conjecture usual Binary was Goldbach communicated Conjecture to comma him thisearlier i s not orally by Goldbach iobvious s clearly with the equivalent first formulation to the open usual parenthesis Binary 2 period Goldbach 1 closing Conjecture parenthesis period , this However i s not comma obvious withhimself the first . So formulationit i s really justified ( 2 . to 1 attribute ) . However the Binary , surprisingly Goldbach Conjec- , this i s really surprisingly commature this i to s really Goldbach . The correspondence of Euler and Goldbach appeared thethe case case period . Let Let us us suppose namely namely that that an even an number even 2number k i s the sum$ 2 of two k $ i s the sum of two primes . Thenalready $ 2 in k 1 $ 843 i ( s cf also . Fuss a sum ( 1 843 of )three ) . While primes the . second One of formulation them has ( to be even , primes period Then2 . 2 2 )k iof s alsoGoldbach a sum of i three s clearly primes equivalent period One to of them the usual has to Binary be even commaGoldbach soso 2 $ k 2 minus k 2 i s− also2 a sum $ ofi twos also primes a period sum of Continuing two primes the procedure . Continuing we see that the procedure we see that every even integerConjecture below , this $ 2 i s not k $ obvious i s the with sum the of first two formulation primes . Since ( 2 . all1 ) . numbers of every even integerHowever below 2 , k surprisingly i s the sum of , two this primes i s really period the Since case all numbers . Let us of suppose namely thethe form form 2 p $ are 2 sums p $of two are primes sums comma of two the primesusual Binary , the Goldbach usual Conjecture Binary Goldbach Conjecture follows fromthat the an existence even number of arbitrarily2k i s the sum large of two primes primes . . Then 2k i s also a follows from thesum existence of three of arbitrarily primes . large One primes of them period has to be even , so 2k −2 i s also a sum Waring statedof Goldbach two primes quoteright . Continuing s Problem the in 1 procedure 770 open parenthesis we see that Waring every open even parenthesis integer 1 770 closingWaring parenthesis stated Goldbachclosing parenthesis ’ s Problem and added in 1 770 ( Waring ( 1 770 ) ) and added that every oddbelow number2k i i s thes either sum of a two prime primes or i. Since s a sum all numbers of three of primes the form . 2p that every oddare number sums i s ofeither two a primesprime or , i thes a sum usual of three Binary primes Goldbach period Conjecture follows It is much less well known that Descartes formulated a related but not \ hspace ∗{\ f i l lfrom} It isthe much existence less of well arbitrarily known large that primes Descartes . formulated a related but not equivalent conjectureWaring much earlier stated than Goldbach Goldbach ’ s since Problem he died in already 1 770 in ( Waring ( 1 770 ) ) and 1 650 period According to this every even number i s the sum of at most 3 primes period \noindent equivalentadded that conjecture every odd much number earlier i s either than Goldbach a prime or since i s a hesum died of three already in It i s unclear whyprimes he formulated . this only for even integers comma but it is very 1easy 650 to . show According that this toi s equivalent this every to the even following number comma i mores the natural sum version of at most 3 primes . It i s unclear whyIt he is formulatedmuch less well this known only that for Descartes even integers formulated , but a related it is but very open parenthesisnot where we do not consider one to be prime closing parenthesis : easyDescartes to show quoteright that Conjecture this i s period equivalent Every integer to greaterthe following than one can , be more written natural as version ( where we doequivalent not consider conjecture one to much be primeearlier )than : Goldbach since he died already the sum of at mostin 1 3 primes 650 . period According to this every even number i s the sum of at most Let us introduce3 the primes following . It i s unclear why he formulated this only for even integers , \noindentDefinitionDescartes period .... An ’ even Conjecture number is called . Every a Goldbach integer number greater open parenthesis than one their can set be will written as the sum of atbut most it is 3 very primes easy . to show that this i s equivalent to the following , more be denoted by Gnatural further version on closing ( parenthesis where we if do it notcan be consider writtenone as theto sum be of prime two primes ) : period Then it i s easy to see that the Descartes quoteright Conjecture i s equivalent to \ centerline { LetDescartes us introduce ’ Conjecture the following . Every} integer greater than one can be written open parenthesisas 2 period the sum 3 closing of at parenthesis most 3 primes .. If N greater . 2 is even comma then N in G or N minus 2 in G period \noindent Definition . \ h f i l l AnLet even us introduce number is the called following a Goldbach number ( their set will It i s worth remarkingDefinition that open . parenthesisAn even as number one canis easily called derive a Goldbach from open parenthesis number ( 2 their period set 3 closing parenthesis closing parenthesis Descartes quoteright \noindent be denotedwill by $ G $ further on ) if it can be written as the sum of two primes . Conjecture i s equivalentbe denoted to a by strongerG further form onof it ) comma if it can namely be written as the sum of two primes Every integer greater than 1 can be written as a sum of three \ centerline {Then. it i s easy to see that the Descartes ’ Conjecture i s equivalent to } open parenthesisThen 2 period it i 4 s closing easy to parenthesis see that .. the primes Descartes comma where ’ Conjecture the third summandi s equivalent comma to if it exists comma can be chosen( 2 as . 3 ) If N > 2 is even , then N ∈ G or N − 2 ∈ G. \noindent2 comma 3( or 2 5 . period 3 ) \quad I f $ N > 2$ is even , then $N \ in G $ or $ N − 2 \ inIt i sG worth . $ remarking that ( as one can easily derive from ( 2 . 3 ) ) Although .. DescartesDescartes quoteright ’ Conjecture .. Conjecture i s ..equivalent i s .. not .. to equivalent a stronger .. to form .. Goldbach of it , quoteright namely s comma .. the Every integer greater than 1 can be written as a sum of three ( 2 . 4 ) Itquestion i s worth arises remarking : .. could Euler that or Goldbach ( as one have can been easily .. aware derive of Descartes from quoteright ( 2 . 3 ) ) Descartes ’ Conjecture iprimes s equivalent , where the to a third stronger summand form , if of it exists it , namely , can be chosen as Conjecture ? .. Theoretically yes comma .. since some2 , 3 copies or 5 of . his notes and manu hyphen scripts circulated inAlthough Europe period Descartes However comma ’ Conjecture the above two hyphen i s not line long equivalent conjecture to \noindentwas not includedEvery in integer his collected greater works which than appeared 1 can be in 1 written 70 1 in Amster as a hyphen sum of three ( 2 . 4 ) \quadGoldbachprimes ’ s , , where the the question third arises summand : , could if it Euler exists or Goldbach , can be have chosen as dam period It isbeen only contained aware in of the Descartes edition of ’ Descartes Conjecture open parenthesis? Theoretically 1 908 closing yes parenthesis , since comma under Opuscula \ centerline {2some , 3 or copies 5 . of} his notes and manu - scripts circulated in Europe . However Posthuma comma, the Excerpta above Mathematica two - line open long parenthesis conjecture Vol was period not 1 0 included comma p periodin his 298 collected closing parenthesis period Although \quadworksDescartes which appeared ’ \quad inConjecture 1 70 1 in Amster\quad -i dam s \quad . It isnot only\quad containede q u i v in a l e n t \quad to \quad Goldbach ’ s , \quad the question arisesthe edition: \quad ofcould Descartes Euler ( 1 or 908 Goldbach ) , under have Opuscula been \quad aware of Descartes ’ Conjecture $Posthuma ? $ \quad , ExcerptaTheoretically Mathematica yes ,( Vol\quad . 1since 0 , p . some 298 ) copies. of his notes and manu − scripts circulated in Europe . However , the above two − line long conjecture was not included in his collected works which appeared in 1 70 1 in Amster − dam . It is only contained in the edition of Descartes ( 1 908 ) , under Opuscula

\noindent Posthuma , Excerpta Mathematica ( Vol . 1 0 , p . 298 ) . 3 60 J´anos Pintz Apart from a numerical veriﬁcation by Desboves ( 1855 ) up to 1 0 , 0 and Ripert ( 1903 ) up to 50 , 0 actually no result was proved before Landau ’ s lecture . We have to mention however the conjectural asymptotic formula of J . J . Sylvester ( 1 871 ) for the number P2(n) of representation of an even n as the sum of two primes :

) −γ n Y 1 P2(n) ∼ 4e C0 2n (1 + p , (2.5) log −2 p|n p > 2 where C0 i s the so - called twin prime constant ,

Y 1 ) C = (1 − = 0.66016... . (2.6) 0 (p − 1)2 p>2

Here and later p( as further on p0, p00, pi) will always denote primes , P will denote the set of all primes . It was proved later by Hardy and Littlewood ( 1923 ) that this formula −γ i s deﬁnitely not correct . They made the same conjecture with 4e C0 replaced by 2C0. By now , we know that the analogue of ( 2 . 5 ) with 2C0 is true for almost all even numbers . The only area where some non - trivial results existed before Landau ’ s lecture was Problem 4 about large gaps between primes . Bertrand ( 1 845 ) stated the assertion – called Bertrand ’ s Postulate – that there i s always a prime between n and 2n. The same assertion – also without any proof – appeared about 1 0 years earlier in one of the unpublished manuscripts of Euler ( see Narkiewicz ( 2000 ) , p . 104 ) . Bertrand ’ s Postulate was proven already 5 years later by Cˇ ebyˇsev( 1 850 ) . He used elementary tools to show

x x 0.92129 < π(x) < 1.1555 forx > x , (2.7) log x log x 0 where π(x) denotes the number of primes not exceeding x. Further on we will use the notation

. dn = pn + 1 − pn (2.8)

Cˇ ebyˇsev’ s proof implies

6 d < ( + ε)pn forn > n (ε). (2.9) n 5 0 The next step

dn = o(pn) (2.10) 3 60 .. Jaacutenos Pintz \noindentApart from3 a 60 numerical\quad veriﬁcationJ \ ’{ a} nos by PintzDesboves open parenthesis 1855 closing parenthesis up to 1 0 comma 0 and \ hspaceRipert∗{\ openf iparenthesis l l }Apart 1903from closing a numerical parenthesis verification up to 50 comma by 0 Desbovesactually no (result 1855 was ) proved up to before 1 0 , 0 and Landau quoteright s \noindent Ripert ( 1903 ) up to 50 , 0 actually no result was proved before Landau ’ s lecture period Wewas have a consequence to mention however of the the Prime conjectural Number asymptotic Theorem formula ( PNT ) lectureof J period . We J period have Sylvester to mention open however parenthesis the 1 871 conjectural closing parenthesis asymptotic for the number formula P sub 2 open parenthesisof J . J n closing. Sylvester parenthesis ( 1 of 871representation ) for the of an number even $ P { 2 } ( n ) $ of representation of an even $ n $ as the sum of two primes : n as the sum of two primes : x Equation: open parenthesis 2 period 5 closing parenthesisx .. P subZ 2dt open parenthesis n closing parenthesis \ begin { a l i g n ∗} π(x) ∼ ∼ lix := , (2.11) thicksim 4 e to the power of minus gamma C sub 0 sublog logx to the powerlog oft hline to the power of n 2 n product p\ tag bar∗{ n parenleftbigg$ ( 2 1 . plus 5 p 1 divided ) $} P by minus{ 2 } 2 to( the npower ) of parenrightbigg\0 sim 4 comma e ˆ{ p− greater \gamma 2 } C { 0 ˆ{\ r u l e {3em}{0.4 pt }} {\ log }}ˆ{ n } 2where n C\prod sub 0 i s{ thep so hyphen\mid calledn } twin( prime 1 constant + p comma\ f r a c { 1 }{ − 2 }ˆ{ ) } , \\ p > Equation:2 open parenthesis 2 period 6 closing parenthesis .. C sub 0 = product p greater 2 parenleftbigg 1 minus\end{ 1a divided l i g n ∗} by open parenthesis p minus 1 closing parenthesis to the power of 2 to the power of parenrightbigg = 0 period 660 1 6 period period period period \noindentHere and laterwhere p open $ C parenthesis{ 0 }$ as i further s the on so p to− thecalled power twin of prime prime comma constant p to the power, of prime prime comma p i closing parenthesis will always denote primes comma P will \ begindenote{ a the l i g setn ∗} of all primes period \ tagIt∗{ was$ proved ( 2 later . by Hardy 6 )and $} LittlewoodC { 0 open} = parenthesis\prod 1923{ p closing> parenthesis2 } ( that 1 this− formula \ f r a i c s{ 1 }{ ( p deﬁnitely− 1 not ) correct ˆ{ 2 period}}ˆ{ They) } made=0. the same conjecture 66016. with 4 e to the power . of. minus . gamma C sub 0 replaced\end{ a l i g n ∗} by 2 C sub 0 period By now comma we know that the analogue of open parenthesis 2 period 5 closing parenthesis\noindent withHere 2 C and sub 0 later is true for $p ( $ as further on $p ˆ{\prime } , p ˆ{\prime \primealmost} all even, numbersp i period )$ will always denote primes $ , P$ will denoteThe only the area set where of some all non primes hyphen . trivial results existed before Landau quoteright s lecture was Problem 4 about large gaps between primes period Bertrand open parenthesis 1 845 closing parenthesisIt was proved later by Hardy and Littlewood ( 1923 ) that this formula i s definitelystated the assertion not correct endash called . They Bertrand made quoteright the same s Postulate conjecture endash with that there $ 4 i s always e ˆ{ − \gamma } C a{ prime0 }$ between r e p l a n c and e d 2 n period The same assertion endash also without any proof byendash $ 2 appeared C { about0 } 1. 0 years $ Bynow earlier in , one we of know the unpublished that the manuscripts analogue of ( 2 . 5 ) with $ 2 C of{ Euler0 }$ open i s parenthesis true f o r see Narkiewicz open parenthesis 2000 closing parenthesis comma p period 104 closingalmost parenthesis all even period numbers Bertrand . quoteright s Postulate was proven already 5 years later by Ccaron ebyscaronev open parenthesis 1 850 closing parenthesis period He used elementaryThe only tools area to where show some non − trivial results existed before Landau ’ s lectureEquation: was open Problem parenthesis 4 2about period large 7 closing gaps parenthesis between .. 0 primesperiod 92 . 1 29 Bertrand x divided ( by 1 log 845 x less ) pi open parenthesisstated the x closing assertion parenthesis−− lesscalled 1 period Bertrand 1 555 x divided ’ s Postulate by log x for x−− greaterthat x subthere 0 comma i s always awhere prime pi betweenopen parenthesis $ n x $ closing and parenthesis $ 2 n denotes . $ the The number same of primes assertion not exceeding−− also x period without Further any proof on−− weappeared about 1 0 years earlier in one of the unpublished manuscripts ofwill Euler use the ( notation see Narkiewicz ( 2000 ) , p . 104 ) . Bertrand ’ s Postulate was proven alreadyEquation: 5 open years parenthesis later 2by period $ \ 8check closing{C parenthesis} $ eby\ ..v d{ s sub}ev n = ( p 1 n 850 plus 1) minus . He p used n to the elementary power of tools to show period \ beginCcaron{ a ebyscaronevl i g n ∗} quoteright s proof implies \ tagEquation:∗{$ ( open 2 parenthesis . 7 2 ) period $} 0 9 closing . parenthesis92 1 .. 29 d sub\ f r n a lessc { x parenleftbigg}{\ log 6 dividedx } < by 5\ pluspi epsilon( x parenrightbigg ) < 1 p n for . n greater 1 555 n sub\ 0f r open a c { parenthesisx }{\ log epsilonx closing} f o parenthesisr x > periodx { 0 } , \endThe{ a next l i g n step∗} Equation: open parenthesis 2 period 1 0 closing parenthesis .. d sub n = o open parenthesis p n closing parenthesis\noindent where $ \ pi ( x ) $ denotes the number of primes not exceeding $xwas a . consequence $ Further of the onwe Prime Number Theorem open parenthesis PNT closing parenthesis willx Equation: use the open notation parenthesis 2 period 1 1 closing parenthesis .. pi open parenthesis x closing parenthesis thicksim x divided by log x thicksim li x : = integral 0 dt divided by log t comma \ begin { a l i g n ∗} \ tag ∗{$ ( 2 . 8 ) $} d { n } = p n + 1 − p n ˆ{ . } \end{ a l i g n ∗}

\noindent $ \check{C} $ eby\v{ s }ev ’ s proof implies

\ begin { a l i g n ∗} \ tag ∗{$ ( 2 . 9 ) $} d { n } < ( \ f r a c { 6 }{ 5 } + \ varepsilon ) p n f o r n > n { 0 } ( \ varepsilon ). \end{ a l i g n ∗}

\noindent The next step

\ begin { a l i g n ∗} \ tag ∗{$ ( 2 . 1 0 ) $} d { n } = o ( p n ) \end{ a l i g n ∗}

\noindent was a consequence of the Prime Number Theorem ( PNT )

\ begin { a l i g n ∗} x \\\ tag ∗{$ ( 2 . 1 1 ) $}\ pi ( x ) \sim \ f r a c { x }{\ log x } \sim l i x : = \ int { 0 }\ f r a c { dt }{\ log t } , \end{ a l i g n ∗} Landau quoteright s problems on primes .. 36 1 \ hspaceshown∗{\ simultaneouslyf i l l }Landau using ’ different s problems .. arguments on primes by J period\quad .. Hadamard36 1 .. open parenthesis 1 896 closing parenthesis \noindentand de la Valleacuteeshown simultaneously Poussin open parenthesis using different1896 closing parenthesis\quad arguments period by J . \quad Hadamard \quad ( 1 896 ) andThe de last la step Vall before\ ’{ 1e 9}e 1 2 Poussin comma the ( inequality 1896 ) . Equation: open parenthesis 2 period 1 2 closing parenthesis .. d sub n less p n exponent open parenthesis \ centerline {The last step before 1 9 1 2 , the inequalityLandau ’ s problems} on primes 36 1 minus c radicalbig-lineshown of p simultaneously n sub log closing using parenthesis different arguments by J . Hadamard ( was a consequence of the Prime Number Theorem with remainder term \ begin { a l i g n ∗}1 896 ) and de la ValléePoussin ( 1896 ) . Equation: open parenthesis 2 periodThe 1last 3 closing step before parenthesis 1 9 .. 1 pi2 , open the parenthesis inequality x closing parenthesis = li x\ tag plus∗{ O$ open ( parenthesis 2 . x 1 exponent 2 parenleftbig ) $} d { subn } minus< c radicalbig-linep n \exp of x sub( log− parenrightbigc \ sqrt closing{ p parenthesisn } {\ log comma} ) \end{ a l i g n ∗} √ proved by de la Valleacutee Poussin open parenthesisdn < pn 1exp( 899− closingc pnlog parenthesis) period (2.12) Finally we mention that H period .. Brocard open parenthesis 1 897 closing parenthesis .. gave an incorrect proof\noindent of waswas a consequence a consequence of of the the Prime Prime Number Number Theorem Theorem with with remainder remainder term the closely related conjecture that there exists a prime between any two \ beginconsecutive{ a l i g n triangular∗} numbers period This shows that Problem 4 of Landau was \ tag ∗{$ ( 2 . 1 3 ) $}\ pi (x)=lix+O(x√ \exp examined before 1 91 2 comma although inπ( ax) slightly = lix + differentO(x exp( form−c periodxlog)), (2.13) ( As{ −we } mentionedc \ sqrt already{ x comma} {\ log the Twin} )), Prime Conjecture appeared in print \endalready{ a l i g the n ∗} firstproved time in by a de more la generalValléePoussin form comma ( 1 due899 to ) . de Polignac open parenthesis 1 849 closing parenthesis : Finally we mention that H . Brocard ( 1 897 ) gave an incorrect \noindentEvery evenproved numberproof bycan of debe the written la closely Vall in\ an’ related{ infinie}e hyphen Poussin conjecture ( 1 that 899 there ) . exists a prime between open parenthesisany 2 period two consecutive14 closing parenthesis triangular .. tude numbers of ways . as This the difference shows that of two Problem consecutive 4 of Finallyprimes periodwe mentionLandau that was H examined . \quad beforeBrocard 1 91 ( 2 1 , 897although ) \quad in a slightlygave an different incorrect form proof of theKronecker closely open related. parenthesis conjecture 1 901 closing that parenthesis there mentioned exists athe prime same conjecture between open any parenthesis two with referenceconsecutive to an triangularAs we mentioned numbers already. This shows , the Twin that Prime Problem Conjecture 4 of Landau appeared was in examinedunnamed writer beforeprint closing 1 already91 parenthesis 2 , the although in first a weaker time in formin a a slightly asmore general different form , dueform to . de Polignac ( Equation: open1 parenthesis 849 ) : 2 period 1 5 closing parenthesis .. Every finitude to the power of even of number waysAs we as mentionedthe to theEvery power already of even can number difference , the Twin can to the be Prime power written Conjectureof be in expressed an infini appeared of - in ( 2 2 . primes 14 in ) sub print tude period of toways the power ofalready an in hyphen the firstas the time difference in a of more two consecutivegeneral form , due to de Polignac ( 1 849 ) : Maillet open parenthesis 1 905 closing parenthesis commentedprimes . on de Polignac quoteright s conjecture that \noindentopen parenthesisEvery 2 evenKronecker period number 1 6 closing ( 1 can 901 parenthesis )be mentioned written .. Every the in even an same number infini conjecture is− the difference ( with referenceof two primes to period (When 2 . 14 the )even\quadan numbertude is 2 orof 4 ways then open as the parenthesis difference 2 period of 14 closing two consecutive parenthesis and open parenthesis 2 period 1 5 closingunnamed parenthesis writer are equivalent ) in a weaker comma form as \ centerlineotherwise open{ primes parenthesis . } 2 period 1 5 closing parenthesis is weaker than open parenthesis 2 period 14 closing parenthesis comma while open parenthesis 2 period 16 closing parenthesis i s weaker than open parenthesis 2 \ hspace ∗{\ f i l l } Kroneckereven ( 1 901 ) mentionedcan the samebeexpressed conjectureanin− ( with reference to an period 1 5 closing parenthesisEvery periodfinitudeofnumberwaysasthedifference ofin2primes. (2.15) The form open parenthesis 2 period 1 6 closing parenthesis is trivial for every concrete small even number and\noindent today unnamedMaillet writer ( ) 1 905in a ) weakercommented form on as de Polignac ’ s conjecture that we know its truth( 2 for . 1 almost 6 ) all Every even numbers even number period In is strong the difference contrast to of this two primes . \ beginwe do{ nota l i g know n ∗} whetherWhen there ithe s any even number number for which is 2 openor 4 parenthesisthen ( 2 . 2 14 period ) and 1 5 ( closing 2 . 1 5 parenthesis ) are or open\ tag parenthesis∗{$ ( 2 2equivalent period . 14 1 closing , 5 parenthesis ) $} Every i s { f i nitude }ˆ{ even } o f number { ways astrue the period}ˆ{ canotherwise} di ( ff 2 . erence 1 5 ) is ˆ weaker{ be than expressed ( 2 . 14} ) ,o while f in ( 2 .{ 162 ) i sprimes weaker}ˆ{ an in The− Goldbach } { . } andthan Twin ( 2 Prime . 1 5 Conjecture) . were mentioned in the cele hyphen \endbrated{ a l i gaddress n ∗} ofThe Hilbert form at ( the 2 . International 1 6 ) is trivial Congress for everyof Mathematicians concrete small even number and in Paris commatoday 1900 open we parenthesis know its see truth Hilbert for open almost parenthesis all even 1935 numbers closing parenthesis . In strong closing contrast parenthesis period\ centerline In his Problem{ Mailletto this No period ( we 1 do 905 8 he not mentioned) know commented whether them on there de Polignac i s any number ’ s conjecture for which ( that 2 . 1} 5 together with the) or Riemann ( 2 . 14 Hypothesis ) i s true comma . using the following words : \noindentquotedblleft( After 2 . 1aThe comprehensive 6 Goldbach) \quad Every discussionand Twin even of Prime Riemann number Conjecture quoteright is the were difference s prime mentioned number of formula in two the primes cele - . we might be somebrated day in address the position of Hilbert to give at a rigorous the International answer on Gold Congress hyphen of Mathematicians \ hspacebach quoteright∗{\ f i l lin} sWhenProblem Paris the , 1900 comma even ( see whether number Hilbert every is ( 1935 even2 or )number ) 4 . then In hiscan ( Problem be 2 expressed . 14 No ) asand . 8the he ( sum mentioned 2 . 1 5 ) are equivalent , of two primes commathem further together on the with problem the Riemann whether there Hypothesis exist infinitely , using many the following words \noindent otherwise: ( 2 . 1 5 ) is weaker than ( 2 . 14 ) , while ( 2 . 16 ) i s weaker than ( 2 . 1 5 ) . “ After a comprehensive discussion of Riemann ’ s prime number for- \noindent Themula form we( 2 might . 1 6 be ) some is trivial day in the for position every toconcrete give a rigorous small even answer number on and today we know its truthGold - for bach almost ’ s Problem all even , whether numbers every . In even strong number contrast can be toexpressed this we do not knowas whetherthe sum there of two i primes s any , number further for on the which problem ( 2 . whether 1 5 ) orthere ( 2 exist . 14 ) i s true . infinitely many

The Goldbach and Twin Prime Conjecture were mentioned in the cele − brated address of Hilbert at the International Congress of Mathematicians in Paris , 1900 ( see Hilbert ( 1935 ) ) . In his Problem No . 8 he mentioned them together with the Riemann Hypothesis , using the following words :

‘‘ After a comprehensive discussion of Riemann ’ s prime number formula we might be some day in the position to give a rigorous answer on Gold − bach ’ s Problem , whether every even number can be expressed as the sum of two primes , further on the problem whether there exist infinitely many 3 62 .. Jaacutenos Pintz \noindentprimes with3 difference 62 \quad 2 orJ \ on’{ thea} nos more Pintz general problem whether the dio hyphen primesphantine with equation difference 2 or on the more general problem whether the dio − Equation: open parenthesis 2 period 1 7 closing parenthesis .. ax plus by plus c = 0 \noindenti s always solvablephantine in primes equation x comma y if the coefficients a comma b comma c are given pairwise relatively prime3 integers 62 János periodPintz quotedblrightprimes with difference 2 or on the more general problem \ beginThere{ a are l i g close n ∗} t ies between Landau quoteright s problems period These connections de hyphen \ tag ∗{$ ( 2whether . 1 the 7 dio - ) $} ax + by + c = 0 pend strongly uponphantine which equation formulation of the Conjectures open parenthesis 2 period 14 closing parenthesis endash\end{ a open l i g n parenthesis∗} 2 period 1 6 closing parenthesis we consider period The first two are really generalizations of the Twin Prime Con hyphen \noindent i s always solvable in primes $ x , y $ if the coefficients $ a j ecture comma the third one comma open parenthesisax + 2by period+ c = 1 0 6 closing parenthesis comma (2 i s.17) obviously trivial, b if the , difference c$ i are s two given period pairwise As the cited lines ofi sHilbert always quoteright solvable s in lecture primes alsox, indicate y if the comma coefficients both Goldbacha, b, c are quoteright given pairwise s Conjec hyphen \noindentture and therelatively Twinrelatively Prime prime Conjecture prime integers integers are special . . ” cases ’’ of linear equations of type open parenthesisThere 2 are period close 1 7t iesclosing between parenthesis Landau for primes ’ s problems period Using . These the connections formulation of open parenthesisThere are 2 period closede 1 t - 6 closing iespend between strongly parenthesis Landau upon there which is ’ really s problems formulation a . of These the Conjectures connections ( 2 de .− 14 ) pendvery strongstrongly similarity– upon ( 2 .between 1which 6 ) wethe formulation equations consider p . plus The of p the firstto the Conjectures two power are of really prime ( = generalizations 2 N .and 14 p minus) −− p( of to 2 the the . 1power 6 ) we ofconsider prime = N . TheTwin first Prime two Con are - really j ecture generalizations , the third one , of ( 2 the . 1 Twin 6 ) , Prime i s obviously Con − jfor ecture even values , thetrivial of N third period if the one In difference fact , comma( 2 . i most s1 two 6 of ) . the As, i results the s obviously cited for Goldbach lines of trivial Hilbert quoteright if’ s s lecture theConjecture difference also i s two . As theare citedtransferable linesindicate to the of other Hilbert , both equation Goldbach ’ s comma lecture ’ toos Conjec period also - indicate ture and the , both Twin Goldbach Prime Conjecture ’ s Conjec − tureOn the and other the handare Twin comma special Prime the cases ConjectureTwin of Prime linear Conjecture equations are special is also of typeconnected cases ( 2 of . with 1 linear 7 ) for primes equations . Using ofProblem type (4 period 2 .the 1 .. 7formulation The ) former for primes one of refers ( 2 . . to Using 1 the 6 ) smallest there the is formulation possible really gapsa between of ( 2 . 1 6 ) there is really a consecutive primesvery comma strong the similarity latter one between to the largest the possible equations gapsp period+ p0 = N and p − p0 = N for \noindentFinally commaveryeven .. strong the values .. Twin similarity of PrimeN. In Conjecture fact between , most and Problemtheof the equations results .. 1 .. admit for Goldbach $ .. p a common + ’ s p Conjecture ˆ{\prime } = N $generalization and $ p commaare− transferable formulatedp ˆ{\prime first to the by} A other period= equation N Schinzel $ open , too parenthesis . Schinzel comma Sierpinacuteski 1for 958 closing even parenthesis valuesOn of : the $other N . hand $ In , the fact Twin , mostPrime of Conjecture the results is also for connected Goldbach with ’ s Conjecture areif f 1 transferable comma periodProblem period to the 4 period . other The comma former equation f k .. oneare irreducible refers , too to . the polynomials smallest in possible Z open square gaps bracket between X closing square bracket .. andconsecutive their product primes .. does , the latter one to the largest possible gaps . Onnot the have other a fixed hand factorFinally , commathe , Twin then the Prime for Twin infinitely Conjecture Prime many Conjecture integers is also n all and connected values Problem f i open with parenthesis 1 admit n closing parenthesisProblem 4 . \aquad commonThe former generalization one refers , formulated to the firstsmallest by A possible . Schinzel gaps ( Schinzel between , consecutive primes , the latter one to the largest possible gaps . are prime periodSierpiński1 .. Bateman 958 and ) Horn : if ..f open1, ..., fkparenthesisare irreducible 1 962 closing polynomials parenthesis .. in formulatedZ[X] a quantitative form ofand their product does not have a fixed factor , then for infinitely F i nit a period l l y , ..\quad Themany specialthe integers\ casequad f i openTwinn all parenthesis Prime values Conjecturefi x( closingn) are parenthesis prime and . Problem = Bateman x plus\quad h sub and i1 comma\ Hornquad hadmit sub ( i in\ Zquad a common ofgeneralization Schinzel quoteright1 962 ,s conjecture formulated ) formulated was first a by quantitative A . Schinzel form ( of Schinzel it . The , Sierpi special\ ’{ casen} s k i 1 958 ) : if$f1 , . . . , fk$ \quad are irreducible polynomials in formulated by Lfi period(x) = Ex period+ hi, Dickson hi ∈ openZ of parenthesis Schinzel 1904 ’ s conjecture closing parenthesis was formulated more than a by hundred years$ Z ago [ comma X whileL ] . $ E\ .quad Dicksonand ( their 1904 product) more than\quad a hundreddoes years ago , while the notthe have quantitative a fixedquantitative version factor of it iversion , s thendue to of for Hardy it i infinitely s and due Littlewood to Hardy many open and integers parenthesis Littlewood $ 1 n 923 ( $1 closing923 all ) . valuesparenthesis In $ f periodi ( In the n )the $ simplest case k = 2, Dickson ’ s conjecture i s clearly equivalent aresimplest prime case . k\toquad = 2 comma (Bateman 2 . 1 .. 5 Dickson ) and . On Hornquoteright the\ otherquad s conjecture hand( 1 962 , if i sk ) clearly\quad= equivalent 1 formulatedand f to(x ..) open a = ax quantitative parenthesis+ h, 2 form of periodi t . 1\ 5quad closingThespecialcase parenthesisthen this periodi s Dirichlet $f ’ s i theorem (x ( see ) Dirichlet = x ( 1 + 837 )h .{ Landaui } ’, s h { i } \ inOn theZ $other of handProblem Schinzel comma No if ’ k . s = 1 conjecture1 is .. the and simplest f open was parenthesis case of Schinzel x closing parenthesis ’ s conjecture = ax if plusk h= comma 1 then thisformulated i s Dirichlet by quoterightand L . deg E s .f Dickson > 1. (There 1904 is ) no more single than non a - hundred linear polynomial years ago for , which while thetheorem quantitative open parenthesiswe know version thesee Dirichlet answer of it open to i Schinzel s parenthesis due to ’ s Hardy conjecture1 837 closing and Littlewood, parenthesis even for periodk (= 1. LandauHowever 923 ) quoteright . , In the s Problemsimplest No case periodif primes 1 $k is the simplest =are substituted 2 case ,$ \ byquad almostDickson primes ’ , s then conjecture Schinzel i ’ s conjecture clearly equivalent to \quad ( 2 . 1 5 ) . Ontheof Schinzel other quoterightis hand true s , inconjecture if the $k case if kk = == 1 1 .. 1$for and an deg\quad arbitrary f greaterand$f 1 polynomial period .. (Theref x( is see no ) single Section = non ax 1 hyphen 9 + hlinear , $ polynomial thenfor this for the which i case s we Dirichlet when know thef i sanswer ’ an s irreducible to Schinzel polynomial quoteright s conjecture ) . comma theoremeven for k ( = see 1 period DirichletAccording However ( comma to 1 the 837 if above primes ) . Landau connections are substituted ’ sbetween Problem by almost Landau No primes . 1 comma’ is s problems the then simplest we case ofSchinzel Schinzel quoteright ’will s s conjecture organizeconjecture the is true if material in $ the k case into = k four= 1 1 $ for areas an\quad arbitrary as followsand polynomial deg ( the $ firstf f > three1 dis- . $ \quad There is no single non − linearopen parenthesis polynomialcussed see Section for in detail which 1 9 for , wethe thecase know fourth when the one f i answer s briefly an irreducible to) : Schinzel polynomial ’ sclosing conjecture parenthesis , period evenAccording for to $ the k above = connections 1 . $ between However( i ) Large Landau , gapsif quoteright primes between s are problems primes substituted we will by almost primes , then Schinzelorganize the ’ material s conjecture( ii into ) Small four isareas gaps true as between follows in the open primes case parenthesis and $ k the the prime = first 1threek $− tuple discussed for an conjecture in arbitrary of polynomial $ fdetail $ comma theDick fourth - son one , briefly Hardy closing and parenthesisLittlewood : (open see parenthesis Section 1i( closing 9iii for ) Goldbach parenthesis the case ’ Large s when Conjecture gaps $ between f $and iprimes numbers s an irreducible of the form polynomialN = p1 − p2 ) . open parenthesis ii closing parenthesis( iv ) SmallApproximations gaps between to primes Problem and the No prime . 1 . k hyphen tuple conjecture ofAccording Dick hyphen to the above connections between Landau ’ s problems we will organizeson comma the Hardy material and Littlewood into four areas as follows ( the first three discussed in detailopen parenthesis , the fourth iii closing one parenthesis briefly Goldbach ) : quoteright s Conjecture and numbers of the form N = p 1 minus p 2 \ centerlineopen parenthesis{( i iv ) closing Large parenthesis gaps between Approximations primes to} Problem No period 1 period ( ii ) Small gaps between primes and the prime $ k − $ tuple conjecture of Dick − son , Hardy and Littlewood

\ centerline {( iii ) Goldbach ’ s Conjecture and numbers of the form $ N = p 1 − p 2 $ }

\ centerline {( iv ) Approximations to Problem No . 1 . } Landau quoteright s problems on primes .. 363 \ hspace3 period∗{\ ..f Upper i l l }Landau bounds for ’ slarge problems gaps between on primes consecutive\quad primes363 As mentioned in Section 2 comma the only area where significant results existed \ centerlinebefore 1 9 1{ 23 was . \ thequad upperUpper estimation bounds of the for differences large gaps d sub between n = p n plus consecutive 1 minus p n toprimes the power} of period As mentioned in Section 2 , the only area where significant results existed These estimations were trivial consequences of the deep results openLandau parenthesis ’ s problems 2 period on primes 7 closing parenthesis363 commabefore open 1 parenthesis 9 1 2 was 2 period the upper 1 1 closing estimation parenthesis of the differences $ d { n } = p n + 1 − p3 . n ˆ{ Upper. }$ bounds for large gaps between consecutive primes and open parenthesisAs 2 mentioned period 1 3 inclosing Section parenthesis 2 , the concerning only area estimation where and significant asymptotics results of pi open parenthesisThese estimations x closing parenthesis were periodtrivial However consequences comma this of the deep results ( 2 . 7 ) , ( 2 . 1 1 ) and ( 2 . 1 3existed ) concerning before 1 estimation9 1 2 was the and upper asymptotics estimation of of the $ \ differencespi ( xdn = ) . $ approach has it s natural limits. period The Riemann endash Von Mangoldt Prime Num hyphen However , thispn + 1 − pn These estimations were trivial consequences of the deep results ber Formula open( 2 parenthesis . 7 ) , ( 2 cf . period 1 1 ) andDavenport ( 2 . 1 open 3 ) parenthesis concerning 1 980estimation closing parenthesis and asymptotics comma Chapter 1approach 7 closing parenthesis has it s natural limits . The Riemann −− Von Mangoldt Prime Num − ber Formula (of cfπ( .x). DavenportHowever , (this 1 980 approach ) , Chapter has it s 1 natural 7 ) limits . The Riemann Equation: open– parenthesis Von Mangoldt 3 period Prime 1 closing Num parenthesis - ber Formula .. Capital ( Delta cf . open Davenport parenthesis ( 1 x 980 closing ) , parenthesis : = psi openChapter parenthesis 1 7 x ) closing parenthesis minus x : = sum n less or equal x Capital Lambda open parenthesis\ begin { a l n i g closing n ∗} parenthesis minus x = minus sum gamma bar bar less or equal T x to the power of rho divided\ tag ∗{ by$ (rho plus 3 O . open 1 parenthesis ) $}\ xDelta log to the( power x of 2 ) x divided : by= T closing\ psi parenthesis( x ) − x X X x% x log2 x :open = parenthesis\sum { wheren \ rholeq∆( =x beta) :=x ψ}\ plus(x) −Lambda i gammax := denotesΛ((n) − n thex = zeros− ) of− Riemann+xO( = quoteright−) \ ssum Zeta (3.1){\ hyphengamma {\mid } \mid \ leq T }\ f r a c { x ˆ{\varrho }}{\varrho } +% O ( T\ f r a c { x \ log ˆ{ 2 } function comma T less or equal x n≤x γ||≤T x }{andT Capital} ) Lambda open parenthesis n closing parenthesis = log p if n = p to the power of m comma Capital Lambda\end{ a l open i g n ∗} parenthesis( where n closing% = β + parenthesisiγ denotes = the 0 otherwise zeros of closing Riemann parenthesis ’ s Zeta tells - functionus that any,T zero≤ x rho and Λ(n) = log p if n = pm, Λ(n) = 0 otherwise ) tells us that any zero % it self implies an expected oscillation of size x to the power of beta slashβ bar rho bar for the remainder term \noindentCapital Delta( where openit self parenthesis implies $ \varrho an x closing expected= parenthesis\ oscillationbeta period+ of size ..i Answeringx\gamma/ | % | for a$ question the denotes remainder of Littlewood the term zeros comma of Riemann ’ s Zeta − f u n c t i o n this$ , was T proved\ leq rigorously∆(x)x. $ inAnswering a question of Littlewood , this was proved rigorously in andan effective $ \Lambda wayan first effective by( Turaacuten n way ) first open =$log$p$ by parenthesis Turán( 1 1 950 950 closing if ) , later$n parenthesis in = an comma improvedpˆ{ m later} form in, an by improved\Lambda form( byn Pintz ) =Pintz 0$ ( 1 otherwise 980 a ) and ) in tells the sharpest us that ( any in some zero sense $ \ optimalvarrho $ ) form open parenthesis 1 980 a closing parenthesis and in the sharpest open parenthesis in some sense optimal closing\noindent parenthesisit self form implies an expected oscillation| ∆(x) | ofπ size $ x ˆ{\beta } / \mid \varrho \mid $ for the remainder termsup β ≥ (3.2) Equation: open parenthesis 3 period 2 closing parenthesisx x / | ..%0 supremum| 2 , x bar Capital Delta open parenthesis $ \Delta ( x ) . $ \quad Answering a question of Littlewood , this was proved rigorously in x closing parenthesisby bar Révész( divided by1 988 x to ) the . power of beta slash bar rho 0 bar greater equal pi divided by 2 sub commaan effective way first by Tur \ ’{ a}n ( 1 950 ) , later in an improved form by Pintz ( 1 980 a ) andThe in thecrucial sharpest observation ( in that some helps sense to produce optimal improvements ) form of the es by Reacuteveacutesz- t imate open parenthesis ( 2 . 1 2 ) 1 i 988 s that closing , subtracting parenthesis periodthe two formulas of type ( 3 . 1 The crucial observation) for x + thaty and helpsx, towe produce obtain improvements of the es hyphen \ begint imate{ a openl i g n ∗} parenthesis 2 period 1 2 closing parenthesis i s that comma subtracting the two formulas of type open\ tag parenthesis∗{$ ( 3 3 period . 1 2 closing ) $parenthesis}\sup for{ x x plus}\ f y r a c {\mid \Delta ( x ) \mid }{ x ˆ{\beta } / \mid \varrho 0 \mid }\geq \ f rX a c {\(xpi+ y)}{% −2x% } { ,x log} 2 x and x comma we obtain ψ(x + y) − ψ(x) = y − + O( ) (3.3) \endEquation:{ a l i g n ∗} open parenthesis 3 period 3 closing parenthesis .. psi open% parenthesisT x plus y closing parenthesis γ||≤T minus psi open parenthesis x closing parenthesis = y minus sum gamma bar bar less or equal T open parenthesis x plus\noindent y closingby parenthesis Rand\ ’{ e in} tov (\ the 3’{ .e power} 3sz ) any (of rho1 single 988 minus ) zero x . to the% has power only of anrho effect divided of by size rho plus O open parenthesis x log to the power of 2 x divided by T closing parenthesis Theand crucial in open parenthesis observation 3 period that 3 closing helps(x + parenthesisy to)% − producex% any single improvements(x + zeroy)β rho has of only the an effect es − of size ≤ min(2 yxβ−1), (3.4) tEquation: imate ( open 2 . parenthesis 1 2 ) i3 s period that 4 , closing subtracting parenthesis the .. open two parenthesis formulas x plus of type y closing ( 3 parenthesis . 1 ) for to % | % | , the$ x power + of rhoy $ minus x to the power of rho divided by rho less or equal minimum open parenthesis 2 open parenthesisand $x x plus , ywhich $ closing weobtain is parenthesis alone always to the inferior power ofto betay. So divided , unlike by barin the rho problem bar sub comma of estimating yx to the power of beta minus 1 closing∆(x) parenthesisas in ( 3 . comma 2 ) , one single zero it self can never destroy everything . It \ beginwhich{ a is l alonei g n ∗} alwaysis the inferior number to yof period zeros So with comma large unlike real in part the problemβ and not of estimating too large imaginary \ tagCapital∗{$ ( Delta 3 openparts . parenthesis| 3γ | that ) $x influences}\ closingpsi parenthesis the( estimation x as + in open y of parenthesisdn ). The− earliest 3 \ periodpsi of 2 such( closing results x parenthesis ) = commay − one \ singlesum zero,{\ calledgamma it selfdensity can{\ nevermid theorems destroy}\ everythingmidtoday\ leq , period was provedT It}\ is thef by r a c Carlson{ ( x ( 1 920 + ) y : ) ˆ{\varrho } − numberx ˆ{\ ofvarrho zeros with}}{\ largevarrho real part} beta+ and O not too ( large\ f r a imaginary c { x \ partslog ˆ{ 2 } x }{ T } ) \end{ a l i g n ∗} X bar gamma bar that influences the estimationN(α, T ) of = d sub n period1 TheT 4α earliest(1−α)+ε. of such results comma(3.5) called density theorems today comma was proved by Carlson open parenthesis 1 920 closing parenthesis : \noindentEquation:and open in parenthesis ( 3 . 3 3 )period any 5 single closing parenthesis zero $ \ ..varrho N openζ($% parenthesis) = has 0 only alpha an comma effect T of closing size parenthesis = sum 1 ll T to the power of 4 alpha open parenthesisβ 1≥ minusα, | γ alpha|≤ T closing parenthesis plus epsilon \ begin { a l i g n ∗} period zeta open parenthesisBased on rho this closing , Hoheisel parenthesis ( 1 930 = 0 beta ) could greater reach equal the alpha first comma result bar of gamma type bar less or equal\ tag ∗{ T $ ( 3 . 4 ) $}\ f r a c { ( x + y ) ˆ{\varrho } − x ˆ{\varrho }}{\varrho } \ leqBased\ onmin this comma( 2 Hoheisel\ f r a c open{ ( parenthesis x + 1 930 y closing ) ˆ{\ parenthesisbeta }}{\ couldmid reach the\varrho first result\ ofmid type} { , } yx ˆ{\beta − 1 } ), ϑ1 Equation: open parenthesis 3 period 6 closing parenthesisdn pn .. d(ϑ sub1 < n1) ll p sub n to the power of vartheta (3.6) sub 1\end open{ a parenthesis l i g n ∗} vartheta sub 1 less 1 closing parenthesis with the value ϑ1 = 1 − 1/330. with the value vartheta sub 1 = 1 minus 1 slash 33 0 period \noindent which is alone always inferior to $ y . $ So , unlike in the problem of estimating $ \Delta ( x ) $ as in ( 3 . 2 ) , one single zero it self can never destroy everything . It is the number of zeros with large real part $ \beta $ and not too large imaginary parts $ \mid \gamma \mid $ that influences the estimation of $ d { n } . $ The earliest of such results , called density theorems today , was proved by Carlson ( 1 920 ) :

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 5 ) $} N( \alpha , T ) = \sum 1 \ l l T ˆ{ 4 \alpha ( 1 − \alpha ) + \ varepsilon } . \\\zeta ( \varrho ) = 0 \\\beta \geq \alpha , \mid \gamma \mid \ leq T \end{ a l i g n ∗}

\noindent Based on this , Hoheisel ( 1 930 ) could reach the first result of type

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 6 ) $} d { n }\ l l p ˆ{\vartheta { 1 }} { n } ( \vartheta { 1 } < 1 ) \end{ a l i g n ∗}

\noindent with the value $ \vartheta { 1 } = 1 − 1 / 33 0 . $ 3 64 .. Jaacutenos Pintz \noindentHis result was3 64 improved\quad laterJ \ ’{ significantlya} nos Pintz as Line 1 vartheta sub 1 = 3 slash 4 plus epsilon Ccaron udakov open parenthesis 1 936 closing parenthesis comma\ centerline Line 2 vartheta{ His result sub 1 = was 5 slash improved 8 plus epsilon later Ingham significantly open parenthesis as 1} 937 closing parenthesis comma Line 3 vartheta sub 1 = 3 slash 5 plus epsilon Montgomery open parenthesis 1969 closing parenthesis comma Line\ [ \ begin 4 vartheta{ a l i gsub n e 1d }\ = 7vartheta slash 1 2 plus{ epsilon1 } Huxley= 3 open / parenthesis 4 + 1 972\ varepsilon closing parenthesis\check period{C} udakov ( 13 64 936János Pintz ) , \\ The result of Ingham shows thatHis we result always was have improved primes between later n significantly to the power of as 3 and \openvartheta parenthesis{ 1 n plus} = 1 closing 5 parenthesis / 8 + to the\ powervarepsilon of 3 if n isIngham sufficiently large ( period 1 937 These results ) , \\ \vartheta { 1 } = 3 / 5 + \ varepsilon Montgomery ( 1969 ) all showed beyond open parenthesis 3 period 6 closingϑ1 = 3 parenthesis/4 + ε Cudakovˇ that (1936), , \\the PNT i s valid in intervals of length x to the power of vartheta sub 1 period ϑ1 = 5/8 + ε Ingham(1937), \Equation:vartheta open{ parenthesis1 } = 3 7 period / 7 closing 1 2 parenthesis + \ varepsilon .. pi open parenthesisHuxley x plus y ( closing 1 parenthesis 972 ) ϑ = 3/5 + ε Montgomery(1969), minus. \end pi{ opena l i g n parenthesis e d }\ ] x closing parenthesis1 thicksim y divided by log x parenleftbig y = x to the power of vartheta sub 1 parenrightbig period ϑ1 = 7/12 + ε Huxley(1972). Riemann quoteright s Hypothesis open parenthesis RH closing parenthesis implies open parenthesis3 cf period \noindent TheThe result result of of Ingham Ingham shows shows that that we we always always have have primes primes between betweenn $and n ˆ{ 3 }$ v period Koch open(n parenthesis+ 1)3 if n 1is 90 sufficiently 1 closing parenthesis large . These closing results parenthesis all showed beyond ( 3 . 6 andEquation: open parenthesis 3 period 8 closing parenthesis .. Capital Delta open parenthesis x closing paren- ) that the PNT i s valid in intervals of length xϑ1 . thesis$ ( ll square n root + of 1 x sub ) log ˆ{ to3 the}$ power if of $ 2 nx comma $ is sufficiently large . These results all showed beyond ( 3 . 6 ) that theand PNT thereby i s valid in intervals of length $ x ˆ{\vartheta { 1 }} . $ y Equation: open parenthesis 3 period 9 closingπ(x + y parenthesis) − π(x) ∼ .. d sub( ny ll= squarexϑ1 ). root of x sub log to(3 the.7) power of\ begin 2 x period{ a l i g n ∗} log x \ tagThis∗{$ result ( was 3 improved . 7 under ) $RH}\ bypi Crameacuter( x open + parenthesis y ) 1− 920 closing \ pi parenthesis( x to ) \sim \ f rEquation: a c { y }{\ openlog parenthesisRiemannx } 3( ’period s Hypothesis y 1 0 = closing x ( parenthesis ˆ RH{\ )vartheta implies .. d sub ( cf{ n .1 ll v square}} . Koch). root ( of 1 90x sub 1 ) log ) x comma \endwhich{ a l i still g n ∗} falls short of answering Landau quoteright s question No period 4 positively comma even assuming the RH period √ 2 ∆(x) x x, (3.8) \ centerlineIn the case of{Riemann the unconditional ’ s Hypothesis estimates the ( exponent RH ) implies 7 slashlog 1 ( 2 is cf still . the v . Koch ( 1 90 1 ) ) } best one knownand for thereby which open parenthesis 3 period 7 closing parenthesis holds period However comma concerning\ begin { a lopen i g n ∗} parenthesis 3 period 6 closing parenthesis comma a break hyphen \ tagthrough∗{$ ( occurred 3 when . Iwaniec 8 ) and $}\ JutilaDelta open parenthesis( x 1 ) 979 closing\ l l parenthesis\ sqrt { x obtained} {\ log by an}ˆ ingenious{ 2 } √ xcombination , of analytic and sieve methods the result 2 dn xlogx. (3.9) \endopen{ a l parenthesis i g n ∗} 3 period 1 1 closing parenthesis vartheta sub 1 = 1 3 slash 23 period An important theoreticalThis result consequence was improved of the results under of RH Iwaniec by Cramér( and Jutila 1 920 ) to \noindentwas to overcomeand thereby the quoteleft parity obstacle quoteright open parenthesis to be discussed later in Sections 1 0 endash \ begin { a l i g n ∗} √ 1 1 closing parenthesis which in general prevents sievedn methods xlog fromx, revealing the existence (3.10) \ tagof∗{ primes$ ( in a3 suitable . set 9 open ) $ parenthesis} d { n cf}\ periodl l Greaves\ sqrt { openx } parenthesis{\ log 2001}ˆ{ 2 closing} x parenthesis . p period\end{ 1a l 71 i g closingn ∗} which parenthesis still periodfalls short of answering Landau ’ s question No . 4 positively , The later developmentseven all used both analytic and sieve methods and \noindentshowed commaThis similarlyassuming result to wasthe Iwaniec RH improved and . Jutila under open parenthesis RH by Cram 1 979\ ’ closing{ e} r parenthesis ( 1 920 ) comma to an inequality weaker than In the case of the unconditional estimates the exponent 7 / 1 2 is still \ beginopen{ parenthesisa l i g n ∗}the 3 period best 7 one closing known parenthesis for which but stronger ( 3 . 7 than ) holds open .parenthesis However 3 , period concerning 6 closing ( parenthesis\ tag ∗{ comma$ ( namely 33 . . 6 ) 1 , a break 0 ) - $} throughd { n occurred}\ l l when\ sqrt Iwaniec{ x } {\ andlog Jutila} (x 1 979 , ) \endEquation:{ a l i g n ∗} openobtained parenthesis by 3 period an ingenious 1 2 closing combination parenthesis .. piof open analytic parenthesis and sieve x plus methods y closing parenthesis the minus pi open parenthesisresult x closing parenthesis gg y divided by log x parenleftbig y = x to the power of vartheta \noindent which still falls short of answering Landau ’ s question No . 4 positively , even sub 1 parenrightbig(3 period.11) ϑ1 = 13/23. An important theoretical consequence of the results of Iwaniec and \noindent assumingJutila the was RH to .overcome the ‘ parity obstacle ’ ( to be discussed later in Sections 1 0 – 1 1 ) which in general prevents sieve methods from revealing In the case ofthe the existence unconditional of primes estimates in a suitable the set exponent ( cf . Greaves 7 / 1 ( 2 2001 is ) still p . 1 71 the ) best one known. for which ( 3 . 7 ) holds . However , concerning ( 3 . 6 ) , a break − through occurredThe when later Iwaniec developments and Jutila all used ( 1 both 979 analytic ) obtained and sieve by an methods ingenious and combination ofshowed analytic , similarly and sieve to Iwaniec methods and the Jutila result ( 1 979 ) , an inequality weaker than ( 3 . 7 ) but stronger than ( 3 . 6 ) , namely \noindent $ ( 3 . 1 1 ) \vartheta { 1 } = 1 3 / 23 . $ y An important theoretical consequenceπ(x + y) of− π( thex) results( ofy = Iwaniecxϑ1 ). and Jutila(3.12) was to overcome the ‘ parity obstacle ’ ( to belog x discussed later in Sections 1 0 −− 1 1 ) which in general prevents sieve methods from revealing the existence of primes in a suitable set ( cf . Greaves ( 2001 ) p . 1 71 ) .

The later developments all used both analytic and sieve methods and showed , similarly to Iwaniec and Jutila ( 1 979 ) , an inequality weaker than ( 3 . 7 ) but stronger than ( 3 . 6 ) , namely

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 1 2 ) $}\ pi ( x + y ) − \ pi ( x ) \gg \ f r a c { y }{\ log x } ( y = x ˆ{\vartheta { 1 }} ). \end{ a l i g n ∗} Landau quoteright s problems on primes .. 365 \ hspacevartheta∗{\ subf i l 1 l } =Landau 1 1 slash ’ 20s problems .. Heath hyphen on primes Brown\quad comma365 Iwaniec open parenthesis 1 979 closing parenthesis comma \ centerlinevartheta sub{ 1$ =\ 1vartheta 7 slash 3 1 ..{ Pintz1 } open= parenthesis 1 1 1 / 98 1 comma 20 $ 1984\quad closingHeath parenthesis− Brown comma , Iwaniec ( 1 979 ) , } vartheta sub 1 = 23 slash 42 .. Iwaniec endash Pintz open parenthesis 1984 closing parenthesis comma \ centerline { $ \vartheta { 1 } = 1 7 / 3 1 $ \quad Pintz ( 1 98 1 , 1984 ) , } open parenthesis 3 period 1 3 closing parenthesis vartheta sub 1 =Landau 1 1 slash ’ s problems 20 minus on 1 slashprimes 384 ..365 Mozzochi open parenthesis 1 986 closing parenthesis comma \noindent $ \vartheta {ϑ11=} 11/=20 Heath 23 / - Brown 42 $ , Iwaniec\quad Iwaniec ( 1 979 )−− , Pintz ( 1984 ) , vartheta sub 1 = 6 slash 1 1 .. Louϑ endash1 = 17/31 Yao openPintz parenthesis ( 1 98 1 ,1 1984 992 comma ) , 1 993 closing parenthesis comma$ ( 3 . 1 3 ) \vartheta { 1 } = 1 1 / 20 − 1 / 384 $ \quad Mozzochiϑ1 (= 1 23 986/42 )Iwaniec , – Pintz ( 1984 ) , (3.13) ϑ1 = 11/20−1/384 Mozzochi vartheta sub 1( = 1 7 986 slash ) 1, 3 .. Lou endash Yao open parenthesis 1 992 comma 1 993 closing parenthesis comma \ centerline { $ \vartheta { ϑ11 =} 6/=11 6Lou /– Yao 1 ( 1 1 992 $ ,\ 1quad 993 )Lou , −− Yao ( 1 992 , 1 993 ) , } vartheta sub 1 = 1 7 slash 200 .. Bakerϑ1 = 7endash/13 Lou Harman – Yao open ( parenthesis 1 992 , 1 993 1 996 ) closing , parenthesis period Finally the best known result i s \ centerline { $ \vartheta { ϑ1 =} 17/=200 7Baker / – 1 Harman 3 $ (\ 1quad 996 )Lou . −− Yao ( 1 992 , 1 993 ) , } Theorem openFinally parenthesis the Baker best endash known Harman result i endash s Pintz open parenthesis 200 1 closing parenthesis closing parenthesis period d sub n ll p 2 n to the power of 1 slash 40 sub period 1/40 \ centerline { $Theorem\vartheta( Baker{ 1 } – Harman= 1 – 7 Pintz /(2001)) 200. $ dn\quad p2Bakern . −− Harman ( 1 996 ) . } If we are contented with resultsIf we are which contented guarantee with the existence results ofwhich primes guarantee the existence of in almost all short intervals of type \noindent Finallyprimes the best known result i s Equation: openin parenthesis almost all 3 period short 14 intervals closing parenthesis of type .. open square bracket x comma x plus y closing square bracket comma y = x to the power of vartheta sub 2 comma \noindentthen the methodTheorem of Huxley ( Baker open−− parenthesisHarman −− 1 972Pintz closing $( parenthesis 200 yields 1 this ) with ) open . parenthesis d { n 3} period\ l l 7p closing 2 parenthesis{ n }ˆ{ for1 / 40 } { . [x,}$ x + y], y = xϑ2 , (3.14) Equation: open parenthesis 3 period 1 5 closing parenthesis .. vartheta sub 2 = 1 slash 6 plus epsilon period \ hspaceFurther∗{\ commaf i l lthen} ..If the we the combination are method contented of of Huxley analytic with ( and 1 results 972 sieve ) methods yields which this led guarantee towith .. open ( 3 . parenthesis 7 the ) for existence 3 period of 1 2 primes closing parenthesis .. in \noindent in almost all short intervals of type almost all short intervals with ϑ2 = 1/6 + ε. (3.15) vartheta sub 2 = 1 slash 1 0 plus epsilon Harman open parenthesis 1982 closing parenthesis Equation: open parenthesis\ begin { a l 3 i g period n ∗}Further 1 6 closing , parenthesisthe combination .. vartheta of analytic sub 2 to andthe power sieve of methods vartheta led sub to 2 = to ( the3 . power of\ tag = 1∗{ to$ the ( power 31 of2 . ) 1 sub 14 in slash almost ) 1 $ 5} to all[x,x+y],y=xˆ the short power intervals of slash 14 with plus to the power of plus epsilon to{\ thevartheta power { 2 }} of, epsilon H sub period to the power of Ch period Z sub period to the power of Jia Li to the power of open parenthesis\end{ a l i g sub n ∗} open parenthesis 1997 closing parenthesis to the power of 1995 a closing parenthesis comma Watt ϑ2 = 1/10 + ε Harman(1982) open parenthesis 1995 closing parenthesis \noindent then the method ofϑ2 Huxley= 1/14 (+ 1ε 972Ch. ) yieldsJia (1995 thisa), with ( 3 . 7 ) for vartheta sub 2 = 1 slash 20 plusϑ epsilon2 = 1 ../ Ch15 + periodε H Jia. openZ. parenthesisLi (1997) W 1996 att(1995) a closing parenthesis (3.16) period These results will also have later significance in the examination of gaps ϑ = 1/20 + ε \ beginbetween{ a l consecutivei g n ∗} Goldbach numbers2 in Section 14 periodCh . Jia ( 1996 a ) . \ tagLandau∗{$ ( quoteright 3These . s Problem 1 results 5 No period will ) $}\ also 4 canvartheta have be approximated later significance{ 2 } in= other in 1 ways the / examination if primes 6 + of\ varepsilon gaps . are substitutedbetween by almost consecutive primes period Goldbach We call an numbersinteger a P in sub Section r .. number 14 . if it has \endat{ mosta l i g r n prime∗} factorsLandau open ’ parenthesis s Problem counted No . 4 with can bemultiplicity approximated closing parenthesis in other ways period if primes Already Viggo P Brun are substituted by almost primes . We call an integer a r number if r \noindentopen parenthesisFurtherit 1 has 920 , closing\ atquad most parenthesistheprime combination showed factors that ( of therecounted analytic i s a with P sub and multiplicity 1 1 sieve number methods in ) any . Already interval led toof type\quad ( 3 . 1 2 ) \quad in openalmost parenthesis all short xViggo comma intervals Brun x plus square with root of x sub closing parenthesis P for x greater x sub( 1 0 920 open√ ) parenthesis showed that consequently there i between s a 11 neighboringnumber squares in any comma interval if n of greater type n sub 0 (x, x + x x > x ( closing\ begin parenthesis{ a l i g n ∗} period After) for 0 consequently between neighboring squares , if \vartheta { 2 } = 1 / 1 0 + \ varepsilon Harman ( 1982 ) \\\ tag ∗{$ ( various improvementsn > n0) comma. After J various period R improvements period Chen open , J parenthesis . R . Chen 1975 ( 1975 closing ) showed parenthesis this showed for this 3 . 1 6 ) $}\vartheta ˆ{\vartheta { 2 }} { 2 } = ˆ{ = } 1 ˆ{ 1 }ˆ{ / for P sub 2 numbersP2 commanumbers too ,period too . Another approach√ i s to show that we have a number 14 } { / 1 5 } + ˆ{ + }\ varepsilon ˆ{\ varepsilon } H ˆ{ Ch . } { . } Z ˆ{ Jia } { . } Another approachn in i s every to show interval that we of have type a number(x, x + n inx) everysuch interval that the greatest prime factor Liof ˆ{ type( open}ˆ{ parenthesis1995of it a x comma ) ,x plus} { square( 1997 root of x sub) } closingWatt parenthesis ( 1995 such that ) the greatest prime factor\end{ ofa l it i g n ∗} Equation: open parenthesis 3 period 1 7 closing parenthesis .. P open parenthesis n closing parenthesis greater P (n) > nc1 (3.17) n\ centerline to the power{ of$ c sub\vartheta 1 { 2 } = 1 / 20 + \ varepsilon $ \quad Ch . Jia ( 1996 a ) . } with a c sub 1 lesswith 1 comma a c1 < possibly1, possibly near to near 1 period to 1 The . The first firstand the and last the results last of results this type of this Theseare resultstype will are also have later significance in the examination of gaps betweenEquation: consecutive open parenthesis Goldbach 3 period 1 numbers 8 closing parenthesis in Section .. c 14 sub .1 to the power of c sub 1 = sub = 1 5 0 sub period 738 to the power of slash 26 = 0 period 5 769 period period period Ramachandra H period Q period cc1 = 150/26 = 0.5769... Ramachandra(1969 ), (3.18) LiuLandau comma ’ J s period Problem Wu to No the . power1 4 can= of beopen.738 approximated parenthesis 1 969 in sub other openH.Q.Liu,J.W parenthesis ways if u(1999) primes1 999. closing parenthesis periodare tosubstituted the power of closing by almost parenthesis primes comma . We call an integer a $ P { r }$ \quad number if it has at most $ r $It prime i s interesting factors to ( observe counted that with if we multiplicity consider the slightly ) . Already larger interval Viggo Brun It i s interesting[x, to x observe+x1/2+ε that] then if we we consider have already the slightly numbers larger intervalwith much larger prime factors open square bracket, x comma x plus x to the power of 1 slash 2 plus epsilon closing square bracket then we have\noindent already numbers( 1 920 with ) showedmuch larger that prime there factors i comma s a $ P { 1 1 }$ number in any interval of type $ ( x , x + \ sqrt { x } { ) }$ f o r $ x > x { 0 } ( $ consequently between neighboring squares , if $ n > n { 0 } ) . $ After various improvements , J . R . Chen ( 1975 ) showed this for $ P { 2 }$ numbers , too . Another approach i s to show that we have a number $ n $ in every interval oftype $( x , x + \ sqrt { x } { ) }$ such that the greatest prime factor of it

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 1 7 ) $} P ( n ) > n ˆ{ c { 1 }} \end{ a l i g n ∗}

\noindent with a $ c { 1 } < 1 , $ possibly near to 1 . The first and the last results of this type are

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 1 8 ) $} c ˆ{ c { 1 }} { 1 } = { = } 1 5 { 0 }ˆ{ / 26 } { . 738 } = 0 . 5 769 . . . Ramachandra { H.Q. Liu , J . Wu }ˆ{ ( 1 969 }ˆ{ ), } { ( 1 999 ) . } \end{ a l i g n ∗}

It i s interesting to observe that if we consider the slightly larger interval $ [ x , x + x ˆ{ 1 / 2 + \ varepsilon } ] $ then we have already numbers with much larger prime factors , 3 66 .. Jaacutenos Pintz \noindentnamely P3 open 66 parenthesis\quad J \ n’{ closinga} nos parenthesis Pintz greater n to the power of c sub 2 comma where again the earliestnamely and the$P latest ( results n are ) the> n ˆ{ c { 2 }} , $ where again the earliest and the latest results are the following \noindentEquation:f open o l l o wparenthesis i n g 3 period 1 9 closing parenthesis .. c sub 2 to the power of c sub 2 = sub = 2 c slash 25 sub slash 263 66 to theJános powerPintz of 3namely to the powerP (n) of> minus n 2 , where epsilon again minus the epsilon earliest Ch sub and period the latest to the power of\ begin Jutila{ opena l i g parenthesis n ∗} Jia sub comma to the power of 1 973 sub M period Ch period to the power of closing \ tag ∗{$ ( 3results . are1 the 9 ) $} c ˆ{ c { 2 }} { 2 } = { = } 2 / { 25 }ˆ{ 3 } { / parenthesis commafollowing Liu open parenthesis 2000 closing parenthesis period 26 We}ˆ{ mention − } that\ varepsilon the methods{ leading − } to \ varepsilon the strong resultsCh about ˆ{ PJ openu t i l a parenthesis} { . } n closing( { Jia parenthesis}ˆ{ 1 comma973 } { , }ˆ{ ), } { M . Ch . } Liu ( 2000 ) . \end{ a l i g n ∗} c2 3 − Jutila 1973), use comma .. similarly to the workc2 = of= Iwaniec2/25/26 andε−ε .. Jutila Ch. comma(Jia , ..M.Ch. a combinationLiu(2000). of ana hyphen(3.19) lytic and sieve methodsWe mention including that the linear the methods sieve with leading Iwaniec quoterightto the strong s bilinear results about P (n), Weexpression mention of that the error the term methods period leading to the strong results about $ P ( n ) , $ use , similarly to the work of Iwaniec and Jutila , a combination 4 period .... Theof expected ana - lytic size of and large sieve gaps methods period Crameacuter including quoteright the linear s probabilistic sieve with Iwaniecmodel ’ useEmpirical , \quad datasimilarly suggest that to the thelargest work gaps of between Iwaniec primes and are much\quad J u t i l a , \quad a combination of ana − lytic and sieves bilinear methods including the linear sieve with Iwaniec ’ s bilinear smaller than theexpression size d sub n of less the 2 squareerror term root of . p n which would imply Landau quoteright s conjecture open parenthesis and i s of about the same strength closing parenthesis period It was Crameacuter who first \noindent expression4 . The of the expected error size term of . large gaps . Cramér’ s probabilistic used a prob hyphenmodel abilistic approach toEmpirical predict the data size suggestof the largest that possible the largest gaps between gaps between primes are much \noindentconsecutive4 primes . \ h f period i l l The His expected probabilistic size model√ of open large parenthesis gaps . Crameacuter Cram\ ’{ e} openr ’ s parenthesis probabilistic 1 935 model smaller than the size dn < 2 pn which would imply Landau ’ s conjecture comma 1 936 closing( and parenthesis i s of aboutclosing parenthesisthe same strength is a good ) . It was Cramérwho first used a Empiricalstarting point data to formulatesuggest conjectures that the about largest the asymptotic gaps between behaviour primes of are much smaller thanprob the size- abilistic $ d approach{ n } < to predict2 \ sqrt the{ sizep of n the}$ largest which possible would imply gaps Landau ’ s conjecture primes period Basedbetween on the consecutive Prime Number primes Theorem . His open probabilistic parenthesis 2model period ( 1 Cramér( 1 closing parenthesis 1 935 he defined the inde hyphen \noindent ( and, 1 i 936 s of ) ) about is a good the same starting strength point to ) formulate . It was conjectures Cram\ ’{ e} r about who first the used a prob − pendent randomasymptotic variables xi openbehaviour parenthesis of primes n closing . parenthesisBased on thefor n Prime greater Number equal 3 by Theorem abilisticEquation: open approach parenthesis to 4 predict period 1 closing the sizeparenthesis of the .. P open largest parenthesis possible xi sub n gaps = 1 closing between parenthesis consecutive primes( 2 . 1 1 . ) His he defined probabilistic the inde - model pendent ( Cramrandom\ ’{ e} variablesr ( 1 935ξ(n ,) for 1 936n ≥ )3 ) is a good = 1 divided by logby n sub comma P open parenthesis xi sub n = 0 closing parenthesis = 1 minus 1 divided by log nstarting sub period point to formulate conjectures about the asymptotic behaviour of primesOn the basis. Based of his on model the he Prime conjectured Number Theorem ( 2 . 1 1 ) he defined the inde − pendent random variables $ \ xi ( n1 ) $ f o r $ n 1\geq 3 $ by Equation: open parenthesis 4 periodP 2(ξ closingn = 1) parenthesis= P ..(ξn limint= 0) n= right 1 − arrow infinity supremum(4.1) d sub n divided by log to the power of 2 p n = 1 comma log n , log n . \ begin { a l i g n ∗} which would beOn true the with basis probability of his 1 model in his modelhe conjectured period \ tagThis∗{$ model ( would 4 predict. 1 the ) truth $} P( of all four conjectures\ xi { n of} Landau= 1 and ) = \ f r a c { 1 }{\ log n } { , } P( \ xi { n } = 0 ) = 1 − \ f r a c { 1 }{\ log n } { . } seemed to agree with our knowledge about primes when usedd for appropri hyphen \endate{ problemsa l i g n ∗} period The Crameacuter model openlim parenthesissup n CM= closing 1, parenthesis predicts asymptotically(4.2) n→∞ 2 the same log pn \noindentnumber ofOn even thewhich and basis odd would primes of be belowhis true model a with given he probability bound conjectured comma 1 which in his is model clearly . not true period That wasThis not model an i ssue would as long predict as the mathematical the truth of community all four conjectures be hyphen of Landau \ beginlieved{ ita l knewi g n ∗} whichand were seemed the appropriate to agree with problems our for knowledge the model period about Crameacuter primes when quoteright used for s \ tagmodel∗{$ predicted ( 4appropri the . truth 2 - of PNT ) ate $}\ inproblems shortlim intervals{ n . The of\rightarrow Cramérmodelsize open parenthesis\ (infty CM log ) x}\ predicts closingsup parenthesis asymp-\ f r a c { tod the{ n }}{\ log ˆ{ 2 } powerp n of} lambda= for 1totically any , the same number of even and odd primes below a given bound , \endlambda{ a l i g greater n ∗} which 2 comma is clearly for example not true comma . That that iswas comma not an the issue relation as long open as parenthesis the mathemat- cf period open parenthesis 3 periodical 7 closing community parenthesis be - closing lieved parenthesis it knew which were the appropriate problems \noindentEquation:which openfor parenthesis would the model be 4 period true . Cramér’ 3 with closing probability s parenthesis .. pi 1 open in his parenthesis model x . plus y closing parenthesis minus pi open parenthesismodel predicted x closing parenthesisthe truth thicksim of PNT y divided in short by logintervals x comma of y size = open ( log parenthesisx)λ for log x closingThis model parenthesis wouldany to the predict power of the lambda truth comma of lambda all four greater conjectures 2 period of Landau and seemedIt .. i s to .. naturally agreeλ > 2 with quite, for exampledifficult our knowledge .. , to that check is about numerically , the relation primes conjectures ( when cf . ( used .. 3 like . 7 ..for ) )open appropri parenthesis− 4 period 3 closingate problems parenthesis . The Cram\ ’{ e} r model ( CM ) predicts asymptotically the same numberfor .. all of .. short even .. and intervals odd .. primes for really below large .. a values giveny .. of bound x period , ..which However is comma clearly .. the not general true . That was not an i ssueπ(x + asy) − longπ(x) as∼ the, mathematical y = (log x)λ, λ community > 2. be −(4.3) belief was that this is an appropriate problem and CMlog canx be applied to lieved it knew which were the appropriate problems for the model . Cram\ ’{ e} r ’ s predict relationsIt like open i s parenthesis naturally 4 period quite3 difficult closing parenthesis to check despite numerically the obvious conjectures deficiencies of the model period It like ( 4 . 3 ) for all short intervals for really large values \noindentwas thereforemodel a great predicted surprise when the Maier truth open of PNT parenthesis in short 1985 closing intervals parenthesis of size showed ( log that comma $ x ) ˆ{\lambda }of$x. f o rHowever any , the general belief was that this is an appropriate taking an problem and CM can be applied to predict relations like ( 4 . 3 ) despite \noindent $ \thelambda obvious> deficiencies2 , $ of for the model example . It , that is , the relation ( cf . ( 3 . 7 ) ) was therefore a great surprise when Maier ( 1985 ) showed that , taking \ begin { a l i g n ∗}an \ tag ∗{$ ( 4 . 3 ) $}\ pi ( x + y ) − \ pi ( x ) \sim \ f r a c { y }{\ log x } , y = ( \ log x ) ˆ{\lambda } , \lambda > 2 . \end{ a l i g n ∗}

\noindent I t \quad i s \quad naturally quite difficult \quad to check numerically conjectures \quad l i k e \quad ( 4 . 3 ) f o r \quad a l l \quad short \quad i n t e r v a l s \quad for really large \quad values \quad o f $ x . $ \quad However , \quad the g e n e r a l belief was that this is an appropriate problem and CM can be applied to predict relations like ( 4 . 3 ) despite the obvious deficiencies of the model . It

\noindent was therefore a great surprise when Maier ( 1985 ) showed that , taking an Landau ’ s problems on primes 367 arbitrarily large ﬁxed λ, the relation ( 4 . 3 ) will be always false for suitable

λ valuesxn, yn = (log xn) → ∞. As explained in Granville ( 1994 , 1 995 ) the reason why CM makes an incorrect prediction for ( 4 . 3 ) is the same as mentioned already : the model does not contain the trivial information that primes have no small divisors . If CM is corrected in the way that all numbers having a divisor below a log z(x) given parameter z = z(x)( with log log x → ∞) are a priori excluded from the set of possible primes ( and the remaining numbers are chosen with a probability proportional to 1 / log n), then the contradiction discovered by Maier disappears . The corrected CM ( CCM ) will predict falsity of ( 4 . 3 ) for a suitable ( rare ) set of short intervals . On the basis of this corrected model , Granville ( 1 993 ) conjectures that ( 4 . 2 ) holds with 1 replaced by 2e−γ . However , the present author has shown that any type of modiﬁcation preserving the independence of the variables ξn will still be in a ‘ non - trivial ’ contradiction with the true distribution of primes . Speciﬁcally , we still have a contradiction with the global result

Z 0 ) 1 ( X 1 X π(x) − 2dx 2 (4.4) X X log n log X <2n≤x , valid on RH . If RH is not true , then we have a much more significant contradiction with CM - type probabilistic models , since then we have much √ larger oscillation than X as shown by the result ( 3 . 2 ) of Révész. We remark that in case of existence of zeros with β0 > 1/2, also the average size of the error is larger , as shown first by Knapowski ( 1959 ) and later in a stronger form by Pintz ( 1 980 b , c ) . What makes the contradiction between ( 4 . 4 ) and Cramér’ s model more peculiar i s the fact that the result ( 4 . 4 ) was proved 1 5 years before the discovery of Cramér’ s model and the mathematician who showed ( 4 . 4 ) was Harald Cramér( 1920 ) himself . The theorem below shows that in order to avoid conflict with reality , our set from which we choose our ‘ possible primes ’ ( which was the set of numbers without prime divisors of size O(logλ x) for any λ earlier ) has to coincide nearly exactly with the set of primes . Our freedom i s just to add a thin set of composite numbers to the primes whose cardinality is less than that of the primes by a factor at least c log x. This means that any reasonable new model has to give up the simple condition of independence . Theorem 1 ( J . Pintz (200?)). Let x be a large even number ,I = ∗ (x/2, x]∩ Z. Let Sx be arbitrary with Landau quoteright s problems on primes .. 367 Landauarbitrarily ’ s large problems fixed lambda on primes comma\quad the relation367 open parenthesis 4 period 3 closing parenthesis will be alwaysarbitrarily false for suitable large fixed $ \lambda , $ the relation ( 4 . 3 ) will be always false for suitable values x sub n comma y n = open parenthesis log x sub n closing parenthesis to the power of lambda right arrow\ begin infinity{ a l i g period n ∗} valuesAs explained x in{ Granvillen } , .. open y parenthesis n = 1994 ( comma\ log .. 1x 995{ closingn } parenthesis) ˆ{\lambda .. the reason}\rightarrow why CM makes\ infty an . \end{ a l i g n ∗} incorrect prediction for open parenthesis 4∗ period 3 closing∗ parenthesis| I is| the same as mentioned already : Px := P ∩ I ⊆ Sx ⊆ I,A = ∗ (4.5) the model | Sx | . Asdoes explained not contain in the Granville trivial information\quad that( 1994 primes , have\quad no1 small 995 divisors ) \quad periodthe reason why CM makes an incorrectIf CM is corrected prediction in the way for that ( 4all . numbers 3 ) is having the a same divisor as below mentioned a already : the model doesgiven not parameter contain z = z the open trivial parenthesis information x closing parenthesis that parenleftbig primes have with logno z small open parenthesis divisors x closing . parenthesis divided by log log x right arrow infinity parenrightbig .... are a priori excluded from \noindentthe set of possibleIf CM primes is corrected open parenthesis in the and way the that remaining all numbers numbers are having chosen with a divisor a below a probability proportional to 1 slash log n closing parenthesis comma then the contradiction discovered by \noindentMaier disappearsgivenparameter period The corrected $z CM = open z parenthesis ( x CCM ) closing ($ parenthesis with $\ willf r a c predict{\ log falsityz of open( parenthesisx ) }{\ 4 periodlog 3\ closinglog parenthesisx }\rightarrow for \ infty ) $ \ h f i l l are a priori excluded from a suitable open parenthesis rare closing parenthesis set of short intervals period On the basis of this corrected model\noindent commathe set of possible primes ( and the remaining numbers are chosen with a probabilityGranville open proportional parenthesis 1 993 to closing 1 / log parenthesis $ n conjectures ) , $ that then open the parenthesis contradiction 4 period 2 discovered closing by parenthesisMaier disappears holds with 1 replaced. The corrected by 2 e to the CM power ( CCM of minus ) will gamma predict period falsity of ( 4 . 3 ) for aHowever suitable comma ( rare the present ) set .. ofauthor short has shown intervals that any . type On theof modification basis of this corrected model , Granvillepreserving the ( 1 independence 993 ) conjectures of the variables that xi sub ( n4 will . 2 still ) be holds in a quoteleft with 1 non replaced hyphen trivial by quoteright$ 2 e ˆ{ − \gammacontradiction} . $ with the true distribution of primes period Specifically comma we still have a contradiction with the global result \ hspaceEquation:∗{\ openf i l l } parenthesisHowever 4 , period the present4 closing parenthesis\quad author .. 1 divided has by shown X integral that from any 0 to type X to ofthe powermodification of parenleftbigg pi open parenthesis x closing parenthesis minus sum less 2 n less or equal x 1 divided by log n to\noindent the power ofpreserving parenrightbigg the 2 sub independence dx ll X divided of by logthe to variables the power of 2 $ X\ subxi comma{ n }$ will still be in a ‘ non − t r i v i a l ’ valid on .. RH period .. If RH is .. not .. true comma .. then we .. have .. a much more .. significant \noindentcontradictioncontradiction with CM hyphen with type theprobabilistic true distribution models comma since of primesthen we have . Specifically much , we still have larger oscillation than .. square root of X as .. shown by the .. result .. open parenthesis 3 period 2 closing parenthesis\noindent ..a of contradictionReacuteveacutesz period with .. the We global result remark that in case of existence of zeros with beta 0 greater 1 slash 2 comma also the average \ beginsize of{ a the l i g error n ∗} is larger comma as shown first by Knapowski open parenthesis 1959 closing parenthesis and later\ tag in∗{$ ( 4 . 4 ) $}\ f r a c { 1 }{ X }\ int ˆ{ 0 } { X }ˆ{ ( }\ pi ( x ) a− stronger \sum form{ by< Pintz{ open2 } parenthesisn \ leq 1 980x b}\ commaf r a c c{ closing1 }{\ parenthesislog n period}ˆ{ ) } 2 { dx } \ l lWhat\ f r makes a c { X the}{\ contradictionlog ˆ{ 2 between} X } open{ , parenthesis} 4 period 4 closing parenthesis and Crameacuter quoteright\end{ a l i g s n model∗} more peculiar i s the fact that the result .. open parenthesis 4 period 4 closing parenthesis .. was proved .. 1 5 years\noindent before thev a l i d on \quad RH . \quad I f RH i s \quad not \quad true , \quad then we \quad have \quad a much more \quad significant contradictiondiscovery of Crameacuter with CM quoteright− type s probabilistic model and the mathematician models , sincewho showed then open we parenthesis have much 4 period 4 closing parenthesis was \noindentHarald Crameacuterlarger oscillation open parenthesis than 1920\ closingquad parenthesis$\ sqrt { himselfX }$ period as \quad shown by the \quad r e s u l t \quad ( 3 . 2 ) \quad o f R\ ’{ e}v \ ’{ e} sz . \quad We remarkThe theorem that below in case shows of that existence in order to avoid of zeros conflict with with reality $ \beta comma 0 > 1 / 2 , $ alsoour the set from average which we choose our quoteleft possible primes quoteright .. open parenthesis which was the set ofsize of the error is larger , as shown first by Knapowski ( 1959 ) and later in anumbers stronger without form prime by Pintzdivisors (of 1size 980 O open b , parenthesis c ) . log to the power of lambda x closing parenthesis for any lambda earlier closing parenthesis has to Whatcoincide makes nearly the exactly contradiction with the set of between primes period ( 4. Our 4 freedom ) and Cram i s just\ ’ to{ e add} r ’ s model more peculiara thin set of i composite s the fact numbers that to the the primes result whose\quad cardinality( 4 . is 4 less ) \quad was proved \quad 1 5 years before the discoverythan that of of the Cram primes\ ’{ bye} ar factor ’ s at model least c and log x the period mathematician This means that who any showed ( 4 . 4 ) was Haraldreasonable Cram new\ ’{ modele} r has ( 1920 to give ) up himself the simple . condition of independence period Theorem 1 open parenthesis J period Pintz open parenthesis 200 ? closing parenthesis closing parenthesis periodThe theorem .. Let x be below a large shows even number that incomma order I = toopen avoid parenthesis conflict x slash with 2 comma reality x closing , square bracket capour set from which we choose our ‘ possible primes ’ \quad ( which was the set of Z period Let S sub x to the power of * .. be arbitrary with \noindentEquation:numbers open parenthesis without 4 period prime 5 closing divisors parenthesis of size .. P sub $ O x to (the power\ log ofˆ *{\ : =lambda P cap I} subsetx equal) $ S f subo r xany to the $ power\lambda of * subset$ earlier equal I comma ) has A to = bar I bar divided by bar S sub x to the power of * barcoincide sub period nearly exactly with the set of primes . Our freedom i s just to add a thin set of composite numbers to the primes whose cardinality is less than that of the primes by a factor at least $ c $ log $ x . $ This means that any reasonable new model has to give up the simple condition of independence .

\noindent Theorem1(J.Pintz $( 200 ? ) ) .$ \quad Let $ x $ be a large even number $,I=(x/2,x] \cap $ $ Z . $ Let $ S ˆ{ ∗ } { x }$ \quad be arbitrary with

\ begin { a l i g n ∗} \ tag ∗{$ ( 4 . 5 ) $} P ˆ{ ∗ } { x } : = P \cap I \subseteq S ˆ{ ∗ } { x } \subseteq I , A = \ f r a c {\mid I \mid }{\mid S { x }ˆ{ ∗ } \mid } { . } \end{ a l i g n ∗} 3 68 J´anos Pintz Let us deﬁne independent random variables ηn for all n ∈ I as

∗ ηn = 0 ifnelement − slashSx; (4.6)

∗ while for n ∈ Sx let

A A P(ηn = 1) = P(ηn = 0) = 1 − (4.7) log n , log n . Then the truth of the relation

2 P x D ( ∈I ηn) (4.8) n log2 x implies

x | S∗ \P∗ | . (4.9) x x log2 x 5 . Lower bounds for large gaps between primes . The Erd ohungarumlaut s – Rankin Problem The Prime Number Theo- rem ( 2 . 1 1 ) obviously implies

d λ := lim sup n ≥ 1. (5.1) n→∞ log n

This was improved to λ ≥ 2 by Backlund ( 1 929 ) and λ ≥ 4 by Brauer , Zeitz ( 1 930 ) . Soon after this , further improvements were made . Westzyn - thius ( 193 1 ) proved just one year later that λ = ∞, by showing

d lim sup n ≥ 2eγ , (5.2) n→∞ log pn log3 pn/ log4 pn where logν x denotes the ν− fold iterated logarithmic function . Erd ohungarumlaut s ( 1 935 ) succeeded in improving log log log pn to log log pn. More precisely , he proved

sup dn lim ∞ > 0. (5.3) n→ 2 log pn log2 pn/(log3 pn)

Rankin ( 1 938 ) was able to add a further factor log4 pn to it :

sup dn lim ∞ ≥ c0withc0 = 1/3. (5.4) n→ 2 log pn log2 pn log4 pn/(log3 pn)

1 γ The value of c0 = 3 was increased to e by Ricci ( 1952 ) and Rankin 3 68 .. Jaacutenos Pintz \noindentLet us define3 68 independent\quad J random\ ’{ a} nos variables Pintz eta n for all n in I as LetEquation: us define open parenthesis independent 4 period random 6 closing variables parenthesis .. $ eta\eta n = 0 ifn n $ element-slash f o r a l l S sub$ n x to\ thein powerI $ ofas * semicolon while for n in S sub x to the power of * let \ beginEquation:{ a l i g open n ∗} parenthesis 4 period 7 closing parenthesis .. P open parenthesis eta n = 1 closing parenthesis \ tag ∗{$ ( 4( 1 . 962 6 / 63 ) ) .$}\eta n = 0 if n element−s l a s h S ˆ{ ∗ } { x } = A divided by log n subIn 1 comma 979 Erd P openohungarumlaut parenthesis etas n offered = 0 closing a prize parenthesis of USD = 1 1 minus 0 , 0 A for divided the by log n; sub period \end{ a l i g n ∗} proof that ( 5 . 4 ) i s true with c0 = ∞, the highest prize ever offered by Then the truthErd of theohungarumlaut relation s for any mathematical problem . Two improvements Equation: open parenthesis 4 period 8 closing parenthesis .. D to the power of 2 parenleftbigg sub n sub in I \noindent whileof thef o r constant $ n \ inc0 wereS ˆ achieved{ ∗ } { inx the}$ past l e t 28 years . Maier and to the power of sum eta n parenrightbigg ll x divided by log to the power of 2 x γ Pomerance ( 1990 ) showed this with c0 = 1.31...e , while the best implies \ begin { a l i g n ∗}known result i s the following Equation: open parenthesis 4 period 9 closing parenthesis .. bar S sub x to the power of * backslashγ P sub x Theorem ( Pintz ( 1 997 ) ) . ( 5 . 4 ) is true with c0 = 2e . to\ tag the∗{ power$ ( of * 4 bar ll . x divided 7 ) by $} logP( to the power\eta of 2 xn period = 1 ) = \ f r a c { A }{\ log n }5 period{ , } .. LowerP( bounds\eta for largen gaps = between 0 ) primes = period 1 The− \ f r a c { A }{\ log n } { . } \endErd{ a ohungarumlaut l i g n ∗} s endash Rankin Problem The Prime Number Theorem open parenthesis 2 period 1 1 closing parenthesis obviously implies \noindentEquation:Then open theparenthesis truth 5 of period the 1 relation closing parenthesis .. lambda : = limint n right arrow infinity supremum d sub n divided by log n greater equal 1 period \ beginThis{ wasa l i improvedg n ∗} to lambda greater equal 2 by Backlund open parenthesis 1 929 closing parenthesis and lambda\ tag ∗{ greater$ ( equal 4 4 . by Brauer 8 ) comma $} D ˆ{ 2 } ( { n }ˆ{\sum } {\ in I }\eta n ) \ l lZeitz\ f open r a c { parenthesisx }{\ log 1 930ˆ{ closing2 } parenthesisx } period .. Soon after this comma further improvements were made\end{ perioda l i g n Westzyn∗} hyphen thius open parenthesis 193 1 closing parenthesis proved just one year later that lambda = infinity comma by showing\noindent i m p l i e s Equation: open parenthesis 5 period 2 closing parenthesis .. limint n right arrow infinity supremum d sub n divided\ begin by{ a logl i g np∗} n log sub 3 p n slash log sub 4 p n greater equal 2 e to the power of gamma comma \ tagwhere∗{$ log ( sub 4 nu x . denotes 9 the ) $ nu}\ hyphenmid foldS ˆ iterated{ ∗ } logarithmic{ x }\setminus function periodP Erd ˆ{ ohungarumlaut∗ } { x }\ smid open\ l l parenthesis\ f r a c { x 1 935}{\ closinglog ˆ parenthesis{ 2 } x } . \endsucceeded{ a l i g n ∗} in improving log log log p n to log log p n period More precisely comma he proved Equation: open parenthesis 5 period 3 closing parenthesis .. limint n right arrow sub infinity to the power of supremum\ centerline d sub{5 n divided . \quad by logLower p n log bounds sub 2 p for n slash large opengaps parenthesis between log sub primes 3 to the . power The of} 2 p n closing parenthesis greater 0 period \noindentRankin openErd parenthesis $ ohungarumlaut 1 938 closing parenthesis$ s −− Rankin was able Problem to add a further factor log sub 4 p n to it : TheEquation: Prime open Number parenthesis Theorem 5 period ( 2 4 . closing 1 1 )parenthesis obviously .. limint implies n right arrow sub infinity to the power of supremum d sub n divided by log p n log sub 2 p n log sub 4 p n slash open parenthesis log sub 3 to the power of\ begin 2 p n{ closinga l i g n parenthesis∗} greater equal c sub 0 with c sub 0 = 1 slash 3 period \ tagThe∗{ value$ ( of c 5 sub 0 . = 1 1 divided ) by$}\ 3 ..lambda was increased: to = e to the\lim power{ ofn gamma\rightarrow by Ricci .. open\ parenthesisinfty } 1952\sup closing\ f r a parenthesis c { d { n .. and}}{\ Rankinlog n }\geq 1 . \endopen{ a l parenthesis i g n ∗} 1 962 slash 63 closing parenthesis period In 1 979 Erd ohungarumlaut s offered a prize of USD 1 0 comma 0 for the proof that open parenthesis 5 periodThis was4 closing improved parenthesis to i s $ true\lambda \geq 2 $ by Backlund ( 1 929 ) and $ \lambda \geqwith c4 sub $ 0 =by infinity Brauer comma , the highest prize ever offered by Erd ohungarumlaut s for any mathematical Zeitzproblem ( period1 930 Two ) . improvements\quad Soon of after the constant this c ,sub further 0 were achieved improvements in the past were made . Westzyn − thius28 years ( period193 1 .. ) Maier proved and Pomerance just one open year parenthesis later that 1990 closing $ \lambda parenthesis= .. showed\ infty this with, $ c sub by showing 0 = 1 period 3 1 period period period e to the power of gamma comma \ beginwhile{ thea l i bestg n ∗} known result i s the following \ tagTheorem∗{$ ( open 5 parenthesis . 2 Pintz ) $ open}\lim parenthesis{ n 1 997\rightarrow closing parenthesis\ infty closing}\ parenthesissup \ periodf r a c { .. opend { n }}{\ log parenthesisp n \ 5log period{ 43 closing} p parenthesis n / .. is\ log true with{ 4 c sub} 0p = 2 n e to}\ thegeq power of2 gamma e ˆ{\ periodgamma } , \end{ a l i g n ∗}

\noindent where $ \ log {\nu } x $ denotes the $ \nu − $ fold iterated logarithmic function . Erd $ ohungarumlaut $ s ( 1 935 ) succeeded in improving log log log $ p n $ to log log $ p n . $ More precisely , he proved

\ begin { a l i g n ∗} \ tag ∗{$ ( 5 . 3 ) $}\lim { n \rightarrow }ˆ{\sup } {\ infty }\ f r a c { d { n }}{\ log p n \ log { 2 } p n / ( \ log ˆ{ 2 } { 3 } p n ) } > 0 . \end{ a l i g n ∗}

\noindent Rankin ( 1 938 ) was able to add a further factor $ \ log { 4 } p n $ to i t :

\ begin { a l i g n ∗} \ tag ∗{$ ( 5 . 4 ) $}\lim { n \rightarrow }ˆ{\sup } {\ infty }\ f r a c { d { n }}{\ log p n \ log { 2 } p n \ log { 4 } p n / ( \ log ˆ{ 2 } { 3 } p n ) }\geq c { 0 } with c { 0 } = 1 / 3 . \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }The value of $ c { 0 } = \ f r a c { 1 }{ 3 }$ \quad was increased to $ e ˆ{\gamma }$ by R i c c i \quad ( 1952 ) \quad and Rankin

\noindent ( 1 962 / 63 ) .

In 1 979 Erd $ ohungarumlaut $ s offered a prize of USD 1 0 , 0 for the proof that ( 5 . 4 ) i s true with $ c { 0 } = \ infty , $ the highest prize ever offered by Erd $ ohungarumlaut $ s for any mathematical problem . Two improvements of the constant $ c { 0 }$ were achieved in the past 28 years . \quad Maier and Pomerance ( 1990 ) \quad showed this with $ c { 0 } = 1 . 3 1 . . . e ˆ{\gamma } , $ while the best known result i s the following

\noindent Theorem ( Pintz ( 1 997 ) ) . \quad ( 5 . 4 ) \quad is true with $ c { 0 } = 2 e ˆ{\gamma } . $ Landau quoteright s problems on primes .. 369 \ hspaceThe usual∗{\ wayf i l l to}Landau find lower ’ estimation s problems for d on sub primes n i s by showing\quad a369 lower esti hyphen mate for the function J open parenthesis x closing parenthesis = maximum n less or equal x j open parenthesis nThe closing usual parenthesis way to comma find where lower j open estimation parenthesis for n closing $ d parenthesis{ n }$ stands i s for by the showing maximal a lower esti − mateforgap between the consecutive function integers $J prime ( to n x open ) parenthesis = \max Jacobsthal{ n quoteright\ leq s functionx } j closing ( paren- n thesis) ,$ period where $ j ( n )$ stands for themaximal gap between consecutive integers prime to $ nLandau ( $ ’Jacobsthal s problems on primes ’ s function369 ) . The results before ..The 1 970 usual used way Brun to quoteright find lower s .. estimationsieve and estimates for dn ofi de s by Bru showing ij n a lower for the numberesti of integers - mate below for a giventhe function x composedJ of(x) primes = max lessn≤x thanj(n), where j(n) stands for Thea suitably results chosen before y =\ yquad open1 parenthesis 970 used x closing Brun parenthesis’ s \quad periodsieve The and work estimates of Maier and of Pomerance de Bru $ i j $ n the maximal gap between consecutive integers prime to n( Jacobsthal ’ s relied on function ) . fordeep the analytic number results of about integers the distribution below a of given generalized $ x twin $ primes composed in of primes less than a suitably chosenThe $ results y = before y ( 1 970 x used ) Brun . $ ’ sThe sieve work and of Maier estimates and of Pomerance de relied on arithmetic progressionsBru ij periodn for Finallythe number comma of the integers work of below the author a given neededx beyondcomposed the of primes deeptools analytic of Maier and results Pomerance about a new the result distribution about colorings of graphs generalized comma which twin primes in arithmetic progressionsless than a suitably . Finally chosen , they = worky(x). ofThe the work author of Maier needed and Pomerancebeyond the was shown in Pintzrelied open on parenthesis deep analytic 1997 closing results parenthesis about the by distribution probabilistic methods of generalized period twin tools6 period of .. Maier Small gaps and between Pomerance primes a period new Earlierresult results about colorings of graphs , which was shown inprimes Pintz in ( 1997 arithmetic ) by progressions probabilistic . Finally methods , the . work of the author needed Contrary to thebeyond uncertainty the concerning tools of Maier the size and of possible Pomerance large gaps a new result about colorings of between primes comma .. the .. smallest .. possible gaps .. d sub n .. occurring infinitely often \ centerline {6graphs . \quad , whichSmall was gaps shown between in Pintz primes ( 1997 . Earlier ) by probabilistic results methods} . between consecutive primes6 . are generallySmall gaps believed between to be 2 comma primes as predicted . Earlier by results the Twin Prime ConjectureContrary period to the Hence uncertainty comma we concerning try to give upper the size estimates of possible for the large gaps Contrarysize of the to small the gaps uncertainty in terms of p concerning n to the power the of period size Since of thepossible average valuelarge ofgaps d sub n i s log p n between primesbetween , \quad primesthe ,\quad thes m a smallest l l e s t \quad possiblepossible gaps gapsdn\quadoccurring$ d in-{ n }$ by the Prime Numberfinitely Theorem often comma between analogously consecutive to open primes parenthesis are generally 5 period 1 believed closing parenthesis to be 2 we try to\quad give upperoccurring infinitely often between consecutive, as predicted primes by are the generally Twin Prime believed Conjecture to be . Hence 2 , as , we predicted try to give by bounds for the corresponding quantity . the Twin Primeupper Conjecture estimates . for Hence the size, we of try the to small give gaps upper in terms estimates of pn forSince the the Equation: openaverage parenthesis value 6 period of dn 1i closing s log pn parenthesis .. Capital Delta sub 1 : = limint n right arrow infinitysize inf of d the sub n small divided gaps by log in p n terms less or equalof $ 1 periodp n ˆ{ . }$ Since the average value of $ d { n }$ iby s the l o g Prime $ p Number n $ Theorem , analogously to ( 5 . 1 ) we try to give Progress in theupper case of the bounds analogous for the problem corresponding of the lower estimationquantity of lambda was rather quick period One year after the first non hyphen trivial estimate of Backlund \noindentopen parenthesisby the 1 929 Prime closing Number parenthesis Theorem comma , the analogously bound lambda to greater ( 5 equal . 1 4) was we obtained try to semicolon give upper twobounds years after for that the comma corresponding lambda = infinity quantity was dn ∆1 := lim inf ≤ 1. (6.1) reached period This was not the case with this problemn→∞ openlog parenthesispn cf period open parenthesis 6 period 7\ begin closing{ parenthesisa l i g n ∗} closing parenthesis period The first non hyphen \ tagtrivial∗{$ result ( 6 was reachedProgress . 1 80 in) years $ the}\ ago caseDelta : Hardy of the{ and1 analogous} Littlewood: = problem open\lim parenthesis of{ then lower 1926\rightarrow estimation closing parenthesis\ infty } showed\ inf \ f r a c { dof {λ nwas}}{\ ratherlog quickp . Onen }\ yearleq after1 the . first non - trivial estimate of \endopen{ a l parenthesis i g n ∗} Backlund 6 period 2 ( closing 1 929 parenthesis ) , the bound Capitalλ Delta≥ 4 was sub obtained 1 less or equal ; two 2 slashyears 3 after .. on GRH that by .. the circle method, λ = ∞ commawas .. reached where .. . GRH This .. was stands not .. for the the case .. Generalized with this ..problem Riemann ( cf . ( ProgressHypothesis in period the6 . case.. 7 It ) ..) .ofwas The the .. 14 first analogous years non later - trivial that problem .. Rankin result of was.. theopen reached lowerparenthesis 80 estimation years 1 940 ago closing : of Hardy parenthesis .. improved$ \lambda .. open$ parenthesisand was Littlewood rather 6 period quick ( 2 1926 closing . ) One showed parenthesis year after .. to the first non − trivial estimate of Backlund (Capital 1 929 Delta ) , subthe(6.2) 1 bound less ∆1 or≤ equal2/ $3 \lambda 3on slash GRH 5 comma\geq also4 assuming $ was GRH obtained period In ; thetwo same years year after Erd ohun- that garumlaut$ , \lambda s open parenthesisby= the\ infty circle1 940 closing method$ was parenthesis , where succeeded GRH stands for the General- reachedin obtaining . This theized first was unconditional Riemann not the case Hypothesis estimate with this . It problem was ( 14 cf years . ( later 6 . that 7 ) ) Rankin. The first non − trivialEquation: result open( parenthesis 1 was 940 reached) 6 improved period 80 3 closing years ( 6 parenthesis ago. 2 ) : Hardy to ..∆ Capital1 and≤ 3 Delta Littlewood/5, also sub assuming 1 less ( 1 1926minus GRH c) showed . In with .. an .. unspecifiedthe same .. yearbut .. Erd explicitlyohungarumlaut .. calculables .. ( small 1 940 .. positive ) succeeded .. constant in obtaining .. c period the \noindentSpecifically$ comma (first he6 unconditional could . show 2 that ) estimate values\Delta of d sub{ n1 cannot}\ leq accumulate2 too / strongly 3 $ \quad on GRH around the mean value log p n comma since every even value 2 k appears as a dif hyphen \noindentference of twoby \ primesquad pthe 1 comma circle p 2 atmethod most , \quad where \quad GRH \quad stands \quad f o r the \quad Generalized \quad Riemann ∆ < 1 − c (6.3) HypothesisEquation: open . \ parenthesisquad I t \ 6quad periodwas 4 closing\quad parenthesis14 years1 .. Clater S open that parenthesis\quad 2Rankin k closing\quad parenthesis( 1 940 ) \quad improved \quad ( 6 . 2 ) \quad to $ \Delta { 1 }\ leq 3 / 5 , $ also assumingGRH . In the same year Erd N divided by log towith the power an of 2 unspecified N parenlefttp-parenleftbt but explicitly S open parenthesis calculable 2 k closing small parenthesis posi- : = p bar$ ohungarumlaut product k comma $ p greater s ( 1 2 940 parenleftbigg ) succeeded 1 plus p 1 divided by minus 2 to the power of parenrightbigg in obtainingtive the first constant unconditionalc. Specifically estimate , he could show that values of dn cannot parenrightbt-parenrighttpaccumulate too strongly around the mean value log pn, since every even t imes for p 1 = p 2 plus 2 k less or equal N period \ begin { a l i g n ∗}value 2k appears as a dif - ference of two primes p1, p2 at most \ tag ∗{$ ( 6 . 3 ) $}\Delta { 1 } < 1 − c \end{ a l i g n ∗} ) N Y 1 CS(2k) (S(2k) := p | 2(1 + p ) (6.4) log2 N −2 \noindent with \quad an \quad unspecified \quad butk,p>\quad e x p l i c i t l y \quad c a l c u l a b l e \quad small \quad p o s i t i v e \quad constant \quad $ c . $ Specificallyt , imes he could for p1 show = p2 + that 2k ≤ N. values of $ d { n }$ cannot accumulate too strongly around the mean value log $ p n , $ since every even value $ 2 k $ appears as a dif − ference of twoprimes $p 1 , p 2$ atmost

\ begin { a l i g n ∗} \ tag ∗{$ ( 6 . 4 ) $} C S ( 2 k ) \ f r a c { N }{\ log ˆ{ 2 } N } ( S ( 2 k ) : = p \mid \prod { k , p > } 2 ( 1 + p \ f r a c { 1 }{ − 2 }ˆ{ ) } ) \end{ a l i g n ∗}

\noindent timesfor $p 1 = p 2 + 2 k \ leq N . $ 370 .. Jaacutenos Pintz \noindentThis result370 was\ improvedquad J \ by’{ a Ricci} nos open Pintz parenthesis 1954 closing parenthesis to Capital Delta sub 1 less or equal 1 5 slash 1 6 comma later by Wang comma ThisXie result comma Yu was open improved parenthesis by 1 Ricci 965 closing ( 1954 parenthesis ) to to $ Capital\Delta Delta{ sub1 }\ 1 lessleq or equal1 29 5 slash / 32 period1 6 ,$ laterbyWang, Xie ,Yu( 1 965 ) to $ \Delta { 1 }\ leq 29 / 32 . $ A breakthrough370 cameJános whenPintz Bombieri and Davenport open parenthesis 1 966 closing parenthesis refined and made unconditional the method of Hardy and Littlewood by substituting This result was improved by Ricci ( 1954 ) to ∆1 ≤ 15/16, later by Wang A breakthroughthe Bombieri endash came Vinogradov when Bombieri theorem for and the Davenport GRH and obtained ( 1 966 Capital ) refined Delta sub 1and less or equal 1 slash made unconditional, Xie , theYu ( method 1 965 ) ofto Hardy∆1 ≤ 29 and/32. Littlewood by substituting 2 period A breakthrough came when Bombieri and Davenport ( 1 966 ) re- theThey Bombieri also combined−− Vinogradov their method with theorem that of for Erd theohungarumlaut GRH and s obtained to obtain $ \Delta { 1 }\ leq 1 / 2 .fined $ and made unconditional the method of Hardy and Littlewood by Equation: opensubstituting parenthesis 6 period the Bombieri 5 closing parenthesis – Vinogradov .. Capital theorem Delta subfor1 the less GRH or equal and 2 plus ob- square rootThey of 3 also divided combined by 8 = 0 period their 4665 method period with period that period of period Erd $ ohungarumlaut $ s to obtain tained ∆1 ≤ 1/2. They also combined their method with that of Erd Their result wasohungarumlaut further improveds to to obtain \ begin0 period{ a l i4571 g n ∗} period period period .. Pilt quoteright ai open parenthesis 1972 closing parenthesis comma \ tag0 period∗{$ ( 4542 6 period . period 5 period ) $}\ UchiyamaDelta { open1 }\ parenthesis√ leq 1975\ f r a cclosing{ 2 parenthesis + \ sqrt comma{ 3 }}{ Equation:8 } =0 . 4665 . . . . 2 + 3 open parenthesis 6 period 6 closing parenthesis∆ ..1 0≤ sub period= to 0.4665 the power... . of 0 period sub 4425 to(6 the.5) power of\end 4463{ a period l i g n ∗} to the power of period period to the power8 of period period to the power of period Huxley to the power of Huxley open parenthesis to theTheir power result of open was parenthesis further sub improved 1 977 closing to parenthesis comma to the power\ centerline of 1 973{ closingTheir parenthesis result was comma further0 . 0 4571 period improved . 4393. . period Pilt to ’ period ai} ( 1972 period ) ,Huxley open parenthesis 1 984 closing parenthesis comma \ centerline {0 . 4571 . . . \quad Pilt ’ ai ( 1972 ) , } 0 period 4342 .. Fouvry comma Grupp open parenthesis0.4542... 1986 Uchiyama closing(1975) parenthesis, period Finally comma Maier open parenthesis 1988 closing parenthesis succeeded to apply his celebrated matrix \ begin { a l i g n ∗} 0.4463 . . . Huxley (1973), method 0. 4425. . . Huxley ( 1977), (6.6) 0 . 4542 . . . Uchiyama ( 1975 ) , \\\ tag ∗{$ ( 6 . 6 to improve Huxley quoteright s estimate by a factor0.4393 e to... the Huxley power of(1984) minus, gamma comma where gamma is Euler) $} quoteright0 ˆ{ 0 s constant . } { . period}ˆ{ 4463 } { 4425 } . ˆ{ . } . ˆ{ . } . ˆ{ . } Huxley ˆ{ Huxley } ( ˆHe{ obtained( }ˆ{ 1 973 ) , } 0{ .1 4342 977 Fouvry ) , , Grupp}\\ 0 ( 1986 . ) 4393 . . . . Huxley (Equation: 1 984 open ) parenthesisFinally , 6 , period Maier 7 closing( 1988 parenthesis ) succeeded .. Capital to apply Delta sub his 1 celebrated less or equal matrix e to the power of\end minus{ a l gamma i g n ∗} timesmethod 0 period to 4425 improve times times Huxley times ’ =s estimate0 period 2484 by period a factor periode− periodγ , where commaγ is which was the bestEuler result ’ s untilconstant 2005 period. He obtained \ centerlineThe method{ of0 Bombieri . 4342 and\quad DavenportFouvry open , Grupp parenthesis ( 1986 1 966 ) closing . } parenthesis was also suitable to give an estimate for chains of consecutive primes period−γ They showed ∆1 ≤ e · 0.4425 · ·· = 0.2484..., (6.7) FinallyEquation: , Maier open parenthesis ( 1988 ) 6 periodsucceeded 8 closing to parenthesis apply his .. celebratedCapital Delta sub matrix nu = method limint n right arrow infinityto improve inf p n plus Huxleywhich nu minus was’ s p theestimate n divided best resultby by log a p until nfactor less 2005 or equal $ . e nu ˆ{ minus − 1 \gamma divided by} 2, sub $ comma where $ \gamma $ iswhich Euler was ’ improved s constantThe by Huxley method . open of Bombieriparenthesis and 1 968 Davenport slash 69 comma ( 1 966 1 977 ) closing was also parenthesis suitable to to HeEquation: obtained opengive parenthesis 6 period 9 closing parenthesis .. Capital Delta sub nu less or equal nu minus 5 divided by 8 plus Oan parenleftbigg estimate for 1 divided chains by of nu consecutive parenrightbigg primes period . They showed \ beginFinally{ a comma l i g n ∗} similarly to the case nu = 1 comma Maier open parenthesis 1 988 closing parenthesis obtained an\ tag improvement∗{$ ( 6 . 7 ) $}\Delta { 1 }\n +leqν − pne ˆ{ −1 \gamma }\cdot 0 . 4425 \cdot \cdot \cdot∆ν ==0 lim inf p . 2484≤ ν .− . . , (6.8) by a factor e to the power of minus gamma : n→∞ log pn 2 , \end{ a l i g n ∗} Equation: openwhich parenthesis was improved 6 period 1 by0 closing Huxley parenthesis ( 1 968 ../ 69 Capital , 1 977 Delta ) to sub nu less or equal e to the power of minus gamma parenleftbigg nu minus 5 divided by 8 plus O parenleftbigg 1 divided by nu parenrightbigg \noindent which was the best result until 2005 . parenrightbigg period 5 1 The method of Huxley open parenthesis 1 968 slash∆ ≤ 69ν − closing+ O parenthesis( ). also yielded an extension(6.9) of the ν 8 ν result\ hspace ∗{\ f i l l }The method of Bombieri and Davenport ( 1 966 ) was also suitable to give open parenthesisFinally 6 period , similarly 9 closing parenthesis to the case toν small= 1, gapsMaier between ( 1 988 consecutive ) obtained primes an improve- in an arithmetic progression\noindent an estimatement by a for factor chainse−γ : of consecutive primes . They showed of a fixed difference q period \ beginFinally{ a wel i g have n ∗} to mention an important conditional result of5 Heath1 endash ∆ ≤ e−γ (ν − + O( )). (6.10) \ tagBrown∗{$ .. ( open 6 parenthesis . 8 1983 ) closing$}\Delta parenthesisν{\nu period} ..=8 He ..\lim provedν { ..n that ..\rightarrow the .. existence of\ infty Siegel } zeros\ inf .. impliesp \ f .. r a the c { n + \nu − p n }{\ log p n }\ leq \nu − \ f r a c { 1 }{ 2 } { , } \end{ a l i g n ∗} The method of Huxley ( 1 968 / 69 ) also yielded an extension of the Twin Prime Conjectureresult ( comma 6 . 9 ) and to smallmore generally gaps between that every consecutive even number primes can be in an arithmetic expressed in infinitely many ways as the difference of two primes period Naturally \noindent whichprogression was improved of a fixed by Huxley difference ( 1q. 968 / 69 , 1 977 ) to Finally we have to mention an important conditional result of Heath \ begin { a l i g n ∗}– Brown ( 1983 ) . He proved that the existence of \ tag ∗{$ ( 6Siegel . zeros 9 ) $implies}\Delta the{\ Twinnu }\ Primeleq Conjecture\nu − , and \ fmore r a c { generally5 }{ 8 } + O( \ f r a c { that1 }{\ everynu } even). number can be expressed in infinitely many ways as the \end{ a l i g n ∗} difference of two primes . Naturally

\noindent Finally , similarly to the case $ \nu = 1 , $ Maier ( 1 988 ) obtained an improvement by a factor $ e ˆ{ − \gamma } : $

\ begin { a l i g n ∗} \ tag ∗{$ ( 6 . 1 0 ) $}\Delta {\nu }\ leq e ˆ{ − \gamma } ( \nu − \ f r a c { 5 }{ 8 } + O ( \ f r a c { 1 }{\nu } )). \end{ a l i g n ∗}

The method of Huxley ( 1 968 / 69 ) also yielded an extension of the result ( 6 . 9 ) to small gaps between consecutive primes in an arithmetic progression of a fixed difference $ q . $

Finally we have to mention an important conditional result of Heath −− Brown \quad ( 1983 ) . \quad He \quad proved \quad that \quad the \quad existence of Siegel zeros \quad i m p l i e s \quad the Twin Prime Conjecture , and more generally that every even number can be expressed in infinitely many ways as the difference of two primes . Naturally Landau quoteright s problems on primes .. 371 \ hspacemost mathematicians∗{\ f i l l }Landau believe ’ thats problems there are no on Siegel primes zeros\quad period371 open parenthesis The truth of GRH trivially implies this comma for example period closing parenthesis So there i s not much hope to prove \noindentthe Twin Primemost Conjecture mathematicians via Heath believe hyphen Brown that quoteright there are s result no period Siegel However zeros comma . ( The this result truth of means that if we try to prove the Twin Prime Conjecture comma or any weaker \noindentversion ofGRH it as the trivially Small Gap implies Conjecture this or Bounded , for Gap example Conjecture . ) open So there parenthesis i s see not much hope to prove the Twin Prime Conjecture via Heath − Brown ’ s resultLandau ’ s. problems However on primes , this result371 Section 7 closingmost parenthesis mathematicians comma for example believe comma that there then we are are no entitled Siegel to zeros assume . ( that The there truth are no meansSiegel thatzeros period if we In try the light to prove of the results the Twin of the Primenext section Conjecture it i s also interesting , or any weaker version of itof as the Small Gap Conjecture or Bounded Gap Conjecture ( see to note that bothGRH trivially implies this , for example . ) So there i s not much hope to open parenthesis i closing parenthesis .. the existence of Siegel zeros comma .. that .. is comma .. extreme \noindent Sectionprove 7 the ) , Twin for Prime example Conjecture , then we via are Heath entitled - Brown to ’ s assume result . Howeverthat there are no irregularities .. in the, this result means that if we try to prove the Twin Prime Conjecture , Siegeldistribution zeros of primes . In in the some light arithmetic of the progressions results open of parenthesis the next AP section closing parenthesis it i s also comma interesting and to note thator both any weaker version of it as the Small Gap Conjecture or Bounded Gap open parenthesisConjecture ii closing parenthesis ( see improvements of the Bombieri endash Vinogradov theorem comma that i s comma a very \ hspace ∗{\ f i l lSection}( i ) 7\quad ) , forthe example existence , then of we Siegel are entitled zeros to , assume\quad thatthat there\quad arei s , \quad extreme irregularities \quad in the regular distributionnoSiegel of primes zeros in most . In AP the quoteright light of s the results of the next section it i s also imply the Bounded Gap Conjecture period \noindent distributioninteresting of toprimes note that in both some arithmetic progressions ( AP ) , and 7 period .. Small gaps between( i primes ) the period existence Recent ofresults Siegel zeros , that is , extreme In the present .. section comma .. extending the discussion of Section .. 2 comma .. we will \ hspace ∗{\ f i l lirregularities}( ii ) improvements in the of the Bombieri −− Vinogradov theorem , that i s , a very formulate .. in moredistribution .. detail .. of several primes conjectures in some .. arithmetic related to the progressions .. Twin .. Prime ( AP ) , and Conjecture and describe( ii ) some improvements recent results of about the Bombieri them period – All Vinogradov these results theorem , that i s , \noindentwere reachedregular in collaboration distribution with D period of primes A period in Goldston most AP and ’ C s period Y period Y i ld i r i m period imply the Boundeda very Gap Conjecture . The regular distribution of primes in most AP ’ s imply the Bounded Gap results of Section 6 raised the goal to prove the \ centerline {7Conjecture . \quad Small . gaps between primes . Recent results } Small Gap Conjecture7 period . Capital Small Delta gaps sub between 1 = 0 comma primes . Recent results as an approximation to the Twin Prime Conjecture period A much better approx hyphen In the present \quadIn thes epresent c t i o n , \ sectionquad extending , extending the the discussion discussion of of Section Section\quad 2 2 , \quad we w i l l imation would be, to we show will the formulate in more detail several conjectures related formulateBounded Gap\quad Conjecturein more period\quad limint nd eright t a i l arrow\quad infinityseveral inf open conjectures parenthesis p n\ plusquad 1 minusrelated p n closing to the \quad Twin \quad Prime Conjecture andto describe the Twin some recent Prime resultsConjecture about and them describe . All some these recent results results parenthesis less infinityabout period them . All these results were reached in collaboration with D . A wereIt turned reached out open in parenthesiscollaboration as it often with happens D . A in . mathematics Goldston closing and C parenthesis . Y . Y that $ \ inimath order$ to ap ld $ \imath $ r. Goldston$ \imath and$ m C . Y The . Y ı ld ı r ı m . The hyphen results of Section 6 raised the goal to prove the proach the above weaker form of the Twin Prime Conjecture it is worth to \noindent resultsSmall of Gap Section Conjecture 6 raised.∆ the1 = goal 0, to prove the examine the muchas strongeran approximation generalizations to of the it comma Twin formulated Prime Conjecture in a qualitative . A much better form by Dickson open parenthesis 1904 closing parenthesis comma and in a quantitative form by Hardy and \noindent Smallapprox Gap Conjecture- imation would $ . be to\Delta show the{ 1 } = 0 , $ Little hyphen Bounded Gap Conjecture . limn→∞ inf (pn + 1 − pn) < ∞. wood open parenthesisIt turned 1 923 closing out ( parenthesis as it often period happens Let H in = mathematics open brace h sub ) that i closing in order brace subto i = 1 to\noindent the power ofas k anbe a approximation set composed of k to distinct the non Twin hyphen Prime negative Conjecture . A much better approx − imation wouldap be - to proach show the the above weaker form of the Twin Prime Conjecture it is integers commaworth and let to us examine whether the much we have stronger infinitely generalizations many natural num of it hyphen , formulated in bers n such thata all qualitative n plus h sub form i are simultaneously by Dickson ( primes 1904 comma) , and that in ia s quantitative form by \noindentEquation:Bounded open parenthesis Gap Conjecture 7 period 1 closing $ . parenthesis\lim ..{ openn brace\rightarrow n plus h sub i\ closinginfty brace}$ sub i n if = $ ( p nHardy + 1 and− Littlep - n ) < \ infty . $ 1 to the power of k in P to the power of k i period o periodk comma wood ( 1 923 ) . Let H = {hi}i=1 be a set composed of k distinct non - where i period onegative period stands for quoteleft infinitely Case 1 quoteright Case 2 period ItDickson turned open out parenthesis ( as it 1 often 904 closing happens parenthesis in mathematics formulated the conjecture ) that in that order if a trivial to apnecessary− condi proach the aboveintegers weaker , and form let us of examine the Twin whether Prime we Conjecturehave infinitely it many is worth natural to num hyphen - bers n such that all n + hi are simultaneously primes , that i s examinet ion i s true the for much H comma stronger then open generalizations parenthesis 7 period of 1 it closing , formulated parenthesis really in a happens qualitative for infinitely manyform values by Dickson n period The ( 1904 ) , and in a quantitative form by Hardy and Little − k k condition is that the number nu sub p open parenthesis{n + hi}i=1 H closing∈ P parenthesisi.o., of residue classes covered(7.1) by H mod\noindent p wood( 1923 ) . Let $H = \{ h { i }\} ˆ{ k } { i = 1 }$ beshould a set satisfy composed of $ k $ distinct non − negative0 where i . o . stands for ‘ infinitely often open parenthesis 7 period 2 closing parenthesis nu sub p open parenthesis. H closing parenthesis less p .. for every\noindent prime pintegers period , and let us examine whether we have infinitely many natural num − bers $n$ suchthatDickson ( all 1 904 $n ) formulated + h the{ i conjecture}$ are simultaneously that if a trivial necessary primes , that i s condi - t ion i s true for H, then ( 7 . 1 ) really happens for infinitely \ begin { a l i g n ∗}many values n. The condition is that the number νp(H) of residue classes \ tag ∗{$ ( 7covered . 1 by H )mod $}\{p shouldn + satisfy h { i }\} ˆ{ k } { i = 1 }\ in P ˆ{ k } i(7 ..2) oνp(H .) < p , for every prime p. \end{ a l i g n ∗}

\noindent where i . o . stands for ‘ infinitely $\ l e f t . o f t e n \ begin { a l i g n e d } &’ \\ &. \end{ a l i g n e d }\ right . $

Dickson ( 1 904 ) formulated the conjecture that if a trivial necessary condi − t ion i s true for $H , $ then ( 7 . 1 ) really happens for infinitely many values $ n . $ The condition is that the number $ \nu { p } ( H ) $ of residue classes covered by $ H $ mod $ p $ should satisfy

\noindent $ ( 7 . 2 ) \nu { p } (H) < p $ \quad for every prime $ p . $ 372 .. Jaacutenos Pintz \noindentSuch set s372 H are\quad called admissibleJ \ ’{ a} nos period Pintz Hardy and Littlewood open parenthesis 1 923 closing parenthesis examined \noindentalso the frequencySuch set of values s $ n H for $ which are open called parenthesis admissible 7 period . 1 Hardy closing and parenthesis Littlewood i s expected ( 1 to 923 be ) examined truealso period the They frequency of values $ n $ for which ( 7 . 1 ) i s expected to be true . They arrived by an analytic method at the conjecture that arrived by an analytic372 János methodPintz at the conjecture that Equation: openSuch parenthesis set s 7H periodare called 3 closingadmissible parenthesis. .. Hardy sum n and less Littlewood or equal N 1 ( thicksim 1 923 ) S open parenthesis\ begin { a l H i g closing n ∗} parenthesis N divided by log to the power of k N sub comma open brace n plus h sub i \ tag ∗{$ ( 7examined . 3 also ) $}\ thesum frequency{ n of\ valuesleq Nn for} which1 \sim ( 7 . 1S(H) ) i s expected \ f r a c { N }{\ log ˆ{ k } closing brace in P toto the be power true . of They k arrived by an analytic method at the conjecture that N }where{ , S}\\\{ open parenthesisn H+ closing h parenthesis{ i }\}\ .. i s thein so hyphenP ˆ{ calledk } s ingular series comma the convergent non\end hyphen{ a l i g n negative∗} X N product defined by 1 ∼ S(H) (7.3) \noindent where $S ( H )$ \quad i s the sok − called s ingular series , the convergent non − negative Equation: open parenthesis 7 period 4 closingn≤ parenthesisN .. Slog openN , parenthesis H closing parenthesis : = product defined by product p parenleftbigg 1 minus nu sub p open parenthesis H closing parenthesisk divided by p to the power of {n + hi} ∈ P parenrightbigg parenleftbigg 1 minus 1 underbar p to the power of parenrightbigg minus k period \ beginIt i s{ easya l i g to n ∗} showwhere that S(H) i s the so - called s ingular series , the convergent non - \ tagopen∗{$ parenthesis ( 7negative 7 period. 4 5 product closing ) $} parenthesisS defined ( byS H open ) parenthesis : = H closing\prod parenthesis{ p } greater( 0 1 arrowdblleft-− \ f r a c {\nu { p } arrowdblright(H) }{ H ..p i s}ˆ admissible{ ) } ( period 1 − 1 {\underline {\}}{ p }ˆ{ ) } − k . \end{ a l i g n ∗} Heuristic reasoning was provided by Poacutelya open parenthesis) 1 959 closing parenthesis for the validity of Y νp(H) open parenthesis 7 period 3 closing parenthesisS(H) := (1 − (1 − 1 p) − k. (7.4) \noindent It i s easy to show that p on a probabilistic basis open parenthesis at leastp for k = 2 closing parenthesis period If the probabilities $Equation: ( 7 open . parenthesis 5 ) 7 S period ( 6 closing H ) parenthesis> 0 .. n\ plusi f f h subH i $ in P\quad commai n splus admissible h sub j in P . were pairwise independentIt i s easy comma to show we that would(7 obtain.5) S open(H) > parenthesis0 ⇐⇒ H 7 periodi s admissible 3 closing parenthesis . without the\ hspace extra∗{\ factorf i l l } HeuristicHeuristic reasoning reasoning was was provided provided by Pólya by P\ ’ ({ 1o} 959lya ) ( for 1 the 959 validity ) for of the validity of ( 7 . 3 ) S open parenthesis( 7 H. 3 closing ) parenthesis period However comma for a fixed p we have \noindentEquation:on open aon parenthesis probabilistic a probabilistic 7 period basis 7 closing ( ( at parenthesis at least least for ..k for P= parenleftbig 2). $If k the= probabilities p nmid 2 open ) parenthesis . $ If n plusthe h probabilities sub i closing parenthesis comma i = 1 comma 2 comma period period period comma k parenrightbig = 1 minus \ begin { a l i g n ∗} nu sub p open parenthesis H closing parenthesis dividedn + h ∈ by P p, n + h ∈ P (7.6) \ tagin∗{ contrast$ ( to 7 open . parenthesis 6 ) 1 $} minusn 1 + slash h p closingi{ i }\ parenthesisin j P to the , power n of + minus h k comma{ j }\ whichin Pwould be the probability if the events \end{ a l i g n ∗} were pairwise independent , we would obtain ( 7 . 3 ) without the extra p bar n plus h subfactor i commaS(H p). barHowever n plus h, forsub a j .... fixed wouldp we .... havebe .... pairwise independent period .... Hence .... we .... have \noindentto multiplywere the naive pairwise probability independent .... open parenthesis , we would log N closingobtain parenthesis ( 7 . 3 to ) the without power of the minus extra k .... factor $S ( H ) .$ However , forafixed $p$ wehaveνp(H) with the product .... of all the P(p - (n + hi), i = 1, 2, ..., k) = 1 − (7.7) correction factors mod p : this is exactly the quantity S open parenthesisp H closing parenthesis in open \ begin { a l i g n ∗} parenthesis 7 periodin 4 contrast closing parenthesis to (1 − 1 period/p)−k, which would be the probability if the events \ tagSince∗{$ the ( conjecture 7 . about 7 the ) infinitude $} P of ( prime p ..\ knmid hyphen tuples ( n is usually + h { i } ) , i p | n + hi, p | n + hj would be pairwise independent . Hence we =1,2,...,k)=1associated with the names of Hardy and Littlewood and they were the− first \ f r a c {\nu { p } (H ) }{ p } have who examined it in greater detail comma we will define the qualitative−k form of it \end{ a l i g n ∗} to multiply the naive probability ( log N) with the product of all the as correction factors mod p : this is exactly the quantity S(H) in ( 7 . 4 ) . Hardy endash Littlewood endash Dickson .... open parenthesis HLD closing parenthesis .... Conjecture period \noindent in contrastSince the to conjecture $ ( 1 about− the1 infinitude / p of ) prime ˆ{ − kk− tuples} , $ is usually which would be the probability if the events .... If H = open braceassociated h sub i closing with the brace names sub i = of 1 Hardy to the powerand Littlewood of k .... is and they were the first admissible comma then all components n plus h sub i are s imultaneously primes for infin hyphen \noindent $ pwho\ examinedmid n it + in greater h { i detail} , , we p will\mid define then qualitative + h { formj }$ of\ h f i l l would \ h f i l l be \ h f i l l pairwise independent . \ h f i l l Hence \ h f i l l we \ h f i l l have itely many naturalit numbers n period Since this conjecture is extremely deep comma we will formulate an easier ver hyphen \noindent to multiplyas the naive probability \ h f i l l ( l o g $ N ) ˆ{ − k }$ \ h f i l l with the product \ h f i l l o f a l l the sion of it as Hardy – Littlewood – Dickson ( HLD ) Conjecture . If HLD open parenthesis k commak nu closing parenthesis .... Conjecture period .... If H = open brace h sub i H = {hi} is closing\noindent brace subcorrection i = 1 to the factors poweri=1 of kmod .... is $ admissible p : $ comma this .... is then exactly there are the at quantity $ S ( H )$ in(7.4).admissible , then all components n + hi are s imultaneously primes for least nu primesinfin among - open itely brace many n plus natural h sub numbers i closing bracen. for infinitely many values of n period In order to see theSince depth thisof this conjecture we may remark is extremely that if there deep are , any we will formulate an easier ver Sincek comma the nu conjecture greater equal about 2 and any the single infinitude H = open brace of prime h sub i\ closingquad brace$ k sub− i =$ 1 to tuples the power is of usually k associated with- sion the of names it as of Hardy and Littlewood and they were the first for which the above conjecture is true comma k who examinedHLD it in greater(k, ν) Conjecture detail , . weIf willH define= {hi}i=1 theis admissible qualitative , then form there of it then the Boundedare Gap at Conjecture i s obviously also true period \noindent as least ν primes among {n + hi} for infinitely many values of n. In order to see the depth of this we may remark that if there are any k \noindent Hardyk, ν−−≥ 2Littlewoodand any single−− HDickson= {hi}i=1\ hfor f i l which l ( HLD the ) above\ h f i conjecture l l Conjecture is true . , \ h f i l l I f $ H = \{ thenh { thei Bounded}\} ˆ{ Gapk } Conjecture{ i = i 1s obviously}$ \ h f i alsol l i s true .

\noindent admissible , then all components $ n + h { i }$ are s imultaneously primes for infin − itely many natural numbers $ n . $

Since this conjecture is extremely deep , we will formulate an easier ver − s i o n o f i t as

\noindent HLD $ ( k , \nu ) $ \ h f i l l Conjecture . \ h f i l l I f $ H = \{ h { i }\} ˆ{ k } { i = 1 }$ \ h f i l l is admissible , \ h f i l l then there are at

\noindent l e a s t $ \nu $ primes among $ \{ n + h { i }\} $ for infinitely many values of $ n . $

\ hspace ∗{\ f i l l } In order to see the depth of this we may remark that if there are any

\noindent $ k , \nu \geq 2$ andany single $H = \{ h { i }\} ˆ{ k } { i = 1 }$ for which the above conjecture is true ,

\noindent then the Bounded Gap Conjecture i s obviously also true . Landau quoteright s problems on primes .. 373 \ hspaceIn the∗{\ nextf section i l l }Landau we will ’sketch s problems an almost onsuccessful primes attempt\quad to prove373 HLD open parenthesis k comma 2 closing parenthesis for sufficiently large values of k comma which will commaIn the however next commasection yield we the will sketch an almost successful attempt to prove HLDtruth $ of ( the kShort , Gap 2 Conjecture ) $ forperiod sufficiently The method will large also yield values HLD of open $k parenthesis , $ k which comma will 2 , however , yield the closingtruth parenthesis of the Short Gap Conjecture . The method will also yield HLD $ ( k , 2 ) $ Landau ’ s problems on primes 373 for sufficiently largeIn values the nextk greater section k open we parenthesis will sketch delta an closing almost parenthesis successful comma attempt and thus to prove the Bounded Gapfor Conjec sufficiently hyphen large values $ k > k ( \ delta ) , $ and thus the Bounded Gap Conjec − ture , if theHLD Bombieri(k, 2) for−− sufficientlyVinogradov large Theorem values ofcank, bewhich improved will , however as to , include yield the ture comma if thetruth Bombieri of the endash Short Vinogradov Gap Conjecture Theorem . The can be method improved will as also to include yield HLD (k, 2) arithmetic progressions with differences up to X to the power of 1 slash 2 plus delta with a fixed delta greater \noindent arithmeticfor sufficiently progressions large values withk differences > k(δ), and thus up tothe Bounded $ X ˆ{ 1 Gap / Conjec 2 + \ delta }$ 0 period - ture , if the Bombieri – Vinogradov Theorem can be improved as to withWe a will fixed introduce $ the\ delta > 0 . $ We will introduceinclude the Definition periodarithmetic .. We .. say progressions that .. vartheta with .. i differences s .. an .. admissible up to ..X level1/2+δ ..with of the a .. fixed distributionδ > 0. .. of primes if for anyWe A greater will introduce 0 comma the epsilon greater 0 we have \noindentEquation:Definition open parenthesis . \ 7quad periodWe 8\ closingquad parenthesissay that ..\quad sum less$ or\vartheta equal q X$ to the\quad poweri of s var-\quad an \quad a d m i s s i b l e \quad l e v e l \quad o f the \quad distribution \quad o f primes if forDefinition any $ A . > 0We , say\ varepsilon that ϑ i s> an0 $admissible we have level theta minus epsilonof a the from open distribution parenthesis a comma of primes q closingif parenthesis for any A = > 10 to, ε maximum > 0 we have to the power of vextendsingle-vextendsingle-vextendsingle-vextendsingle sub p equiv sub a open parenthesis q closing parenthesis\ begin to the{ a power l i g n ∗} of sum log p minus X divided by phi open parenthesis q closing parenthesis vextendsingle- \ tag ∗{$ ( 7 . 8 ) $}\X sum {\| leqP { q }X X ˆ{\varthetaX } − \ varepsilon } vextendsingle-vextendsingle-vextendsingle lla( suba,q)=1 epsilon commalog p − A X| divided by open parenthesis log(7. X8) closing a ˆ{ ( a , q ) = 1 } max{\maxp≡a(q})ˆ{\arrowvertϕ(q) ε,A} {(logp X)A\equiv }ˆ{\sum } { a parenthesis to the power of A sub periodϑ . ≤q X −ε (By q .. the ) ..}\ Bombierilog endashp − Vinogradov \ f r a c { ..X theorem}{\varphi .. open parenthesis( q Bombieri ) }\arrowvert .. open parenthesis\ l l 1965{\ varepsilon closing,A parenthesis}\ f r a c { commaX }{ ..( A period\ log .. IX period ) .. ˆ{ VinoA }} hyphen{ . } \end{ a l i g n ∗} By the Bombieri – Vinogradov theorem ( Bombieri ( 1965 gradov .. open) parenthesis , A . 1 965 I . closing Vino parenthesis - gradov closing ( 1 parenthesis 965 ) ) .. the the number number varthetaϑ = = 1/ 12 slash 2 is an admissible levelis anperiod admissible .. The Elliott level endash . The Elliott – Halberstam Conjecture ( EH ) By Halberstam\quad the Conjecture\quad Bombieri open parenthesis−− Vinogradov EH closing parenthesis\quad theorem asserts that\quad vartheta( Bombieri = 1 is an\ admissiblequad ( 1965 ) , \quad A. \quad I. \quad Vino − gradov \quad asserts( 1 965 that ) )ϑ \=quad 1 is anthe admissible number level $ \vartheta ( see Elliott= – Halberstam 1 / 2$ ( 1 968 is an admissible level . \quad The E l l i o t t −− level open parenthesis/ 69 see ) ) . HalberstamElliott endash Conjecture Halberstam open ( EH parenthesis ) asserts 1 968 that slash 69 $ closing\vartheta parenthesis= closing 1 $ parenthesis is an admissible period level ( see E l l i o t t −− TheHalberstam following ( results 1 968 are / 69 proved ) ) in . Goldston – Pintz – Y ı ld ı r ı m (200?) : The following resultsTheorem are proved 2 ( in Goldston Goldston endash – Pintz Pintz – Y endashı ld Yı r i ldı m i r i(200?)) m open. parenthesisIf the primes 200 ? closing parenthesis : have an ad - missible level ϑ > 1/2 of distribution , then \ centerlineTheorem 2{ openThe parenthesis following Goldston results endash are Pintz proved endash in Y Goldston i ld i r i m−− openPintz parenthesis−− Y 200 $ \ ?imath closing$ ld $ \imath $for rk $ \ >imath C(ϑ) $m$(any admissible 200k− ?tuple ) contains :$ } at least two primes parenthesis closinginfinitely parenthesis o periodf − t ..en If the . If primesϑ > have0.971 an, then ad hyphen this missible level varthetais true greater for k 1≥ slash6. 2 .. of distribution comma .. then for k greater C open parenthesis vartheta\noindent closingTheorem parenthesis 2 ( .. Goldston any admissible−− Pintz −− Y $ \imath $ ld $ \imath $ r $ \imath $ m$( 200 ?Since ) the ) 6 - .$ tuple \(quadn, n + 4If, n + the 6, n primes+ 10, n + have 12, n + an 16) adi s− admissible , the k hyphen tupleElliott contains – at Halberstam least two primes ( EH infinitely ) Conjecture o f-t en period implies If vartheta greater 0 period 971 comma thenmissible this level $ \vartheta > 1 / 2 $ \quad of distribution , \quad then f o r $ kis true> forC( k greater equal\vartheta 6 period ) $ \quad any admissible $ k − $ tuple contains at least two primes infinitely o $ f−t $ en . I f $ \vartheta Since the 6 hyphen tuple open parenthesis n commalim ninf plusdn 4≤ comma16, n plus 6 comma n plus 1(7 0. comma9) n >plus 10 2 comma . n971 plus 1 ,$6 closing thenthis parenthesis i s admissiblen→∞ comma the Elliott endash Halberstam open parenthesis EH closing parenthesis Conjecture implies \noindent is truethat i for s , pn $+1 k−pn\≤geq16 for6 infinitely . $ many n. Unconditionally we are able Equation: opento parenthesis show the 7 truth period of 9 theclosing Short parenthesis Gap Conjecture .. limint n .rightTheorem arrow infinity 3 ( inf Goldston d sub n less or equal 1 6 comma – Pintz – Y ı ld ı r ı m (200?)). Sincethat thei s comma 6 − tuple$( p n plus 1 minus n p n less , or n+4 equal 1 6 for infinitely , n+6 many n period , Unconditionally n+1 we 0are able, n + 1 2 , n + 1 6 )$ isadmissible,the ∆1 = lim inf(dn/ log pn) = 0. Eto l l i show o t t the−− truthHalberstam of the Short ( EH Gap ) Conjecture Conjecture periodn→ implies∞ Theorem 3 open parenthesis Goldston endash Pintz endash Y i ld i r i m open parenthesis 200 ? closing \ begin { a l i g n ∗} ( For a simplified but self - contained proof of this assertion see Goldston parenthesis closing parenthesis period ı ı ı \ tag ∗{$ ( 7– Motohashi . 9 ) – $}\ Pintzlim – Y{ nld \rrightarrowm ( 2006 ) .\ )infty }\ inf d { n } Capital Delta sub 1 = limint n right arrow inf infinity open parenthesis d sub n slash(k, log2) p n closing parenthesis \ leq 1 6Remark , . We have to mention here that the HLD Conjecture = 0 period k \end{ a l i g n ∗} i s for any given value of in some sense much stronger than it s im- open parenthesismediate For a simplified con - sequence but self hyphen , the contained Bounded proof Gap of Conjecture this assertion . see Goldston Specifically endash Motohashi endash, let Pintz us return endash to Y the i ld ithree r i m different open parenthesis analogues 2006 closing of the parenthesis twin period prime closing parenthesis\noindent thatis $, p n + 1 − p n \ leq 1 6 $ for infinitely many $ n . $ Unconditionallyconjecture , wegiven are in able ( 2 . 14 ) – ( 2 . 1 6 ) , more specially to Remark periodthe .. We formulation have to mention ( 2 .here 1 5 that ) of the Kronecker HLD open . parenthesis Let us denote k comma by 2 closingK the setparenthesis of .. Conjecture i s for all even integers which can be expressed in an infinitude of ways as the \noindentany given valueto show of k in the some truth sense much of the stronger Short than Gap it s Conjecture immediate con . hyphen Theorem 3 ( Goldstondifference−− of twoPintz primes−− Y . $ Although\imath $ we ld believe $ \imath that every$ r even $ \ integerimath $ m sequence comma the BoundedK( Gap Conjecture period .. Specifically comma let us return to the $ ( 200 ?belongs ) ) to . $as formulated in ( 2 . 1 5 ) ) , even the assertion that three different ..K = analogues∅ i s very .. of deep the twin . The .. following prime .. conjecture assertion comma is trivial .. given : in .. open parenthesis 2 period 14 closing parenthesis endash \ [ open\Delta parenthesis{ 1 } 2 period= 1\lim 6 closing{ parenthesisn \rightarrow comma more}\ speciallyinf {\ to theinfty formulation} ( open d parenthesis{ n } / 2\ log period 1p 5 closing n parenthesis ) = 0 of Kronecker . \ ] period Let us denote by K the set of all even integers which can be expressed in an infinitude of ways as the difference of two primes period .. Although we believe that every ( Foreven a integer simplified belongs to but K open self parenthesis− contained as formulated proof in openof this parenthesis assertion 2 period see 1 5 closing Goldston parenthesis−− closingMotohashi parenthesis−− commaPintz −− evenY the $ assertion\imath that$ ld $ \imath $ r $ \imath $ m ( 2006 ) . ) K = varnothing i s very deep period The following assertion is trivial : \noindent Remark . \quad Wehave to mention here that theHLD $ ( k , 2 ) $ \quad Conjecture i s for any given value of $ k $ in some sense much stronger than it s immediate con − sequence , the Bounded Gap Conjecture . \quad Specifically , let us return to the three different \quad analogues \quad o f the twin \quad prime \quad conjecture , \quad given in \quad ( 2 . 14 ) −− ( 2 . 1 6 ) , more specially to the formulation ( 2 . 1 5 ) of Kronecker . Let us denote by $ K $ the set of all even integers which can be expressed in an infinitude of ways as the difference of two primes . \quad Although we believe that every even integer belongs to $K ( $ as formulated in ( 2 . 1 5 ) ) , even the assertion that

\noindent $ K = \ varnothing $ i s very deep . The following assertion is trivial : 374 .. Jaacutenos Pintz \noindentProposition374 1 period\quad K =J \ varnothing’{ a} nos isPintz equivalent to the Bounded Gap Conjecture period On the other hand it i s easy to show \noindentPropositionProposition 2 period .. If the $1 HLD open . parenthesis K = \ kvarnothing comma 2 closing $ parenthesis is equivalent .. Conjecture to the is true Bounded for Gap Conjecture . any given k then \noindent On the other hand it i s easy to show the lower asymptotic374 János densityPintz of K comma hline to the power of d open parenthesis K closing parenthesis greaterProposition c open parenthesis 2 . \quad k closingIftheHLD parenthesis $( comma withk an , explicitly 2 )$ calculable\quad Conjecture is true for any given $ k $ then Proposition 1. K = ∅ is equivalent to the Bounded Gap Conjecture . positive constantOn c open the other parenthesis hand k it closing i s easy parenthesis to show commaProposition .. depending 2 only . onIf k period the HLD The same approach yields more generally for blocks of consecutive primes \noindent the( lowerk, 2) Conjecture asymptotic is density true for any of given $ Kk , then\ r u l e {3em}{0.4 pt } ˆ{ d } ( the following d K) > cthe ( lower k asymptotic ) , $ density with an of explicitlyK, ( calculableK) > c(k), with an explicitly Theorem 4 opencalculable parenthesis Goldston endash Pintz endash Y i ld i r i m open parenthesis 200 ? closing parenthesis closingpositive parenthesis constant period .. Ifc(k the), primesdepending have an only ad hyphen on k. \noindentmissible levelpositive vartheta constantof distribution $c comma ( then k for nu ) greater ,$ equal\quad 2 wedepending have only on $ k . $ The same approach yields more generally for blocks of consecutive Equation: openprimes parenthesis 7 period 1 0 closing parenthesis .. Capital Delta sub nu less or equal parenleftbig square root of nu minus square root of 2 vartheta parenrightbig to the power of 2 sub period \ hspace ∗{\ f i l lthe}The following same approachTheorem yields 4 ( Goldston more generally – Pintz – for Y blocksı ld ı r ofı m consecutive(200?)). primes In particular weIf have the on primes EH have an ad - Equation: openmissible parenthesis level 7 periodϑ of 1 distribution1 closing parenthesis , then .. for Capitalν ≥ Delta2 we sub have 2 = limint n right arrow inf infinity\noindent p n plusthe 2 minus following p n divided by log p n = 0 period Theorem 4 ( Goldston −− Pintz −− Y $ \imath $ ld $ \imath $ r $ \imath $ m It i s also possible to combine Maier quoteright s method with√ the method of Gold hyphen $ ( 200 ? ) ) . $ \quad If the primes√ have2 an ad − ston endash Pintz endash Y i ld i r i m open parenthesis∆ν ≤ ( ν 200− ?2ϑ closing). parenthesis to obtain(7 an.10) improved form of the result open parenthesis 6 period 1 0 closing parenthesis \noindentof Maier commamissibleIn generalized particular level for we arithmetic$ \ havevartheta on progressions EH$ of comma distribution where comma , then extending for the $ re\nu hyphen\geq 2 $sult we s of have Huxley open parenthesis 1 968 slash 69 closing parenthesis we may allow q to tend open parenthesis slowly closing parenthesis to infinity with N period pn + 2 − pn \ begin { a l i g n ∗} ∆2 = lim inf = 0. (7.11) Theorem 5 open parenthesis Goldston endash Pintzn→ ∞ endashlog Ypn i ld i r i m open parenthesis 2006 closing parenthesis\ tag ∗{$ ( closing 7 parenthesis . 1 period 0 ..) Let $}\ nuDelta be an arbitrary{\nu fixed}\ leq ( \ sqrt {\nu } − \ sqrt { 2 \varthetapositive integer} ) and ˆ{It epsilon2 i s} also{ and. possible} A be arbitrary to combine fixed positive Maier numbers ’ s method period withLet q andthe method of \endN{ bea l arbitrary i g n ∗} commaGold - .. ston sufficiently – Pintz large – Yintegersı ld commaı r ı m .. satisfying(200?) to obtain an improved form Equation: openof parenthesis the result 7 period ( 6 . 1 1 20 closing ) of Maier parenthesis , generalized .. q 0 open for parenthesis arithmetic A comma progressions epsilon comma nu\noindent closing parenthesisIn particular, where less q , lessextending we open have parenthesis the on re EH - log sult log s ofN closing Huxley parenthesis ( 1 968 / to 69 the ) we power may of allow A comma N greater N sub 0 openq to parenthesis tend ( slowly A comma ) to epsilon infinity comma with nuN. closing parenthesis comma \ beginand let{ a al i gbe n arbitrary∗}Theorem with open 5 ( parenthesis Goldston a– comma Pintz – q closingY ı ld parenthesisı r ı m ( = 2006 1 period ) ) Let. p subLet 1 to the power\ tag ∗{ of$ comma ( 7 toν thebe . an power 1 arbitrary of 1 prime ) fixed p$}\ subDelta positive2 to the{ power integer2 } of= and comma\limε toand the{ powernA be\ arbitrary ofrightarrow prime period fixed}\ periodinf {\ infty }\ f r a c { p periodn + .. denote 2 − thepositive consecutivep n numbers}{\ log . Letpq nand} = 0 . \endprimes{ a l i gequiv n ∗} aN openbe parenthesis arbitrary , mod q sufficiently closing parenthesis large integers period .. , Then satisfying there exists a b lock of nu plus 1 primes p sub n to the power of comma to the power of prime period period period comma p sub n plus nu to It i s also possible to combine Maier ’ s method with the method of Gold − the power of prime A stonsuch−− thatPintz −− Y $ \imathq0(A,$ ε, ν ld) < q $ <\(logimath log N$) , r N $ >\ Nimath0(A, ε, ν$), m $ ((7 200.12) ? ) $ to obtain an improved form of the result ( 6 . 1 0 ) Equation: open parenthesis 7 period 1 3 closing parenthesis .. p sub0 0 n plus nu to the power of prime minus of Maier , generalizedand let a be for arbitrary arithmetic with (a, progressions q) = 1. Let p1, p ,2, where... denote , extending the consecutive the re − p sub n to the power of prime divided by phi open parenthesis q closing parenthesis log p sub n0 to the0 power of sult s of Huxleyprimes ( 1≡ 968a( mod / 69q). ) weThen may there allow exists $ a q b lock $ to of tendν+1 primes ( slowlypn, ..., ) p ton+ν infinity with prime less e to thesuch power that of minus gamma parenleftbig square root of nu minus 1 parenrightbig to the power of 2$ plus N epsilon . $ comma p sub n to the power of prime in open square bracket N slash 3 comma N closing square bracket period \noindent Theorem 5 ( Goldston0 −− 0Pintz −− Y $ \imath $ ld $ \imath $ r $ \imath $ Consequently comma pn+ν − pn −γ √ 2 0 m ( 2006 ) ) . \quad Let $ \nu $0 be< e an( arbitraryν − 1) + ε, fixed pn ∈ [N/3,N]. (7.13) Equation: open parenthesis 7 periodϕ(q 14) log closingpn parenthesis .. Capital Delta sub nu open parenthesis q comma apositive closing parenthesis integer : = and limint $n right\ varepsilon arrow inf infinity$ and p sub $ n A plus $ nu be to the arbitrary power of prime fixed minus positive p sub n tonumbers . Let the$ q power $ and of primeConsequently divided by phi open, parenthesis q closing parenthesis log p sub n to the power of prime less or equal e to the power of minus gamma parenleftbig square root of nu minus 1 parenrightbig to the power of 2 comma\noindent $N$ be arbitrary , \quad sufficiently0 0 large integers , \quad s a t i s f y i n g pn+ν − pn −γ √ 2 and in particular ∆ν (q, a) := lim inf 0 ≤ e ( ν − 1) , (7.14) n→ ∞ ϕ(q) log pn \ beginEquation:{ a l i g open n ∗} parenthesis 7 period 1 5 closing parenthesis .. Capital Delta sub 1 open parenthesis q comma a\ tag closing∗{$ parenthesis ( 7and = . 0in period particular1 2 ) $} q 0 ( A , \ varepsilon , \nu ) < q The< above( results\ log left open\ log the quantitativeN ) ˆ{ A question} ,N : how can> weN esti{ hyphen0 } (A, \ varepsilon , mate\nu d sub), n as a function of p n from above comma beyond the relation d sub n = o open parenthesis log p\end n closing{ a l i g parenthesis n ∗} ∆1(q, a) = 0. (7.15) infinitely often arrowdblleft-arrowdblrightThe above results left Capital open Delta the quantitative sub 1 = 0 period question We mentioned : how thatcan we are esti not able to\noindent show andlet $a$ bearbitrarywith $( a , q ) = 1 .$ Let - mate d as a function of pn from above , beyond the relation d = o( $ pd sub ˆ{\ nprime less or equal} { C1 infinitely ˆ{ n , }} oftenp comma ˆ{\prime for example} { period2 ˆ{ However, }} comma. . we were . $ able\quad ton substantiallydenote the consecutive primesrefine the $ methods\equiv of proofa of ($mod$q Capital Delta sub 1 = ) 0 so .$ as to yield\quad theThen following there result exists period a b lock of $ \nu + 1$ primes $pˆ{\prime } { n ˆ{ , }} . . . , p ˆ{\prime } { n + \nu }$ such that

\ begin { a l i g n ∗} \ tag ∗{$ ( 7 . 1 3 ) $}\ f r a c { p ˆ{\prime } { n + \nu } − p ˆ{\prime } { n }}{\varphi ( q ) \ log p ˆ{\prime } { n }} < e ˆ{ − \gamma } ( \ sqrt {\nu } − 1 ) ˆ{ 2 } + \ varepsilon , p ˆ{\prime } { n }\ in [ N / 3 , N]. \end{ a l i g n ∗}

\noindent Consequently ,

\ begin { a l i g n ∗} \ tag ∗{$ ( 7 . 14 ) $}\Delta {\nu } ( q , a ) : = \lim { n \rightarrow }\ inf {\ infty }\ f r a c { p ˆ{\prime } { n + \nu } − p ˆ{\prime } { n }}{\varphi ( q ) \ log p ˆ{\prime } { n }}\ leq e ˆ{ − \gamma } ( \ sqrt {\nu } − 1 ) ˆ{ 2 } , \end{ a l i g n ∗}

\noindent and in particular

\ begin { a l i g n ∗} \ tag ∗{$ ( 7 . 1 5 ) $}\Delta { 1 } ( q , a ) = 0 . \end{ a l i g n ∗}

The above results left open the quantitative question : how can we esti − mate $ d { n }$ as a function of $ p n $ from above , beyond the relation $ d { n } = o ($log$p n )$ infinitely often $ \ i f f \Delta { 1 } = 0 . $ We mentioned that we are not able to show $ d { n }\ leq C $ infinitely often , for example . However , we were able to substantially refine the methods of proof of $ \Delta { 1 } = 0 $ so as to yield the following result . log pn) infinitely often ⇐⇒ ∆1 = 0. We mentioned that we are not able to show dn ≤ C infinitely often , for example . However , we were able to substantially refine the methods of proof of ∆1 = 0 so as to yield the following result . Landau quoteright s problems on primes .. 375 LandauTheorem ’ s 6 openproblems parenthesis on primes Goldston\quad endash375 Pintz endash Y i ld i r i m open parenthesis 200 ? ? closing parenthesisTheorem closing 6 ( Goldston parenthesis−− periodPintz −− Y $ \imath $ ld $ \imath $ r $ \imath $ m $(Equation: 200 open ? parenthesis ? ) 7 period) .$ 1 6 closing parenthesis .. limint n right arrow infinity inf d sub n divided by open parenthesis log p n closing parenthesis to the power of 1 slash 2 open parenthesis log log p n closing parenthesis\ begin { a l to i g then ∗} power of 2 less infinity period \ tag ∗{$ ( 7Landau . 1 ’ s problems 6 ) $on}\ primeslim { 375n Theorem\rightarrow 6 ( Goldston\ infty –}\ Pintzinf – Y \ıf r a c { d { n }}{ ( We may remarkld thatı r althoughı m (200??)) the result. Capital Delta sub 1 = 0 was proved 75 years \ loglater thanp the n analogous ) ˆ{ 1 lambda / = 2 infinity} ( open\ log parenthesis\ log cf periodp n open ) parenthesis ˆ{ 2 }} 5 period< \ 2infty closing parenthesis. closing parenthesis comma our present understanding open parenthesis 7 period 1 6 closing parenthesis dn \endof{ smalla l i g gaps n ∗} is much better than of largelim gapsinf open parenthesis cf period< open∞. parenthesis 5 period(7.16) 4 closing n→∞ (log pn)1/2(log log pn)2 parenthesis closing parenthesis period We may remark that although the result $ \Delta { 1 } = 0 $ was proved 75 years 8 period .. The ideasWe of may the proof remark on the that small although gaps the result ∆1 = 0 was proved 75 years laterThe idea than i s based thelater analogouson than a variant the of analogous $ Selberg\lambda quoterightλ = ∞=( cf s sieve\ .infty ( 5 comma . 2 ) )( which , $our appearscf present . ( in 5understanding con . 2 hyphen ) ) , our present understanding ( 7 . 1 6 ) ofnection small with gaps almost( 7is . primes much 1 6 ) inbetter of Selberg small than gaps open ofparenthesisis much large better 199gaps 1 than closing ( cf of parenthesis. large ( 5 gaps . 4 in ) ( the cf ) special. . ( 5 . case 4 ) k = 2 comma and ). \ centerlinefor general k{8 in Heath. \quad hyphen8The . Brown ideas The open ideasof parenthesis the of proof the 1proof 997 on closing the on small parenthesis the small gaps period gaps} We can sketch theThe ideas idea leading i s to based Theorems on a 2 variant and 3 as of follows Selberg period ’ s sieve , which appears in TheStep idea .. 1 iperiod s basedcon Instead - on nection of a the variant special with problemalmost of Selberg primesof twin primes in’ s Selberg sieve we consider ( , 199 which the 1 ) in appears the special in con case− nectiongeneral problem withk almost of= Dickson 2, and primes : for .. we general intry to Selberg showk in thatHeath ( for 199 - any Brown 1 admissible ) in ( 1 the 997 set special ) . case $ k = 2 , $H = and open brace h subWe i can closing sketch brace the sub ideas i = 1 to leading the power to Theorems of k we have 2 infinitely and 3 as many follows k hyphen . tuples of primesfor generalin open brace $ n kStep plus $ h in sub 1 Heath . i closingInstead− braceBrown of the sub ( special i 1 = 9971 to problemthe ) . power of comma twin primes to the power we consider of k that i s the general problem of Dickson : we try to show that for any admissible \ centerlineEquation: open{Weset parenthesis can sketch 8 period the 1 ideas closing leading parenthesis to .. openTheorems brace n 2 plus and h sub3 as i closing follows brace . sub} i = k k 1 to the power of kH in= P{ tohi the}i=1 powerwe have of k i infinitely period o period many k− tuples of primes in {n + hi}i=1, StepThe\ simultaneousquad 1 .that Instead primality of all the components special n plusproblem h sub iof is essentially twin primes equivalent we consider the generalto problemi s of Dickson : \quad we try to show that for any admissible set open parenthesis 8 period 2 closing parenthesis Capital Lambda open parenthesis n semicolon H closing \noindent $ H = \{ h { i }\} ˆ{ k } { i = 1 }$ we have infinitely many parenthesis : = d bar sum P sub H open parenthesis nk closingk parenthesis mu open parenthesis d closing {n + hi} ∈ P i.o. (8.1) parenthesis$ k − $ parenleftbigg tuples of log primes n divided in by d $ parenrightbigg\{ n +i=1 k equal-negationslash h { i }\} 0ˆ{ .. ik period} { oi period = comma 1 ˆ{ , }}$ that where The simultaneous primality of all components n + hi is essentially equiva- k Equation: openlent parenthesis 8 period 3 closing parenthesis .. P sub H open parenthesis n closing parenthesis \noindent i s to (8.2) Λ(n; H) := d | P µ(d)( log n )k 6= 0 i . o . , = product open parenthesis n plus h sub i closingPH(n parenthesis) commad i = 1 since the generalizedwhere von Mangoldt function Capital Lambda open parenthesis n semicolon H closing parenthesis\ begin detects{ a l i g numbers n ∗} P sub H open parenthesis n closing parenthesis \ tagwith∗{$ at (most 8 k different . 1 prime ) factors $}\{ periodn If we + were h able{ toi }\} evaluateˆ for{ k any} { i = 1 }\ in k P ˆfixed{ k tuple} i H the . average o of . Capital Lambda open parenthesis n semicolon H closing parenthesis for n in open \end{ a l i g n ∗} Y parenthesis N comma 2 N closing square bracket commaPH(n) that = is(n comma+ hi), n thicksim N comma this (8.3) could answer our question period Unfortunately this i s not the case comma essentially i = 1 \noindentdue to theThe large simultaneous divisors d of P sub primality H open parenthesis of all n components closing parenthesis $ n period + h { i }$ is essentially equivalent Step .. 2 periodsince .. As .. the usual generalized in .. sieve theory von Mangoldt we try to ..approximate function Λ( ..n the; H) detectordetects numbers \noindent to function CapitalPH Lambda(n) with open at parenthesis most k different n semicolon prime H closing factors parenthesis . If we of were the prime able k to hyphen eval- tuples by$ the ( truncated 8 . divisoruate 2 forsum ) any\Lambda fixed tuple(H the n average ; H of )Λ(n; :H) for =n d∈ (N,\2midN], that\sum is { P { H } (Equation: n ) }\ open,mu n parenthesis∼ N,(this d 8 could period ) answer 4 ( closing $ l o our g parenthesis question $\ f r a c ..{ .n Capital Unfortunately}{ d Lambda} ) sub k this R\ i openne s not parenthesis0 the $ \quad n i . o . , semicolon H closingcase parenthesis , essentially : = d bar due sum to P sub the H large open parenthesisdivisors d nof closingPH(n parenthesis). mu open parenthesis d\noindent closing parenthesiswhere parenleftbiggStep 2 . log R As divided usual by d in parenrightbigg sieve theory k period we dtry less to or equal approximate R In this case wethe can detectorevaluate the function average ofΛ( Capitaln; H) of Lambda the prime sub R openk− tuples parenthesis by the n semicolon truncated H closing parenthesis\ begin { a l .... i g n for∗}divisor n thicksim sum N if R less or equal k N\\\ opentag parenthesis∗{$ ( 8 log N . closing 3 parenthesis ) $} P { toH the} power( of n minus ) B = sub\ 0prod open parenthesis( n k + closing h { i } ), \\ i = 1 parenthesis period However comma we do not obtain any directX arithmeticR informa hyphen \end{ a l i g n ∗} ΛR(n; H) := d | µ(d)(log )k. (8.4) t ion from this period The reason for this is that comma although wed believe that Capital Lambda sub R open parenthesis n semicolon H closing parenthesis PH(n) \noindentand Capitalsince Lambda the open generalized parenthesis n von semicolon Mangoldt H closing function parenthesisd ≤ $ areR\Lambda on average( close n to each ; other H comma) $ detects we have no numbers means to prove $ P { H } ( n ) $ with at mostIn $ this k $ case different we can evaluate prime factorsthe average . If of weΛR( weren; H) ablefor n to∼ evaluateN if R ≤ for any −B0(k) fixed tupleN $H$( log N the) average. However of , we $ \ doLambda not obtain( any n direct ; arithmetic H )$for$n informa \ in ( N , 2- t N ion ] from ,$thatis$, this . The reason for this n is\sim that ,N although , $ we t h believei s that could answerΛ ourR(n; questionH) and Λ(n .; H Unfortunately) are on average this close i to s each not other the ,case we have , essentially no means due to the largeto prove divisors $ d $ of $P { H } ( n ) . $

Step \quad 2 . \quad As \quad usual in \quad sieve theory we try to \quad approximate \quad the detector f u n c t i o n $ \Lambda ( n ; H )$ oftheprime $k − $ tuples by the truncated divisor sum

\ begin { a l i g n ∗} \ tag ∗{$ ( 8 . 4 ) $}\Lambda { R } ( n ; H ) : = d \mid \sum { P { H } ( n ) }\mu ( d ) ( \ log \ f r a c { R }{ d } ) k . \\ d \ leq R \end{ a l i g n ∗}

\noindent In this case we can evaluate the average of $ \Lambda { R } ( n ; H ) $ \ h f i l l f o r $ n \sim N $ i f $ R \ leq $

\noindent $N ($ log $N )ˆ{ − B { 0 } ( k ) } . $ However , we do not obtain any direct arithmetic informa − t ion from this . The reason for this is that , although we believe that $ \Lambda { R } ( n ; H ) $ and $ \Lambda ( n ; H ) $ are on average close to each other , we have no means to prove 376 .. Jaacutenos Pintz \noindentthis period376 .. Further\quad commaJ \ ’{ a ..} thenos .. Pintz sum .. open parenthesis 8 period 4 closing parenthesis .. might .. take negative values .. too comma .. which i s .. a \noindenthandicap fort h us i s period . \quad Further , \quad the \quad sum \quad ( 8 . 4 ) \quad might \quad take negative values \quad too , \quad which i s \quad a handicapStep 3 period for More us generally. we look for non hyphen negative weights open parenthesis depending on the given set H376 commaJ´anos barPintz H bar = k closing parenthesis StepEquation: 3 . More open generallyparenthesis 8 we period look 5 closing for non parenthesis− negative .. a open weights parenthesis ( n depending closing parenthesis on greater the given setthis $H . Further, \mid ,H the\mid sum= ( 8 k . 4 ) ) $ might take negative equal 0 for n thicksimvalues N comma too A , : = which sum n i sthicksim a handicap N a open for parenthesis us . n closing parenthesis greater 0 comma w open parenthesis n closing parenthesis = a open parenthesis n closing parenthesis divided by A sub \ begin { a l i g n ∗} Step 3 . More generally we look for non - negative weights ( depending comma on the given set H, | H |= k) \ tagsuch∗{$ that ( the 8 average . number 5 ) of $ primes} a of ( the form n n ) plus\ hgeq sub i comma0 f o r n \sim N,A :k = Equation:\sum open{ n parenthesis\sim 8 periodN } 6a closing ( parenthesis n ) > .. E open0 parenthesis , w ( N semicolon n ) H = closing\ f r a c { a X a(n) parenthesis( n ) =}{ sumA sum} { w open, } a parenthesis(n) ≥ 0 for n closing n ∼ N,A parenthesis:= chia sub(n) P> open0, wparenthesis(n) = n plus h sub(8.5) i closing \end{ a l i g n ∗} A , parenthesis i = 1 n thicksim N n∼N

should be as largesuch as that possible the comma average where number chi sub of P primes open parenthesis of the form n closingn + h parenthesisi, is the characteristic\noindent functionsuch of that the average number of primes of the form $ n + h { i } , $primes comma Equation: open parenthesis 8 period 7 closing parenthesis .. chi sub P openk parenthesis n closing parenthesis \ begin { a l i g n ∗} XX = 1 if n in P comma chi sub P open parenthesisE(N n; H closing) = parenthesisw(n)χP =(n 0+ otherwisehi) period (8.6) k If\\\ wetag obtain∗{$ ( 8 . 6 ) $} E ( N ; H ) = \sum \sum w ( nEquation: ) \ chi open{ parenthesisP } ( 8 n period + 8 closing h { i parenthesis} ) \\ ..ii E= open = 1n ∼ parenthesis 1N n \ Nsim semicolonN H closing \end{ a l i g n ∗} parenthesis greatershould 1 for N begreater as large N sub as 0 possible , where χP (n) is the characteristic function open parenthesisof or primes at least , for a sequence N = N sub nu right arrow infinity closing parenthesis comma we showed\noindent that thereshould are at be as large as possible , where $ \ chi { P } ( n ) $ is the characteristic function of primesleast two , primes in the given k hyphen tuple period If we obtained χ (n) = 1 if n ∈ P, χ (n) = 0 otherwise. (8.7) Equation: open parenthesis 8 periodP 9 closing parenthesisP .. E open parenthesis N semicolon H closing parenthesis\ begin { a l greater i g n ∗}If k weminus obtain 1 for N = N sub nu right arrow infinity comma \ tagthen∗{$ we ( would 8 prove . the 7 HLD ) prime $}\ kchi hyphen{ P tuple} conjecture( n for ) the = given 1 k hyphen i f n \ in P, \ chituple{ periodP } Some( candidates n ) for = a open 0 parenthesis otherwise n closing . parenthesis are the following : E(N; H) > 1 for N > N0 (8.8) \enda sub{ a l 1 i g open n ∗} parenthesis n closing parenthesis = 1 equal-arrowdblright E open parenthesis N comma H closing parenthesis thicksim( or k divided at least by for log a N sequence open parenthesisN = N Nν right→ arrow ∞), we infinity showed closing that parenthesis there are at \noindentopen parenthesisI f weleast 8 obtain period two primes1 0 closing in theparenthesis given ak− subtuple 2 open . If parenthesis we obtained n closing parenthesis = chi sub P open parenthesis n plus h sub j closing parenthesis with a given j equal-arrowdblright E open parenthesis N semicolon\ begin { a H l i closing g n ∗} parenthesis greater equal 1 E(N; H) > k − 1 for N = Nν → ∞, (8.9) \ tagLine∗{$ 1 a ( sub 38 open . parenthesis 8 ) n $} closingE(N;H) parenthesis = 1 if open brace> n plus1 h sub f o r i closing N brace> subN i{ =0 1 } to\end the{ powera l i g n of∗} k inthen P to we the would power of prove k ? Line the 2 HLD a prime prime sub 3k open− tuple parenthesis conjecture n closing for the parenthesis given k =− Capital Lambda open parenthesistuple . n Some comma candidates H closing parenthesis for a(n) are ? the following : \noindentThe first choice( or of at the least uniform for weights a sequence a sub 1 open $N parenthesis = nN closing{\nu parenthesis}\rightarrow shows the difficulty\ infty of ) , $ we showed that there are at k open parenthesis 8 period 8 closing parenthesisa1(n) = 1 period =⇒ E(N, H) ∼ (N → ∞) log N leastAlthough two a primes sub 2 open in parenthesis the given n closing $ k parenthesis− $ tuple nearly . yields If we open obtained parenthesis 8 period 8 closing parenthesis comma(8 we.10) havea2 no(n) idea = χ howP (n + toh proceedj) with further a given periodj =⇒ E(N; H) ≥ 1 \ beginThe other{ a l i g two n ∗} comma nearly equivalent choices a sub 3 open parenthesis n closing parenthesis and a prime a (n) = 1if{n + h }k ∈ Pk ? sub\ tag 3∗{ open$ ( parenthesis 8 . n closing 9 parenthesis) $} E(N;H) could3 lead to openi parenthesisi=1 > 8 periodk − 9 closing1 parenthesis f o r N = N open{\nu parenthesis}\rightarrow in fact in case\ ofinfty a sub 3 open, parenthesisa03(n) = n Λ(closingn, H)? parenthesis we would obtain E open \end{ a l i g n ∗} parenthesis N comma HThe closing first parenthesis choice of = the k closing uniform parenthesis weights if wea1(n had) shows the difficulty of ( Equation: open8 parenthesis . 8 ) . 8 Although period 1 1 closinga (n) nearly parenthesis yields .. A ( sub 8 . 3 open8 ) , parenthesis we have no N closing idea how parenthesis \noindent then we would prove the2 HLD prime $ k − $ tuple conjecture for the given = sum a sub 3 opento parenthesisproceed further n closing . parenthesisThe other greater two , nearly 0 comma equivalent resp period choices sum a primea (n) and sub 3 open $ k − $ 3 parenthesis n closinga0 parenthesis(n) could lead greater to 0 ( period 8 . 9 n ) thicksim ( in fact N n in thicksim case of N a (n) we would obtain tuple . Some candidates3 for $ a ( n ) $ are the following3 : However commaE( AN, subH) =3 openk) if weparenthesis had N closing parenthesis greater 0 for N = N sub nu right arrow infinity i s trivially equivalent to the HLD \ [ a { 1 } ( n ) = 1 \Longrightarrow E(N,H) \sim \ f r a c { k }{\ log conjecture period So all these obvious choicesX clearly seem to be deadX ends period Nev hyphen A (N) = a (n) > 0, resp. a0 (n) > 0. (8.11) N }ertheless(N comma\rightarrow our final choice will\3infty still originate) \3] from a prime sub 3 open3 parenthesis n closing parenthesis period As we have seen in n ∼ N n ∼ N

\noindent $ (However 8 .,A3( 1N) > 00 for )N a= N{ν 2→} ∞ i( s trivially n ) equivalent = \ chi to the{ P HLD} ( n + h { j } conjecture) $ with . a So given all these $ j obvious\Longrightarrow choices clearly seemE(N;H) to be dead ends . \geq 1 $ Nev - ertheless , our ﬁnal choice will still originate from a03(n). As we have seen in \ [ \ begin { a l i g n e d } a { 3 } ( n ) = 1 i f \{ n + h { i }\} ˆ{ k } { i = 1 }\ in P ˆ{ k } ? \\ a{\prime } { 3 } ( n ) = \Lambda ( n , H ) ? \end{ a l i g n e d }\ ]

The first choice of the uniform weights $ a { 1 } ( n ) $ shows the difficulty of ( 8 . 8 ) . Although $ a { 2 } ( n ) $ nearly yields ( 8 . 8 ) , we have no idea how to proceed further . The other two , nearly equivalent choices $ a { 3 } ( n ) $ and $a{\prime } { 3 } ( n )$ couldleadto(8.9) ( in fact in case of $ a { 3 } ( n )$ wewouldobtain $E ( N , H ) = k )$ ifwehad

\ begin { a l i g n ∗} \ tag ∗{$ ( 8 . 1 1 ) $} A { 3 } ( N ) = \sum a { 3 } ( n ) > 0 , resp . \sum a{\prime } { 3 } ( n ) > 0 . \\ n \sim N n \sim N \end{ a l i g n ∗}

\noindent However $ , A { 3 } (N) > 0 $ f o r $ N = N {\nu } \rightarrow \ infty $ i s trivially equivalent to the HLD conjecture . So all these obvious choices clearly seem to be dead ends . Nev − ertheless , our final choice will still originate from $a{\prime } { 3 } ( n ) . $ Aswehave seen in Landau ’ s problems on primes 377 Step 2 , the truncated version ΛR(n, H) of a03(n) can be evaluated on aver - age and we believe that it i s close to a03(n) in some sense . Since it does not satisfy the non - negativity condition , we can try to square it and examine

2 a4(n) = ΛR(n; H). (8.12)

We can still evaluate their sum A4(N) asymptotically , but , due to the squaring , only for

1/2 −B1 R ≤ N (log N) B1 = B1(k). (8.13) If we restrict R further to

R = N ϑ/2−ε (8.14) where ϑ is an admissible level for the distribution of primes ( cf . ( 7 . 8 ) ) , we can also evaluate

X ϑ − ε0(k) Ei(N, H) := w4(n)χP (n + hi) = (8.15) k . n∼N This yields

E(N, H) = ϑ − ε0(k), lim ε0(k) = 0, (8.16) k→∞ which i s much better than the result k/ log N obtained with the trivial choice a(n) = 1 for all n, but still less than 1 , even supposing EH , that i s , ϑ = 1.( However , as we will see later , this approach could already yield ∆1 = 0 on EH with some additional ideas . ) Step 4 . Since we would be very happy to ﬁnd at least two primes in the k k− tuple {n+hi}i=1 for inﬁnitely many n, there is no compelling ( heuristic ) reason to restrict our attention for the approximation of the detector func - t ion Λ(n, H)( cf . ( 8 . 2 ) ) of prime k− tuples . We can try also to approximate the detector function of those values n for which

i=1 Y ω(PH (n)) = ω (n + hi)) ≤ k + `, 0 ≤ ` < k − 2, (8.17) (k where ` is a free parameter , to be chosen later . This leads to the weight

P R a (n) = Λ2 (n; H, `) := ( µ(d)(log )k + `)2 (8.18) 5 R d|P H(n) d . d ≤ R Landau quoteright s problems on primes .. 377 \ hspaceStep 2∗{\ commaf i l l the}Landau truncated ’ version s problems Capital onLambda primes sub R\quad open parenthesis377 n comma H closing parenthesis of a prime sub 3 open parenthesis n closing parenthesis can be evaluated on aver hyphen \noindentage and weStep believe 2 that , the it i truncated s close to a prime version sub 3 $ open\Lambda parenthesis{ R n closing} ( parenthesis n , in H some ) sense $ o f period$a{\prime Since it} does{ not3 } ( n ) $ can be evaluated on aver − satisfy the non hyphen negativity condition comma we can try to square it and examine \noindent ageThis and we approximation believe that is twice it i as s good close as toa4(n $a) in{\ theprime sense} that{ 3 under} ( the n ) $ Equation: openmild parenthesis restriction 8 period 1 2 closing parenthesis .. a sub 4 open parenthesis n closing parenthesis =in Capital some Lambda sense .sub Since R to the it power does of not 2 open parenthesis n semicolon H closing parenthesis period satisfyWe .. can the .. still non evaluate− negativity their .. sum condition .. A sub 4 open , we parenthesis can try N to closing square parenthesis it and .. examine asymptotically comma .. but comma .. due to the ` = `(k) → ∞, ` = o(k), (8.19) \ beginsquaring{ a l i comma g n ∗} only for \ tagEquation:∗{$ ( open 8 parenthesis . 1 8 2 period ) $1} 3a closing{ 4 parenthesis} ( n .. R )less or = equal\Lambda N to theˆ power{ 2 } of{ 1R slash} 2( openn parenthesis ; H ) log N . closing parenthesis to the power of minus B sub 1 B sub 1 = B sub 1 open parenthesis k closing\end{ a parenthesis l i g n ∗} period If we restrict R further to \noindentEquation:We open\quad parenthesiscan \ 8quad periodstill 14 closing evaluate parenthesis their .. R =\quad N to thesum power\quad of vartheta$ A slash{ 4 2} minus( Nepsilon ) $ \quad asymptotically , \quad but , \quad due to the squaringwhere vartheta , only is an for admissible level for the distribution of primes open parenthesis cf period open parenthesis 7 period 8 closing parenthesis closing parenthesis comma we \ begincan also{ a l evaluatei g n ∗} \ tagEquation:∗{$ ( open 8 parenthesis . 1 8 3 period ) 1 $} 5 closingR \ parenthesisleq N ˆ ..{ E1 sub / i open 2 parenthesis} ( \ Nlog commaN H closing ) ˆ{ − Bparenthesis{ 1 }} : =B sum{ n1 thicksim} = N B w sub{ 1 4 open} ( parenthesis k ) n closing . parenthesis chi sub P open parenthesis n plus\end h{ suba l i g i n closing∗} parenthesis = vartheta minus epsilon sub 0 open parenthesis k closing parenthesis divided by k sub period \ centerlineThis yields { If we restrict $ R $ further to } Equation: open parenthesis 8 period 1 6 closing parenthesis .. E open parenthesis N comma H closing parenthesis\ begin { a l = i g vartheta n ∗} minus epsilon sub 0 open parenthesis k closing parenthesis comma limint k right arrow infinity\ tag ∗{ epsilon$ ( sub 8 0 open . parenthesis 14 ) $} kR closing = parenthesis N ˆ{\ =vartheta 0 comma / 2 − \ varepsilon } \endwhich{ a l i i g s n much∗} better than the result .. k slash log N obtained with the trivial choice a open parenthesis n closing parenthesis = 1 for all n comma but still less than 1 comma even supposing EH\noindent comma thatwhere $ \vartheta $ is an admissible level for the distribution of primes ( cf . ( 7 . 8 ) ) , we cani s comma also evaluate vartheta = 1 period open parenthesis However comma as we will see later comma this approach could already yield \ beginCapital{ a lDelta i g n ∗} sub 1 = 0 on EH with some additional ideas period closing parenthesis \ tagStep∗{$ 4 period ( 8 Since . we would 1 5 be very ) $ happy} E to{ findi } at least( twoN primes , H in the ) : = \sum { n \simk hyphenN } tuplew open{ 4 brace} ( n plus n h sub ) i\ closingchi brace{ P } sub( i = 1 n to the + power h { ofi k for} infinitely) = many\ f r a c n{\vartheta −comma \ varepsilon there is no compelling{ 0 } open( parenthesis k ) }{ heuristick } { closing. } parenthesis \endreason{ a l i tog n restrict∗} our attention for the approximation of the detector func hyphen t ion Capital Lambda open parenthesis n comma H closing parenthesis open parenthesis cf period open parenthesis\noindent 8This period y 2 i closing e l d s parenthesis closing parenthesis of prime k hyphen tuples period We can try also to approximate \ beginthe detector{ a l i g n function∗} of those values n for which \ tagEquation:∗{$ ( open 8 parenthesis . 1 8 6 period ) 1 $ 7} closingE ( parenthesis N , .. omega H ) parenleftbig = \vartheta P sub H open− parenthesis \ varepsilon { 0 } n( closing k parenthesis ) , parenrightbig\lim { k = omega\rightarrow product from\ iinfty = 1 to parenleftbigg}\ varepsilon k open parenthesis{ 0 } ( n plus k h sub ) =i closing 0 parenthesis , parenrightbigg less or equal k plus l comma 0 less or equal l less k minus 2 comma \endwhere{ a l i l g is n a∗} free parameter comma to be chosen later period This leads to the weight Equation: open parenthesis 8 period 1 8 closing parenthesis .. a sub 5 open parenthesis n closing parenthesis =\noindent Capital Lambdawhich sub i R s to much the power better of 2 open than parenthesis the result n semicolon\quad H comma$ k l closing/ $ log parenthesis $N$ : = open obtained with the trivial parenthesischoice sub $a d bar ( P sub n H open ) parenthesis= 1$ n for closing all parenthesis $n to ,$ the powerbut still of sum lessmu open than1 parenthesis , even supposingEH, that di closing s $ parenthesis , \vartheta parenleftbigg= log 1 R divided . by ( $d parenrightbigg However , k as plus we l closing will seeparenthesis later 2 sub, this period approach d could already yield less$ or\Delta equal R { 1 } = 0 $ on EH with some additional ideas . ) This approximation is twice as good as a sub 4 open parenthesis n closing parenthesis in the sense that under the\ hspace ∗{\ f i l l } Step 4 . Since we would be very happy to find at least two primes in the mild restriction \noindentEquation: open$ k parenthesis− $ tu 8 pl period e $ 1\{ 9 closingn parenthesis + h { .. li =}\} l openˆ parenthesis{ k } { ki closing = parenthesis 1 }$ for infinitely many right$ n arrow , $ infinity there comma is nol = ocompelling open parenthesis ( heuristic k closing parenthesis ) comma reason to restrict our attention for the approximation of the detector func − t ion $ \Lambda ( n , H ) ($ cf.(8.2))ofprime $k − $ tuples . We can try also to approximate the detector function of those values $ n $ for which

\ begin { a l i g n ∗} \ tag ∗{$ ( 8 . 1 7 ) $}\omega (P { H } ( n ) ) = \omega \prod ˆ{ i = 1 } { ( k } ( n + h { i } )) \ leq k + \ e l l , 0 \ leq \ e l l < k − 2 , \end{ a l i g n ∗}

\noindent where $ \ e l l $ is a free parameter , to be chosen later . This leads to the weight

\ begin { a l i g n ∗} \ tag ∗{$ ( 8 . 1 8 ) $} a { 5 } ( n ) = \Lambda ˆ{ 2 } { R } ( n ; H , \ e l l ) : = ( { d \mid P }ˆ{\sum } { H ( n ) } \mu ( d ) ( \ log \ f r a c { R }{ d } ) k + \ e l l ) 2 { . }\\ d \ leq R \end{ a l i g n ∗}

\noindent This approximation is twice as good as $ a { 4 } ( n ) $ in the sense that under the mild restriction

\ begin { a l i g n ∗} \ tag ∗{$ ( 8 . 1 9 ) $}\ e l l = \ e l l ( k ) \rightarrow \ infty , \ e l l = o ( k ) , \end{ a l i g n ∗} 378 .. Jaacutenos Pintz \noindentwe obtain378 on average\quad twiceJ \ ’ as{ a many} nos primes Pintz as in Step 4 period Namely comma supposing weopen obtain parenthesis on average 8 period 1 twice 9 closing as parenthesis many primes with these as in weights Step we 4 obtain . Namely in place , of supposing open parenthesis 8 period 1 6 closing parenthesis \noindentEquation:( open 8 .parenthesis 1 9 ) with 8 period these 20 closing weights parenthesis we obtain .. E open in parenthesis place of N ( comma 8 . 1 H 6 closing ) parenthesis = 2 vartheta378 minusJános epsilonPintz subwe 1 open obtain parenthesis on average k comma twice l closing as many parenthesis primes comma as in Step limint 4 k right arrow\ begin infinity{ a l i g epsilon n ∗} sub 1 open parenthesis k comma l closing parenthesis = 0 period \ tag ∗{$ ( 8. Namely . 20 , supposing ) $} E ( N , H ) = 2 \vartheta − \ varepsilon { 1 } If vartheta greater( 8 1. slash1 9 ) 2 with we immediately these weights obtain we open obtain parenthesis in place 8 period of ( 8 8 . closing 1 6 ) parenthesis comma which( k implies , Theorem\ e l l 2 period), How\ hyphenlim { k \rightarrow \ infty }\ varepsilon { 1 } (ever k comma , unconditionally\ e l l ) = we must 0 take . vartheta = 1 slash 2 comma so we have only \end{ a l i g n ∗} Equation: open parenthesis 8 periodE(N, 2 1H closing) = 2ϑ − parenthesisε1(k, `), ..lim Eε1 open(k, `) parenthesis = 0. N comma(8.20) H closing k→∞ parenthesis = 1 minus epsilon sub 1 open parenthesis k comma l closing parenthesis comma I fwhich $ \vartheta i s still weakerIf than>ϑ > the1/2 resultwe / immediately by 2 the $ trivial we immediately choice obtain a sub ( 8 2 . open 8 obtain ) , parenthesis which ( 8 implies . n closing 8 ) Theorem , parenthesis which 2 implies . period Theorem 2 . How − everStep 5, period unconditionally WeHow missed - ever by a we , hairbreadth unconditionally must take an unconditional $ we\vartheta must proof take ofϑ= the= 1 ex/ 12, hyphenso / we have 2 only,$ sowehaveonly i stence of at least two primes in any k hyphen tuple which implies the Bounded \ beginGap{ Conjecturea l i g n ∗} comma but what about the Small Gap Conjecture comma Capital Delta sub 1 = 0 ? .. The E(N, H) = 1 − ε (k, `), (8.21) \ tagfact∗{ that$ ( we obtained 8 . on 2 average 1 already ) $} 1E minus ( epsilon N sub , 1 open1 H parenthesis ) = k 1 comma− l closing \ varepsilon parenthesis { 1 } ( k , \ e l l ), primes i s of crucialwhich i s still weaker than the result by the trivial choice a2(n). \endimportance{ a l i g n ∗} periodStep We missed 5 . We the missed proof of by the a Bounded hairbreadth Gap Conjecture an unconditional but the proof of the ex - primes we foundi stenceduring our of at trial least are twostill there primes period in any If wek can− tuple collect which more thanimplies the Bounded \noindent which i s still weaker than the result by the trivial choice $ a { 2 } epsilon sub 1 openGap parenthesis Conjecture k comma , but l closing what about parenthesis the primes Small on Gap average Conjecture among , ∆ = 0? ( n ) . $ 1 Equation: openThe parenthesis fact that 8 period we 22 obtained closing parenthesison average .. nalready plus h comma1 − ε 11( lessk, `) orprimes equal h i less s of or equal H : = eta log N commacrucial h negationslash-equal importance . We h missed sub i comma the proof of the Bounded Gap Conjecture Stepwhere 5 ... We eta greatermissedbut the 0 by.. iprimes sa .. hairbreadth an we .. arbitrary found during an fixed unconditional .. our parameter trial are comma still proof ..there then of . we the If obtainwe ex can− .. collect Capital Delta i stence of at least two primes in any $ k − $ tuple which implies the Bounded sub 1 = 0 period more than ε1(k, `) primes on average among GapSince Conjecture the weights a , sub but 5 open what parenthesis about then closing Small parenthesis Gap Conjecture .. are not especially $ , sensitive\Delta for the{ primality1 } = open0 parenthesis? $ \quad or The fact that we obtained on average already $ 1 − \ varepsilon { 1 } ( k , prime divisors closing parenthesis of nn plus+ h, h for1 ≤ h equal-negationslashh ≤ H := η log N, h h sub6= hi , comma we can expect(8.22) comma similarly\ e l l to) the $ uniform primes i s of crucial importance .where We missedη the > proof0 i s of an the Bounded arbitrary Gap fixed Conjecture parameter but , the then we distribution a subobtain 1 open parenthesis∆ = 0 n. Since closing the parenthesis weights commaa (n) to obtainare not on especially average sensitive primesEquation: we open found parenthesis during 81 our period trial 23 closing are parenthesis still there .. sum5 . n If thicksim we can N w collect sub 5 open more parenthesis than n $ \ varepsilonfor the{ 1 primality} ( k ( or , prime\ e divisorsl l ) $ ) of primesn + h for onh average6= hi, we amongcan expect , closing parenthesissimilarly chi sub P open to the parenthesis uniform n distribution plus h closing parenthesisa (n), to obtain thicksim on 1 average divided by log N primes for any h negationslash-equal h sub i period This would1 yield in total on average \ beginEquation:{ a l i g open n ∗} parenthesis 8 period 24 closing parenthesis .. thicksim H minus k divided by log N thicksim \ tag ∗{$ ( 8 . 22 ) $} n +X h , 1 \ leq1 h \ leq H : = \eta H divided by log N = eta greater epsilon sub 1 openw parenthesis(n)χ (n + kh comma) ∼ l closing parenthesis if k greater(8.23) k sub \ log N , h \not= h { i } , 5 P log N 0 open parenthesis eta closing parenthesis n∼N \endnew{ a primesl i g n ∗} among n plus h for h in open square bracket 1 comma H closing square bracket comma h negationslash-equalprimes h sub i for period any h 6= hi. This would yield in total on average \noindentThis heuristicwhere works\quad in practice$ \eta too comma> with0 $ a slight\quad changei s period\quad Althoughan \quad openarbitrary parenthesis 8 fixed period \quad parameter , \quad then we obtain \quad 23$ \ closingDelta parenthesis{ 1 } = 0 . $ H − k H Since the weights $ a { 5∼ } (∼ n )= $η >\ εquad1(k, `)areif not k > k especially0(η) sensitive (8.24) for the primality ( or i s not true in the exact form given abovelog N commalog N we can show the similar relation primeopen parenthesis divisors h ) equal-negationslash of $n + h h$ sub i for closing $h parenthesis\ne Equation:h { i open} parenthesis, $ we can 8 period expect 25 , similarly to the uniform closingdistribution parenthesisnew .. $ sum a primes n{ thicksim1 among} N( w subn + 5h open )for parenthesish ,$∈ [1,H to], h n obtainonaverage6= closinghi. parenthesis chi sub P open parenthesis n plus h closing parenthesisThis heuristic thicksim S works open parenthesis in practice H toocup open, with brace a slight h closing change brace . closing Although parenthesis divided\ begin by{ a lS i g open n ∗}( parenthesis 8 . 23 ) H i sclosing not true parenthesis in the times exact 1 formdivided given by log above N open , we parenthesis can show N right the arrow infinity\ tag ∗{ closing$ ( parenthesis 8similar . period 23relation ) $}\sum { n \sim N } w { 5 } ( n ) \ chi { P } (After n this + comma h in ) order\sim to show\ f r Capital a c { 1 Delta}{\ sublog 1 =N 0 comma} it is sufficient to show \endEquation:{ a l i g n ∗} open parenthesis 8 period 26 closing parenthesis .. 1 divided by X sum from h = 1 to X S (h 6= hi) open parenthesis H cup open brace h closing brace closing parenthesis divided by S open parenthesis H closing \noindent primes for anyX $ h \not= h S{(Hi ∪} {h}). $1 This would yield in total on average parenthesis greater equal C open parenthesisw (n)χ H( closingn + h) ∼ parenthesis X greater· X sub(N → 0 open ∞). parenthesis(8.25) H closing 5 P S(H) log N parenthesis n∼N \ begin { a l i g n ∗} \ tag ∗{$ ( 8After . this 24 , in ) order $}\sim to show\ f r a∆ c1{=H 0, it− is sufficientk }{\ log to showN }\sim \ f r a c { H }{\ log N } = \eta > \ varepsilon { 1 } ( k , \ e l l ) i f k > k { 0 } ( \eta ) h=1 1 X S(H ∪ {h}) \end{ a l i g n ∗} ≥ C(H) X > X (H) (8.26) X S(H) 0 X \noindent newprimesamong $n + h$ for $h \ in [ 1 , H ] , h \not= h { i } . $

This heuristic works in practice too , with a slight change . Although ( 8 . 23 ) i s not true in the exact form given above , we can show the similar relation

\ begin { a l i g n ∗} ( h \ne h { i } ) \\\ tag ∗{$ ( 8 . 25 ) $}\sum { n \sim N } w { 5 } ( n ) \ chi { P } ( n + h ) \sim \ f r a c { S(H \cup \{ h \} ) }{ S(H) }\cdot \ f r a c { 1 }{\ log N } (N \rightarrow \ infty ). \end{ a l i g n ∗}

\noindent After this , in order to show $ \Delta { 1 } = 0 , $ it is sufficient to show

\ begin { a l i g n ∗} \ tag ∗{$ ( 8 . 26 ) $}\ f r a c { 1 }{ X }\sum ˆ{ h = 1 } { X }\ f r a c { S (H \cup \{ h \} ) }{ S(H) }\geq C(H)X > X { 0 } (H) \end{ a l i g n ∗} Landau quoteright s problems on primes .. 379 Landaufor at least ’ s one problems choice of on an primesadmissible\quad H comma379 bar H bar = k for any .. k open parenthesis or for a forsequence at least k sub nu one right choice arrow infinity of an closing admissible parenthesis $ Hperiod , Let us\mid chooseH H as \mid = k $ f o r any \quad $ kEquation: ( $ open or f parenthesis o r a 8 period 2 7 closing parenthesis .. P = product p comma h sub i = iP comma i = 1 comma 2 comma period period period comma k period p less or equal 3 k \noindentThen we havesequence for any given $ k even{\ hnu }\rightarrow \ infty ) . $ Letus choose $H$ as Landau ’ s problems on primes 379 for at least one choice of an admissible Equation: openH parenthesis, | H | = 8 periodk for any28 closingk parenthesis( or for a .. nu to the power of prime open parenthesis p closing parenthesissequence : = nu subkν → p open ∞). Let parenthesis us choose H cupH openas brace h closing brace closing parenthesis less or\ begin equal{ nua l subi g n ∗} p open parenthesis H closing parenthesis plus 1 : = nu open parenthesis p closing parenthesis plus\ tag 1∗{ less$ (or equal 8 k . plus 21 comma 7 S ) open $} P parenthesis = \prod H cup openp brace , h h closing{ i brace} = closing iP parenthesis , i =1,2,...,k.Y \\ p \ leq 3 k divided by S open parenthesis H closing parenthesisP = greaterp, hi equal= iP, product i = 1, less2, ..., 2 k. p less or equal 3 k 1(8 minus.27) hline from\end p{ a to l i 2g n divided∗} by open parenthesis 1 minus hline from p to 1 closing parenthesis 2 product p greater 3 k p ≤ 3k 1 minus nu open parenthesis p closing parenthesis plus 1 divided by p divided by open parenthesis 1 minus nu \noindent Then we have for any given even $ h $ open parenthesis pThen closing we parenthesis have for divided any given by p even closingh parenthesis open parenthesis 1 minus hline from p to 1 closing parenthesis Equation: open parenthesis 8 period 29 closing parenthesis .. greater equal product less 2\ begin p less{ ora equall i g n ∗} 3 k parenleftbigg 1 minus 1 divided by open parenthesis p minus 1 closing parenthesis to the \ tag ∗{$ ( 8 . 28 ) $}\0 nu ˆ{\prime } ( p ) : = \nu { p } ( power of 2 to the power of parenrightbiggν (p) :=productνp(H ∪ p { greaterh}) ≤ ν 3p( kH 1) + divided 1 := ν( byp) + 1 1plus≤ k p+ nu 1, open parenthesis(8.28) p H \cup \{ h \} ) \ leq \nu { p } ( H ) + 1 : = \nu closing parenthesis divided by open parenthesis p minus 1 minusp nu open parenthesisν(p)+1 p closing parenthesis closing ( p ) + 1 S$Hleq ∪ {h})k +Y 11 − , \\\ f r aY c { S(H1 − p \cup \{ h \} parenthesis greater equal parenleftbig 2≥ C sub 0 plus o sub k open2 parenthesis 1 closing parenthesis parenrightbig p ν(p) p ) }{ S(H) }\S(geqH) \prod (1{−< { 21)2} p (1 −\ leq )(1 −3 k }$ f r a c { 1 − exponent parenleftbigg minus sum p greater<2p≤3 3k k p k divided byp>3 2k sub slashp 2 to the power1 of parenrightbigg greater\ r u l e { equal3em}{ 20.4 C sub pt } 0 plusˆ{ p o sub} { k open2 }}{ parenthesis( 1 1− closing \ r parenthesis u l e {3em}{ comma0.4 pt } ˆ{ p } { 1 } ) 2 } Y 1 ) Y 1 \prodwhich{ clearlyp > proves3 .. open k }\ parenthesisf r a c { 1 8 period≥− \ 26f r(1 a closing c−{\nu parenthesis( p .. with ) .. + C open 1 }{ parenthesis(8p.29)}}{ ( H 1 − \ f r a c {\nu ( p ) }{ p } ) (( 1p − 1)−2 \ r u l e {3emν(}{p) 0.4 pt } ˆ{ p } { 1 } closing parenthesis = C sub 0 = 0 period 66 period< periodp≤3k period open parenthesisp>3k 1 + p (p cf−1 period−ν(p)) .. open parenthesis ) }\\\ tag ∗{$ ( 8 . 29 ) $}\geq2 \prod { < { 2 } p \ leq 3 k } ( 2 period 6 closing parenthesis closing parenthesis period .. We ) 1 − \ f r a c { 1 }{ ( p − 1 ) ˆ{ 2 }}ˆ{ X) }\k prod { p > 3 k }\ f r a c { 1 }{ 1 remark that it i s also easy to show open≥ (2 parenthesisC0 + ok(1)) 8 exp( period− 26 closingp parenthesis≥ 2C0 + ok for(1), any fixed admissible + p \ f r a c {\nu ( p ) }{ ( p − 1 −2/2 \nu ( p ) ) }}\\\geq H and p>3k (with 2 the C better{ 0 lower} + estimate o { 1 plusk } o sub( k open1 parenthesis) ) \ 1exp closing( parenthesis− \sum period{ Howeverp > comma3 thek } exactp analogue\ f r a c {which ofk }{ clearly2 { / proves 2 }}ˆ ({ 8 .) 26}\ )geq with 2C(H C){ =0 }C0 += o 0.66{...k (}cf .( 1 ),open parenthesis( 82 period . 6 ) )29 . closing We parenthesis remark that i s not it true i s for also all easy admissible to show tuples ( 8 H . comma 26 ) for although any some similar\end{ a estimate l i g n ∗} fixed admissible H and with the better lower estimate 1 + ok(1). However can be given if H, thecup open exact brace analogue h closing of brace ( 8 i s. admissible 29 ) i s notperiod true To have for all an idea admissible about open tuples parenthesis 8\noindent period 26 closingwhichH parenthesis, although clearly for proves some general similar\quad estimate( 8 . 26 can ) be\quad givenwith if H\quad ∪ {h} i s$ admissible C ( H ) =H C we may{ 0 note} .=0 that To the have contribution . an idea 66 ofabout . primes . ( 8with . . 26 p greater ($cf. ) for general 3 k is for\quad anyH we single( may 2 . note 6 ) )that . \ thequad We remarkh at least that 1 plus itcontribution o i sub s k also open of easy parenthesis primes to show with 1 closing (p > 8 parenthesis3k .is 26 for ) any for comma single any as fixed inh at open least admissible parenthesis1 + ok(1) 8, periodas $ H $ 29 closingand parenthesisin period ( 8 . On 29 the ) . other On thehand other comma hand the contribution , the contribution of the of the primes p ≤ 3k withprimes the p less better orto equal the lower 3 left k to - estimate handthe left side hyphen of$ 1 ( hand 8 . + 29 side ) o dependsof open{ k parenthesis} only( on 1 8h periodmodulo ) . 29 $ closing However parenthesis , the exact analogue of depends( 8 . only 29 on) ih modulo s not true for all admissible tuples $ H , $ although some similar estimate canEquation: be given open parenthesisif $H 8\ periodcup 30\{ closingh parenthesis\} $ .. i P s subY admissible 0 = P sub 0 open. To parenthesis have an 3 idea k closing about ( 8 . 26 ) for general P = P (3k) = p, (8.30) parenthesis$ H $ = we product may note p comma that p less the or contribution equal 3 k 0 0 of primes with $ p > 3 k$ is for any single $h$and for a atleastfull period we $1 have the + average o open{ k parenthesis} ( 1 with ) nup sub≤ ,3k p $ open as parenthesis in ( 8 .H closing29 ) .parenthesis Onthe other hand , the contribution of the =primes nu open parenthesis $ p \ leq p closing3 parenthesis k$ tothe closing parenthesis left − hand side of ( 8 . 29 ) depends only on $ h $ modulo open parenthesisand 8 period for a 3 full 1 closing period parenthesis we have 1 the divided average by P = ( towith the powerνp(H) of = 0ν( subp)) p sub product to the power of sum from p to P less or equal sub 3 hline k to the power of 0 = 1 sub p open parenthesis to the power \ begin { a l i g n ∗} νp(H∪{h}) of product less or equal sub 1 minus nu open parenthesis p closing1− parenthesis divided by p closing parenthesis p \ tag ∗{$ ( 8 . 30 ) $P} p P { 0Q } = P { 0 } ( 3 k ) = \prod 3k ν(p) p open parenthesis to the power of open1 parenthesisP 0=1 ≤ sub 1 to(1− the power)(1− of 1 minus) sub minus nu open parenthesis p , \\ p \ leq 3 k p( p 1 (8.31) Q ≤ ) = 1. p closing parenthesis plus 1 divided0 by open3 parenthesisk nuν(p) p(1 to theν(p)+1 power+ ofν p(p) closingν parenthesis(p) divided by P 1− p )( 1−− ( pp) − 1 p) (1− p ) \end{ a l i g n ∗} = p ν ) p p closing parenthesis sub open parenthesis 1 to the power of closing parenthesisp (1 to the power of plus sub minus nu open parenthesis p closing parenthesis divided by 1 from hline to p p closing parenthesis open parenthesis 1 minus\noindent nu openand parenthesisWe for remark a pfull closing that period parenthesis , by we changing have divided the bythe average p above closing argument parenthesis ( with $ to\ slightly thenu power{ p , of} even3 k(H 1 minus nu) sub = p open\nu parenthesisthe( de p - H pendence cup ) open ) $ brace of ok h(1) closingon k bracecan be closing omitted parenthesis and we divided can prove by p divided for any by open parenthesis 1 minusgiven nu openH parenthesis p closing parenthesis divided by p closing parenthesis open parenthesis 1\ begin minus{ hlinea l i g from n ∗} p to 1 closing parenthesis = 1 period (We 8 remark . that 3 comma 1 .. ) by\ changingf r a c { 1 the}{ aboveP { argument= }ˆ{ ..0 slightly}} { commap }ˆ{\ ..sum evenˆ the{ dep } hyphen{ P }} {\prod } andk \ leqpendenceˆ{ 0 of{ o= sub k 1 open} { p parenthesis{ ( }}ˆ{\ 1 closingprod parenthesis}\ leq } on{ k3 can\ ber u omittedl e {3em}{ and0.4 we pt can}{ provek }}ˆ for{ any3 k \ f r a c { 1 − \ f r a c {\nu { p } (H h=1\cup \{ h \} ) }{ p }}{ ( 1 given H 1 X S(H ∪ {h}) − \ f r a c {\nu ( p ) }{ p } lim)inf ( 1 − \ r u l≥ e {13em. }{0.4 pt } ˆ{ p(8}.32){ 1 } and k Equation: open parenthesis 8 periodX 32→∞ closingX parenthesisS(H ..) limint X right arrow infinity inf 1 divided by) }} X sum{ 1 from− h = \ 1f r to a c X{\ S opennu parenthesis( p H ) cup}{ openp }X brace) h ( closing ˆ{ ( brace}ˆ{ closing1 } { parenthesis1 } − divided{ − } by \ f r a c {\nu S( open p parenthesis ) + H closing1 }{\ f parenthesis r a c { ( {\ greaternu equal} p 1 ˆ period{ p } ) }{ p } ) }ˆ{ ) } { ( 1 }ˆ{ + } { − } \ f r a c {\nu ( p ) }{ 1 ˆ{\ r u l e {3em}{0.4 pt }} { p }{ p } ) } ( 1 − \ f r a c {\nu ( p ) }{ p } ) } = 1 . \end{ a l i g n ∗}

\noindent We remark that , \quad by changing the above argument \quad s l i g h t l y , \quad even the de − pendence of $ o { k } ( 1 ) $ on $ k $ can be omitted and we can prove for any given $ H $

\ begin { a l i g n ∗} and k \\\ tag ∗{$ ( 8 . 32 ) $}\lim { X \rightarrow \ infty }\ inf \ f r a c { 1 }{ X }\sum ˆ{ h = 1 } { X }\ f r a c { S(H \cup \{ h \} ) }{ S(H) }\geq 1 . \end{ a l i g n ∗} 3 80 .. Jaacutenos Pintz \noindent9 period ..3 Linear 80 \ equationsquad J \ ’ with{ a} nos almost Pintz primes In Section 3 we mentioned the work of Brun open parenthesis 1 920 closing parenthesis comma according to which\ centerline {9 . \quad Linear equations with almost primes } Problem 4 is true if primes are substituted by almost primes of the form P sub 1 1 Inopen Section parenthesis 3 we that mentioned is comma the numbers work with of at Brun most ( 1 1 1 prime 920 ) factors , according closing parenthesis to which period In the Problem 4 is3 true 80 János if primesPintz are substituted by almost primes of the form $ P { 1 same work Brun 9 . Linear equations with almost primes 1 }showed$ the first significant results concerning the Goldbach and Twin Prime ( that is , numbersIn Section with 3 at we most mentioned 1 1 prime the work factors of Brun ) . ( In 1 920the ) same , according work Brun to Conjectures periodwhich Using Problem his sieve method4 is true he wasif primes able to areprove substituted that by almost primes of showedEquation: the open first parenthesis significant 9 period 1 results closing parenthesis concerning .. P sub the 9 = Goldbach P sub 9 to and the power Twin of Prime prime plus 2 Conjectures .the Using form hisP11 sieve( that method is , numbers he was with able at to most prove 1 1 prime that factors ) . In infinitely often the same work Brun showed the first significant results concerning the and that all even numbers N greater 2 can be expressed as \ begin { a l i g n ∗}Goldbach and Twin Prime Conjectures . Using his sieve method he was Equation: openable parenthesis to prove 9 period that 2 closing parenthesis .. N = P sub 9 plus P sub 9 to the power of period to\ tag the∗{ power$ ( of prime 9 . 1 ) $} P { 9 } = P ˆ{\prime } { 9 } + 2 in fi nitely o f tBrun e n quoteright s sieve was used later to show several results of similar type comma where \end{ a l i g n ∗} 0 by open brace a comma b closing brace weP abbreviate9 = P9 + 2 theinfinitelyoften assertion that every large even integer(9 can.1) be written as a sum of type P sub a plus P sub b period The same method leads to results of type \noindentP sub a =and P suband that b plus that all 2 orall even P even sub numbers a numbers = P sub b $N plus N > 2 2can> d for be every2 expressed $ integer can be d as expressed as open brace 7 comma 7 closing brace .. H period Rademacher open parenthesis 1 924 closing parenthesis comma\ begin { a l i g n ∗} \ tag ∗{$ ( 9 . 2 ) $} N = P { 9 } +0 P ˆ{\prime } { 9 ˆ{ . }} open brace 6 comma 6 closing brace .. T period EstermannN = P9 open+ P9 parenthesis. 1 932 closing parenthesis(9.2) comma \endEquation:{ a l i g n ∗} open parenthesis 9 period 3 closing parenthesis .. open brace to the power of open brace sub 5 comma to the power ofBrun 5 comma ’ s 5sieve to the was power used of 7 later sub closing to show brace several to the power results of ofclosing similar brace type comma , open {a, b} braceBrun 4 ’ comma s sieve 9 closingwhere was brace used by comma laterwe open to abbreviate braceshow 3 several comma the assertion 1 5results closing that brace of every similar open large brace type even2 comma , integer where 366 closing P +P . braceby G $ A\{ sub periodacan to , be the b written power\} of as$ period a we sum Ricci abbreviate of A type perioda open theb parenthesisThe assertion same Buhscarontab method that every leads to the to large results power even of 1 936 integer can be P = P + 2 P = P + 2d d commawritten 1 open as parenthesis aof sum type of sub typea 1 938b $ closing Por{ parenthesisa } b+ comma Pfor{ every tob the} integer power. $ of The 937closing same method parenthesis leads comma to results of type $open P brace{ a 4} comma= 4P closing{ b brace} +{ ..7 W , 2 7 period} $ orH Tartakowski . $Rademacher P { opena } parenthesis (= 1 924 P ) 1{ , 939b } a comma+ 2 1939 d$b closing for every integer parenthesis$ d $ comma { 6 , 6 } T . Estermann ( 1 932 ) , A period A period Buhscarontab open parenthesis 1 940 closing parenthesis period \ centerline {\{ 7 , 7 \}\quad H . Rademacher ( 1 924 ) , } These results were further improved{{5,57}, by{4, the9}, { weighted3, 15}{2, sieve366} ofGA P period.RicciA. Kuhn(1936, 1(937), (9.3) open parenthesis 1 94 1 closing parenthesis5, } who was able to show. in that andBuhstabˇ later1938) works, open parenthesis cf \ centerline {\{ 6 , 6 \}\quad T . Estermann ( 1 932 ) , } period Kuhn open parenthesis 1953{ 4 ,comma 4 } 1W 954 . closing Tartakowski parenthesis ( 1 closing 939 a parenthesis, 1939 b ) , that every even number can be written asA a. sum A . of Buhˇstab( two numbers 1 940having ) . \ beginaltogether{ a l i g at n ∗} most 6 prime factors period \ tag ∗{$ ( 9These . 3 results ) $}\{ wereˆ further{\{} improvedˆ{ 5 , by} { the5 weighted , } 5 sieve ˆ{ 7 of} Pˆ{\} . Kuhn , } {\}} The first result( where 1 94 at 1 least ) who one was of the able summands to show could in be that taken and as later a works ( cf . Kuhn ( \{ prime4 open , brace 9 1 comma\} K, closing\{ brace3 was , proved 1 by 5 A period\}\{ Reacutenyi2 open , parenthesis 366 \} 1 947G{ commaA }ˆ{ . } { . } R i c c i { A } .( { Buh \1953check ,{ 1s } 954tab ) )}ˆ that{ 1 every 936 even , number} 1{ ( can}ˆ{ be937 written ) as , a} sum{ 1 of two 938 ) 1 948 closing parenthesisnumbers using having Linnik quoteright s large sieve period , }open parenthesis Here K was a large unspecified constant period closing parenthesis The main novelty of the \end{ a l i g n ∗} altogether at most 6 prime factors . method The first result where at least one of the summands could be taken as was that he opena parenthesis essentially closing parenthesis .. showed that primes have a positive distribution \ centerlinelevel open parenthesis{\{ 4 , cf4 period\}\quad SectionW. 7 closing Tartakowski parenthesis ( period 1 939 a , 1939 b ) , } prime {1,K} was proved by A . Rényi ( 1 947 , 1 948 ) using Linnik ’ s A few years later Selberg .. open parenthesis 1 950 closing parenthesis .. noted that his sieve can yield open \ centerline {Alarge . A . sieve Buh\ .v{ s (} Heretab (K 1was 940 a ) large . } unspecified constant . ) The main brace 2 comma 3 closingnovelty brace of the method was that he ( essentially ) showed that primes without working out the details period In fact comma the results below were reached by These resultshave were a further positive distributionimproved by level the ( weighted cf . Section sieve 7 ) . of P . Kuhn Selberg quoterightA s sieve few :years later Selberg ( 1 950 ) noted that his sieve can yield { (open 1 94 brace 1 ) 3 who comma was 4 closing able brace to show .. Wang in Yuan that open and parenthesis later works 1 956 (closing cf . parenthesis Kuhn ( 1953 comma , 1 954 ) ) that every even2 , 3 number} without can working be written out the as detailsa sum of. In two fact numbers , the results having below were open parenthesisreached 9 period by 4 closing Selberg parenthesis ’ s sieve .. : open brace 3 comma 3 closing brace .. A period I period Vinogradov open parenthesis 1957 closing parenthesis comma \noindent altogether{ 3 , 4 } atWang most Yuan 6 prime ( 1 956 factors ) , ( 9 . . 4 ) { 3 , 3 } A . I . Vinogradov open brace 2 comma( 1957 3 closing ) , brace .. Wang Yuan open parenthesis 1 958 closing parenthesis period For the detailed proof of Selberg see Selberg open parenthesis 1 99 1 closing parenthesis period \ hspace ∗{\ f i l l }The first result{ 2 where , 3 } atWang least Yuan one ( of 1 958 the ) summands . could be taken as a The next developmentsFor the were detailed based proof on the of method Selberg introduced see Selberg by Reacutenyi ( 1 99 1 period ) . The distribution level of primes was proved to be at least vartheta = 1 slash 3 comma later \noindent primeThe $ \{ next1 developments , K \} were$ based was provedon the method by A . introduced R\ ’{ e} nyi by ( Rényi 1 947 , 1 948 ) using Linnik ’ s large sieve . ( Here $ K $. The was distribution a large unspecified level of primes constant was proved . ) Theto be main at least noveltyϑ = 1/3 of, later the method was that he ( essentially ) \quad showed that primes have a positive distribution level ( cf . Section 7 ) .

A few years later Selberg \quad ( 1 950 ) \quad noted that his sieve can yield \{ 2 , 3 \} without working out the details . In fact , the results below were reached by Selberg ’ s sieve :

\noindent \{ 3 , 4 \}\quad Wang Yuan ( 1 956 ) , ( 9 . 4 ) \quad \{ 3 , 3 \}\quad A . I . Vinogradov ( 1957 ) ,

\ centerline {\{ 2 , 3 \}\quad Wang Yuan ( 1 958 ) . }

\noindent For the detailed proof of Selberg see Selberg ( 1 99 1 ) .

The next developments were based on the method introduced by R\ ’{ e} nyi . The distribution level of primes was proved to be at least $ \vartheta = 1 / 3 , $ l a t e r Landau quoteright s problems on primes .. 38 1 \ hspace3 slash∗{\ 8 byf i Pan l l }Landau Cheng Dong ’ s and problems Barban period on primes These results\quad led38 to the 1 strong ap hyphen proximation of the Goldbach and Twin Prime Conjectures as \noindentopen parenthesis3 / 8 9 by period Pan 5 Cheng closing parenthesisDong and open Barban brace . to These the power results of open led brace to sub the 1 comma strong to apthe − power of 1 comma 4 sub 3 closing brace closing brace Pan A period Cheng A period Buhscarontab to the power of\noindent Dong openproximation parenthesis 1 open of parenthesis the Goldbach 965 sub and closing Twin parenthesis Prime period Conjectures to the power as of 1 962 comma 63 $ ( 9 . 5 ) \{ ˆ{\{}ˆ{ 1 , } { 1Landau , } ’ s4 problems{ 3 on\}}\} primes 38Pan 1 { A } closing parenthesis3 comma / 8 by M Pan period Cheng B period Dong Barban and open Barban parenthesis . These 1 963 results closing led parenthesis to the strong comma .The Cheng celebrated{ A .. Bombieri . Buh endash\check Vinogradov{ s } tab } theoremˆ{ Dong comma} ( .. the 1 level ( vartheta{ 965 } =ˆ{ 1 slash1 962 2 open , } { ). } ap63 - ) ,$ M.B.Barban(1963) , parenthesis cf period {1, Dong 1962, proximation of the Goldbach and Twin Prime Conjectures as (9.5) { 1,43}} P anA.ChengA.Buhstabˇ (1(965 ).63), open parenthesisM 7 . period B . Barban 8 closing ( parenthesis 1 963 ) , closing parenthesis enabled a simpler proof of Buhscarontab quoteright\ hspace ∗{\ s resultf i l l open}The brace celebrated 1 comma 3\ closingquad Bombieri brace comma−− butVinogradov did not yield theorem , \quad the l e v e l $ \vartheta = 1The / celebrated2 ( $ c f Bombieri . – Vinogradov theorem , the level K = 2 period ϑ = 1/2 ( cf . The presently best known results were reached by Jing Rum Chen open parenthesis 1 966 comma \noindent ( 7( . 7 8 . 8) ) ) ) enabled enabled a a simpler simpler proof proof of Buhˇstab’ of Buh\v{ ss result} tab{ ’1 s , result 3 } , but\{ did1 , 3 \} , but did not yield 1 973 closing parenthesisnot yield period He used a form of Kuhn quoteright s weighted sieve comma the switching principle and \ beginthe newly{ a l i g invented n ∗} Bombieri endash Vinogradov theorem to show KChen = quoteright 2 . s Theorem period Every sufficiently largeK = even 2. integer can be written as a \endsum{ a of l i g a n prime∗} and a P sub 2 number period Further comma every even number can be written in infinitely many waysThe as presently the difference best of known a prime results and a P were sub 2reached number by period Jing Rum Chen ( 1 \ hspace10 period∗{\ ..f i Small l l966}The gaps , presently between products best of known two primes results period were The reached by Jing Rum Chen ( 1 966 , Hardy endash Littlewood1 973 ) . He endash used Dickson a form Conjecture of Kuhn ’ for s weighted almost primes sieve , the switching principle \noindentChen quoteright1 973and s theorem) . He open used parenthesis a form seeof Section Kuhn ’ 9 closing s weighted parenthesis sieve showed , the that switchingat least one of principle the and equations the newly invented Bombieri – Vinogradov theorem to show \noindentopen parenthesistheChen newly p comma ’ invented s Theorem p to the Bombieri power . Every of prime−− sufficientlyVinogradov comma p large i in theorem P even closing integer to parenthesis show can be Equation: written open parenthesis 1 0 periodas a 1 closing sum of parenthesis a prime ..and p aplusP 22 =number p to the . powerFurther of prime, every comma even Equation: number open parenthesis\noindent 1Chen 0 periodcan ’ 2 s be closing Theorem written parenthesis . in Every infinitely .. p sufficientlyplus many 2 = p 1 ways p 2 large as the even difference integer of a can prime be and written as a sumhas of .. infinitely a primea many andP2 ..number a solutions $ P . period{ 2 } ..$ The number .. phenomenon . Further .. that , .. every we .. cannot even .. number specify can be written inwhich infinitely one of the many two10 . equations ways Small as open the gaps parenthesis difference between 1 0 period of products a 1 prime closing of and parenthesis two a primes $ P and{ open .2 The} parenthesis$ number 10 . period 2 closing parenthesisHardy has – Littlewood infinitely many – solu Dickson hyphen Conjecture for almost primes \ centerlinet ions open{ parenthesis10 . \quadChen in realitySmall ’ s mosttheorem gaps probably between ( see both Section products closing 9 )parenthesis showed of two that is primes the at most least . significant The one} of the particular case equations \ centerlineof the parity{Hardy problem−− periodLittlewood This i s a heuristic−− Dickson principle Conjecture open parenthesis for observed almost and primes formulated} by Selberg in the 1 950 quoteright s comma see p(p, period p0, pi ∈ 204 P) of Selberg open parenthesis 1 991 closing \ hspace ∗{\ f i l l }Chen ’ s theorem ( see Section 9 ) showed that at least one of the equations parenthesis for example closing parenthesis stating p + 2 = p0, (10.1) that sieve methods cannot differentiate between integers with an even and \ beginodd number{ a l i g n of∗} prime factors period p + 2 = p1p2 (10.2) (Whereas p Chen, p quoteright ˆ{\prime s theorem} , i s relatively p i close\ in to theP) Twin\\\ Primetag Conjecture∗{$ ( comma 1 0 . 1 ) $} p + 2has = infinitely p ˆ{\prime many} solutions, \\\ tag .∗{$ The ( 1 phenomenon 0 . 2 that ) $} p we + 2 it might be surprisingcannot to note specify that .. thewhich seemingly one of much the easier twoequations assertion ( 1 0 . 1 ) and ( 10 =that p for 1 infinitely p many 2 primes p comma p plus 2 has an odd open parenthesis or even closing parenthesis \end{ a l i g n ∗} . 2 ) has infinitely many solu - t ions ( in reality most probably both ) number of is the most significant particular case of the parity problem . This i s a prime factors commaheuristic i s still principle open period ( observed The reason and for formulated this i s the parity by Selberg obstacle openin the parenthesis 1 950 ’ s cf , period \noindentHildebrandhas open\quad parenthesisinfinitely 2002 closing many parenthesis\quad closings o l u t i parenthesis o n s . \quad periodThe \quad phenomenon \quad that \quad we \quad cannot \quad s p e c i f y which one ofsee the p two . 204 equations of Selberg ( ( 1 1 9910 . ) 1 for ) exampleand ( 10 ) stating . 2 ) has that infinitely sieve methods many solu − According to thiscannot comma differentiate until very recently between comma integers problems with involving an even numbers and with odd number of texactly ions (two in distinct reality .. prime most factors probably .. seemed both to be ) as is difficult the most.. as problems significant particular case of the parityprime problem factors . This . i s a heuristic principle ( observed and formulated involving primes periodWhereas We will Chen introduce ’ s theorem the i s relatively close to the Twin Prime Con- byDefinition Selberg period in the .. We 1 call 950 a natural ’ s , number see p an . E 204 sub of 2 hyphen Selberg number ( 1 if 991it i s the) for product example ) stating that sieve methodsjecture cannot, it might differentiate be surprising tobetween note that integers the seemingly with an even much and easier of two distinct primes period p, p + 2 odd number ofassertion prime factors that for . infinitely many primes has an odd ( or even ) number of prime factors , i s still open . The reason for this i s the parity Whereas Chen ’obstacle s theorem ( cf . i s relatively close to the Twin Prime Conjecture , it might be surprising to note that \quad the seemingly much easier assertion that for infinitely many primes $ pHildebrand , p(2002)) +. 2 $ has an odd ( or even ) number of prime factors , i s still open . The reason for this i s the parity obstacle ( cf . According to this , until very recently , problems involving numbers \ begin { a l i g n ∗}with exactly two distinct prime factors seemed to be as difficult Hildebrandas ( problems 2002 involving) ) . primes . We will introduce the \end{ a l i g n ∗} Definition . We call a natural number an E2− number if it i s the product of two distinct primes . According to this , until very recently , problems involving numbers with exactly two distinct \quad prime factors \quad seemed to be as difficult \quad as problems involving primes . We will introduce the

\noindent Definition . \quad We call a natural number an $ E { 2 } − $ number if it i s the product of two distinct primes . 3 82 .. Jaacutenos Pintz \noindentDenoting the3 82 consecutive\quad J E\ sub’{ a 2} nos hyphen Pintz numbers by q 1 less q 2 less period period period .. we may remark that the analogue of Capital Delta sub 1 = 0 open parenthesis cf period open parenthesis 6 period 1 closing parenthesisDenoting closing the consecutive parenthesis comma $ E the{ relation2 } − $ numbersby $q 1 < q 2 < . .open . $ parenthesis\quad we 1 0 period may remark 3 closing parenthesis limint n right arrow infinity inf q n plus 1 minus q n divided that the analogue of $ \Delta { 1 } = 0 ($ cf .(6.1)) ,therelation by log q n slash log3log 82 q nJános = 0Pintz was not known open parenthesis the function in the denominator corresponds to the average Denoting the consecutive E2− numbers by q1 < q2 < ... we may remark \noindentdistance between$ ( E 1sub 2 0 hyphen . numbers 3 ) as the\lim function{ n log p\ inrightarrow case of primes closing\ infty parenthesis}$ i n f period $\ f r a c { q that the analogue of ∆1 = 0( cf . ( 6 . 1 ) ) , the relation nIn + collaboration 1 − withq D nperiod}{\ Goldstonlog qnq S+1 period− nqn W / period\ log Graham\ log commaq C period n } Y= period 0 Y $ i ld i r i m (10.3) limn→∞ inf log qn/ log log qn = 0 open parenthesis Goldwas hyphen not known ( the function in the denominator corresponds to the aver- \noindentston commawas Graham not known comma( Pintz the comma function Y i ld in i r i the m 200 denominator ? comma 200 corresponds ? ? to be abbreviated to the later average by distance betweenage distance $ E { between2 } − E$2− numbersnumbers as as the the function function log logp in case $ p of $ primes in case of primes ) . GGPY ). 200 ? .. and GGPY 200 ? ? closing parenthesis .. we examined various problems period .. The results ob In collaboration with D . Goldston S . W . Graham , C . Y . Y ı ld hyphenIn collaboration with D . Goldston S . W . Graham , C . Y . Y $ \imath $ ld $ \imath $ ı r ı m ( Gold - ston , Graham , Pintz , Y ı ld ı r ı m 200?, 200?? to rtained $ \imath open parenthesis$ m ( Gold cf period− the present and the next section closing parenthesis .. showed that the ston , Grahambe , abbreviated Pintz , Y later $ \imath by GGPY$ ld200? $ \imathand GGPY$ r200??) $ \imathwe$ examined m $ 200 method of various problems . The results ob - tained ( cf . the present and ?Section , 200 8 is suitable ? to ? discuss $ to these be abbreviated problems as well later period byGGPY We are not only able $ 200 ? $ the\quad nextandGGPY section ) $200 showed that ? the? method )$ \quad of Sectionwe examined 8 is suitable various to problems . \quad The results ob − to prove analogousdiscuss results these for E problems sub 2 numbers as well comma . We but are we not obtain only much able stronger to prove analogous tainedones period ( cf .. For . the the difference present of and E sub the 2 .. next numbers section we obtained ) \quad the analogueshowed of that the the method of results for E numbers , but we obtain much stronger ones . For the SectionBounded 8 Gap is Conjecture suitable in to the2 discuss following strong these form problems period as well . We are not only able to prove analogousdifference results of E2 fornumbers $ E we{ obtained2 }$ numbers the analogue , but of we the obtain Bounded much Gap stronger Theorem 7 openConjecture parenthesis in GGPY the following 200 ? closing strong parenthesis form . period limint n right arrow infinity inf open parenthesisones . \ qquad n plusFor 1 minus the q difference n closing parenthesis of $ less E or{ equal2 }$ 6 period\quad numbers we obtained the analogue of the Bounded Gap ConjectureTheorem 7 in( theGGPY following200?). lim strongn→∞ inf form(qn .+ 1 − qn) ≤ 6. Let us consider moreLet generally us consider the appearance more generally of almost the primes appearance in admis hyphenof almost primes in ad- sible k hyphen tuples comma the qualitative analogue of the Hardy endash Littlewood endash Dickson mis - sible k− tuples , the qualitative analogue of the Hardy – Littlewood \noindentopen parenthesisTheorem7 HLD closing (GGPY parenthesis $200 .. conjecture ? ) for . almost\lim .. primes{ n period\rightarrow .. open parenthesis\ infty As we}$ i n f $ ( q– n Dickson + 1 ( HLD− )q conjecture n ) \ forleq almost6 . primes $ . ( As we will will see comma .. Theoremsee , .. Theorem 7 is the 7 is the consequence of such a result – Theorem 9 – consequence ofas such well a result . ) Chenendash ’ Theorem s theorem 9 endash ( Sec as - well t ion period 9 ) closinggives a parenthesis complete Chen answer quoteright ( s theoremLet us open consider parenthesis more Sec generally hyphen the appearance of almost primes in admis − for the qualitative case ) : for any integer d, we have s it b ion l e 9 closing $ k parenthesis− $ tuples gives a , complete the qualitative answer open parenthesis analogue for of the the qualitative Hardy case−− closingLittlewood parenthesis−− Dickson (10.4) p + 2d ∈ P i . o . :( for HLD any integer ) \quad conjecture for2 almost \quad primes . \quad ( As we will see , \quad Theorem \quad 7 i s the This trivially implies that we have infinitely often at least two P num consequenced comma we have of such a result −− Theorem 9 −− as well . ) Chen ’ s theorem2 ( Sec − - bers in any admissible k− tuple . In other words , the HLD (k, 2) topen ion parenthesis 9 ) gives 1 0 a period complete 4 closing answer parenthesis ( for p plus the 2d qualitative in P sub 2 .. i caseperiod ) o period : for any integer Conjecture , formulated in Section 7 i s true for P numbers for any $This d trivially , $ implies we have that we have infinitely often at least two P sub 2 num hyphen2 k ≥ 2. bers in any admissible k hyphen tuple period In other words comma the HLD open parenthesis k comma 2 We will examine the problem of whether we can guarantee for every ν closing\noindent parenthesis$(10.4)p+2d Conjecture comma \ in P { 2 }$ \quad i . o . the existence of νP − numbers ( or at least νP − numbers with a given formulated in Section 7 i s true for P sub2 2 numbers for any k greater equalr 2 period fixed r, independent of ν) in any admissible k− tuple if k i s sufficiently ThisWe triviallywill examine the implies problem that of whether we have we can infinitely guarantee for often every nu at least two $ P { 2 }$ num − large , that i s , k ≥ C (ν). Such a result seems to be unknown for any bersthe existence in any of admissible nu P sub 2 hyphen $ k numbers− $0 open tuple parenthesis . In other or at least words nu P , sub theHLD r hyphen $numbers ( k with , a fixed value of r. given2 ) fixed $ Conjecture , formulated inThe Section strongest 7 i result s true in this for direction $ P { i2 s} due$ numbersto Heath for- Brown any ($ 1 k 997\ )geq r comma independent. of nu closing parenthesis in any admissible k hyphen tuple if k i s sufficiently He large comma2 . that $ showed for any admissible k− tuple {h }k i s comma k greater equal C sub 0 open parenthesis nu closingi i parenthesis=1 period Such a result seems to be Weunknown will for examine any fixed the value problem of r period of whether we can guarantee for every $ \nu $ theThe existence strongest result of in $ this\nu directionP { i s2 due} to− Heath$ numbers hyphen Brown( or at .... least open parenthesis $ \nu 1P 997{ closingr } max 1 ≤ i ≤ kω(n + hi) < C log k, (10.5) −parenthesis$ numbers period with .... He a given fixed $ r , $ independent of $ \nu ) $ in any admissible $ k − $ tu pl e i f showed for anywhere admissibleω(n k) hyphenstands tuple for the open number brace h ofsub distinct i closing prime brace sub divisors i = 1 to of then. power of k $ k $ i s sufficiently large , that Equation: open parenthesisOur method 1 0 period enables 5 closing us to parenthesis prove that .. maximumthe HLD 1( lessk, ν) orConjecture equal i less or i s equal k i s $ , k \geq C { 0 } ( \nu ) . $ Such a result seems to be unknown for any fixed value of omega open parenthesistrue n for plusk h sub > i closing C(ν) parenthesisif primes less are C replaced log k comma by E − numbers ( or $ r . $ 2 where omega openEr− parenthesisnumbers , n for closing any parenthesis fixed r). This stands will for imply the number the existence of distinct of prime infinitely divisors of n period many blocks of ν consecutive E − numbers with a bounded diameter ( \noindent The strongest result in this direction2 i s due to Heath − Brown \ h f i l l ( 1 997 ) . \ h f i l l He Our method enablesdepending us to prove on thatν) for the any HLD given open parenthesisν. We remark k comma that nu in closing the case parenthesis of primes Conjecture i s true we cannot prove ∆ = 0 if ν > 2( cf . ( 6 . 8 ) ) and Theorem 5 ) . In the \noindent showed for any admissibleν $ k − $ tu pl e $ \{ h { i }\} ˆ{ k } { i for k greater Ccase open parenthesisν = 2 we were nu closing able parenthesisto show .. if primes are replaced by E sub 2 hyphen numbers =.. open 1 parenthesis}$ or E sub r hyphen numbers comma for any fixed r closing parenthesis period This will imply the existence of infinitely many blocks of nu \ beginconsecutive{ a l i g n E∗} sub 2 hyphen numbers with a bounded diameter open parenthesis depending on nu closing parenthesis\ tag ∗{$ ( for any 1 0 . 5 ) $}\max{ 1 }\ leq i \ leq k \omega ( n +given h nu{ periodi } ) We remark< C that\ inlog the casek of primes , we cannot prove Capital Delta sub nu = 0 if \endnu{ greatera l i g n ∗} 2 open parenthesis cf period open parenthesis 6 period 8 closing parenthesis closing parenthesis and Theorem 5 closing parenthesis period In the case nu = 2 we were able to show \noindent where $ \omega ( n ) $ stands for the number of distinct prime divisors of $ n . $

Our method enables us to prove that the HLD $ ( k , \nu ) $ Conjecture i s true f o r $ k > C( \nu ) $ \quad if primes are replaced by $ E { 2 } − $ numbers \quad ( or $ E { r } − $ numbers , for any fixed $ r ) . $ This will imply the existence of infinitely many blocks of $ \nu $ consecutive $ E { 2 } − $ numbers with a bounded diameter ( depending on $ \nu ) $ f o r any given $ \nu . $ We remark that in the case of primes we cannot prove $ \Delta {\nu } = 0 $ i f $ \nu > 2 ($ cf . (6 . 8) )andTheorem5) . In the case $ \nu = 2 $ we were able to show Landau ’ s problems on primes 383 ∆2 = 0( but not bounded gaps pn + 2 − pn i . o . ) only under the very deep Elliott – Halberstam Conjecture ( cf . ( 7 . 1 0 ) – ( 7 . 1 1 ) ) . We can prove the result in a more general form . Let

Li(x) = aix + bi (1 ≤ i ≤ k) ai, bi ∈ Z, ai > 0 (10.6) be an admissible k− tuple of distinct linear forms , that i s , we suppose that Q Li(x) has no ﬁxed prime divisor . Theorem 8 ( GGPY 200?). Let D be any constant ,Li(x) as above . Then there are ν + 1 forms among them , which take simultaneously E2− number values with bo th prime factors greater than D if

−γ ν k ≥ C1(ν) = (e + o(1))e . (10.7) −γ ν lim inf(qn + ν − qn) ≤ C2(ν) = (e + o(1))νe . Corollary. n→∞ Theorem 8 does not specify how many forms we need in order to ﬁnd among them ν +1E2− number for given small values of ν. The most impor - tant particular case i s the following , which clearly implies Theorem 7 . Theorem 9 ( GGPY 2 ) . Let Li(x) be an admissible triplet of linear forms . Among these there exist two forms Li,Lj such that Li(n) and Lj(n) are both E2− numbers inﬁnitely o f − t en . 1 1 . Small gaps between almost primes and some conjectures of Erd ohungarumlaut s on consecutive integers

Erd ohungarumlaut s had many favourite problems on consecutive integers ( see the work of Hildebrand ( 2002 ) ) . Let us denote by Ω(n) and ω(n), resp . , the number of prime factors of n with and without multiplicity . Let d(n) stand for the divisor - function . The celebrated Erd ohungarumlaut s – Mirsky Conjec- ture ( 1 952 ) refers to the divisor function , the others to consecutive values of Ω and ω. Conjecture C 1 ( Erd ohungarumlaut s – Mirsky ). d(x) = d(x + 1) infinitely o f − t en . Conjecture C 2 ( Erd ohungarumlaut s (1983)). Ω(x) = Ω(x + 1) infinitely o f − t en . Conjecture C 3 ( Erd ohungarumlaut s (1983)). ω(x) = ω(x+1) infinitely o f − t en . These conjectures would follow from the Twin Prime Conjecture with x + 1 replaced by x + 2 in the following sharp form : Landau quoteright s problems on primes .. 383 \ hspaceCapital∗{\ Deltaf i l sub l }Landau 2 = 0 open ’ parenthesiss problems but on not primes bounded\ gapsquad p n383 plus 2 minus p n i period o period closing parenthesis only under the very deep \noindentElliott endash$ \ HalberstamDelta { Conjecture2 } = open 0 parenthesis ($ butnotboundedgaps cf period open parenthesis 7 $p period 1 n 0 closing + paren- 2 − thesisp n endash $ i open . o parenthesis . ) only 7 under period 1 the 1 closing very parenthesis deep closing parenthesis period We can prove the result in a more general form period Let \noindentEquation:E open l l i o parenthesis t t −− Halberstam 1 0 period 6 Conjectureclosing parenthesis ( cf .. L. sub ( 7 i open . 1 parenthesis 0 ) −− ( x 7 closing . 1 1 parenthesis ) ) . = a sub i x plus b sub id open(x) = parenthesisd(x + 2) = 1 2, lessΩ( orx) equal = Ω(x i less+ 2) or = equalω(x) = kω closing(x + 2) parenthesis = 1 i.o. a sub(11 i. comma1) b sub\ centerline i in Z comma{We a sub can i greater prove 0 the result in a more general form . Let } be an admissible kIn hyphen their tuple original of distinct forms linear Conjectures forms comma 1 1 –that 1 1 i woulds comma follow we suppose from thatan ana- \ beginproduct{ a l L i g sub n ∗} ilogue open parenthesis of the conjecture x closing parenthesis of Sophie hasGermain no fixed .prime Sophie divisor Germain period conjectured \ tagTheorem∗{$ ( 8 open 1that parenthesis 02p .+ 1 GGPY∈ 6 P for ) 200 $ infinitely} ?L closing{ i many parenthesis} ( primes x period )p. ..It = Let i s D apossible ..{ bei any} to constantx show + by comma b { L i } sub( i open1 \ parenthesisleq Cheni x closing ’\ sleq method parenthesisk that ) .. , asa similarly above{ i period} to, ( .. 10 Then b . 1{ )i –}\ ( 1 0in . 2 )Z , either , a { i } > there0 are nu plus 1 forms among them comma .. which take simultaneously E sub 2 hyphen number \endvalues{ a l i with g n ∗} bo th prime factors greater than D if Equation: open parenthesis 1 0 period 7 closing parenthesis2p + 1 ∈ P .. k greater equal C sub 1 open(11 parenthesis.2) nu\noindent closing parenthesisbe an admissible = open parenthesis $ k e to− the$ power tuple of of minus distinct gamma plus linear o open forms parenthesis , that 1 closing i s , we suppose that parenthesis closing parenthesis e to the power of nu period Equation: Corollary period .. limint n right arrow infinity\noindent inf open$ parenthesis\prod L q n{ plusi nu} minus( x q n closing ) $ parenthesishas no fixed less or prime equal C divisor sub 2 open . parenthesis nu closing parenthesis = open parenthesis e to the power of minus gamma plus o open parenthesis 1 closing parenthesis\noindent closingTheorem8 parenthesis (GGPY nu e to $200 the power ?of nu ) period . $ \quad Let $ D $ \quad be any constant $ ,Theorem L { 8 doesi } not( specify x how ) $ many\quad formsas we aboveneed in order . \quad to findThen theamong re are them $nu\ plusnu 1 E+ sub 12 hyphen $ forms number among for given them small , \ valuesquad ofwhich nu period take The simultaneously most impor hyphen $ E { 2 } − $tant number particular case i s the following comma which clearly implies Theorem 7 period valuesTheorem with 9 open bo parenthesis th prime GGPY factors 2 closing greater parenthesis than period $ D .... $ Let if L sub i open parenthesis x closing parenthesis .... be an admissible triplet of linear forms period \ beginAmong{ a lthese i g n ∗} there exist two forms L sub i comma L sub j .... such that L sub i open parenthesis n closing parenthesis\ tag ∗{$ ( .... and 1 L 0sub j open. 7 parenthesis ) $} nk closing\geq parenthesisC { ....1 } are ( \nu ) = ( e ˆ{ − \gammaboth E} sub+ 2 hyphen o ( numbers 1 infinitely ) ) o f-t e en ˆ{\ periodnu } . \\\ tag ∗{$ Corollary . $}\lim { n \rightarrow1 1 period .. Small\ infty gaps between}\ inf almost( primes q and n some + conjectures\nu − of q n ) \ leq C { 2 } ( Erd\nu ohungarumlaut) = s ( on consecutive e ˆ{ − integers \gamma } + o ( 1 ) ) \nu e ˆ{\nu } . Erd ohungarumlaut s had many favourite problems on consecutive integers open parenthesis see the work \endof{ Hildebranda l i g n ∗} open parenthesis 2002 closing parenthesis closing parenthesis period Let us denote by Capital Omega open parenthesis n closing parenthesis and omega open parenthesis n closing parenthesis comma resp periodTheorem comma 8 does the number not specify how many forms we need in order to find amongof prime them factors $ of\nu n with+ and without 1 E multiplicity{ 2 } − period$ number Let d open for parenthesis given small n closing values parenthesis of stand $ \nu for. $ the The most impor − tantdivisor particular hyphen function case period i s The the celebrated following Erdohungarumlaut , which clearly s endash implies Mirsky Theorem Conjecture 7 open . parenthesis 1 952 closing parenthesis refers to \noindentthe divisorTheorem function comma 9 ( GGPY the others 2 ) to . consecutive\ h f i l l Let values $ of L Capital{ i } Omega( and x omega ) $ period\ h f i l l be an admissible triplet of linear forms . Conjecture C 1 .. open parenthesis Erd ohungarumlaut s endash Mirsky closing parenthesis period d open parenthesis\noindent xAmong closing parenthesis these there = d open exist parenthesis two forms x plus $ 1 closing L { parenthesisi } ,L .. infinitely{ j }$ o f-t\ enh f period i l l such that $ LConjecture{ i } C( 2 .. openn parenthesis) $ \ h f i Erd l l and ohungarumlaut $ L { sj open} parenthesis( n ) 1 $983\ closingh f i l l parenthesisare closing parenthesis period Capital Omega open parenthesis x closing parenthesis = Capital Omega open parenthesis x plus\noindent 1 closingboth parenthesis $ E .. infinitely{ 2 } o− f-t$ en numbers period infinitely o $ f−t $ en . Conjecture C 3 open parenthesis Erd ohungarumlaut s open parenthesis 1 983 closing parenthesis closing parenthesis\ begin { c e period n t e r } omega open parenthesis x closing parenthesis = omega open parenthesis x plus 1 closing parenthesis1 1 . \quad .. infinitelySmall o gaps f-t en betweenperiod almost primes and some conjectures of ErdThese $ conjectures ohungarumlaut would follow $ s from on the consecutive Twin Prime Conjecture integers with \endx plus{ c e n 1 t replaced e r } by x plus 2 in the following sharp form : Equation: open parenthesis 1 1 period 1 closing parenthesis .. d open parenthesis x closing parenthesis = d open\ hspace parenthesis∗{\ f i l x l } plusErd 2 closing $ ohungarumlaut parenthesis = 2 $ comma s had Capital many Omega favourite open parenthesis problems x closing on consecutive parenthesis integers ( see the work = Capital Omega open parenthesis x plus 2 closing parenthesis = omega open parenthesis x closing parenthesis =\noindent omega openof parenthesis Hildebrand x plus ( 2 closing2002 ) parenthesis ) . Let = us 1 i denote period o byperiod $ \Omega ( n ) $ and $ \Inomega their original( forms n )Conjectures ,$ resp1 1 endash . ,thenumber 1 1 would follow from an analogue of the conjecture of Sophie Germain period Sophie Germain conjectured that 2 p plus \noindent1 in P for infinitelyof prime many factors primes p of period $ n It $ i s possible with and to show without by Chen multiplicity quoteright s method . Let $ d ( nthat ) comma$ stand similarly for to the open parenthesis 10 period 1 closing parenthesis endash open parenthesis 1 0 period 2 closing parenthesis comma either \noindentEquation:d open i v i s parenthesis o r − function 1 1 period . 2The closing celebrated parenthesis Erd .. 2 p $plus ohungarumlaut 1 in P $ s −− Mirsky Conjecture ( 1 952 ) refers to the divisor function , the others to consecutive values of $ \Omega $ and $ \omega . $

\noindent Conjecture C 1 \quad ( Erd $ ohungarumlaut $ s −− Mirsky $ ) . d (x)=d(x+1)$ \quad infinitely o $ f−t $ en .

\noindent Conjecture C 2 \quad (Erd $ohungarumlaut$ s $( 1 983 ) ) . \Omega ( x ) = \Omega ( x + 1 ) $ \quad infinitely o $ f−t $ en .

\noindent ConjectureC3 ( Erd $ohungarumlaut$ s $ ( 1 983 ) ) . \omega ( x ) = \omega ( x + 1 ) $ \quad infinitely o $ f−t $ en .

These conjectures would follow from the Twin Prime Conjecture with $x + 1$ replaced by $x + 2$ in the following sharp form :

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 1 . 1 ) $} d(x)=d(x+2)=2 , \Omega ( x ) = \Omega ( x + 2 ) = \omega ( x ) = \omega (x+2)=1i.o. \end{ a l i g n ∗}

In their original forms Conjectures 1 1 −− 1 1 would follow from an analogue of the conjecture of Sophie Germain . Sophie Germain conjectured that $ 2 p + $ $ 1 \ in P $ for infinitely many primes $ p . $ It i s possible to show by Chen ’ s method that , similarly to ( 10 . 1 ) −− ( 10 . 2 ) , either

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 1 . 2 ) $} 2 p + 1 \ in P \end{ a l i g n ∗} 3 84 .. Jaacutenos Pintz \noindentor 3 84 \quad J \ ’{ a} nos Pintz orEquation: open parenthesis 1 1 period 3 closing parenthesis .. 2 p plus 1 = p 1 p 2 p i in P i s true for infinitely many primes p period .. Now if .. open parenthesis 1 1 period 3 closing parenthesis .. it self\ begin holds{ a infinitely l i g n ∗} \ tag ∗{$ ( 1 1 . 3 ) $} 2p+1=p1p2pi \ in often comma then3 84 ConjecturesJános Pintz C 1 endashor C 3 hold comma namely comma P open parenthesis 1 1 period 4 closing parenthesis \endd open{ a l i g parenthesis n ∗} 2 p closing parenthesis = d open parenthesis 2 p plus 1 closing parenthesis = 4 comma omega open parenthesis 2 p closing parenthesis =2p Capital+ 1 = p Omega1p2 pi open∈ P parenthesis 2 p closing parenthesis(11.3) = Capital\noindent Omegai open s true parenthesis for infinitely 2 p plus 1 closing many parenthesis primes $ = p omega . $open\quad parenthesisNow 2 i f p\ plusquad 1 closing( 1 1 . 3 ) \quad it self holds infinitely often , theni Conjectures s true for infinitely C 1 −− manyC 3 primes hold ,p. namelyNow ,if ( 1 1 . 3 ) it self holds parenthesis = 2 periodinfinitely often , then Conjectures C 1 – C 3 hold , namely , ( 1 1 . 4 ) (Due 1 1 .. . to 4 .. ) this .. connection .. also .. Conjectures .. C 1 endash C 3 .. were .. considered .. ex hyphen tremely difficult comma .. if not hopeless period .. open parenthesis Problem .. 1 1 period 3 is believed to d(2p) = d(2p + 1) = 4, ω(2p) = Ω(2p) = Ω(2p + 1) = ω(2p + 1) = 2. be\ [d(2p)=d(2p+1)=4, of the \omega ( 2 psame ) depth = ..\Omega as ..Due the ..( Twin to 2 .. this Prime p connection.. ) Conjecture = \Omega comma also .. Conjecturesas( .. remarked 2 p by + C Hildebrand 1 – 1 C 3 ) were = \omega (open 2 parenthesis p +considered 2002 1 closing ) = parenthesis ex -2 tremely . closing\ ] difficult parenthesis , ifperiod not hopeless . ( Problem 1 It was a great surprise1 . 3 is when believed C period to Spiro be of open the parenthesis same depth 1 98 1 closing as the parenthesis Twin was able Prime to prove Equation: openConjecture parenthesis 1 , 1 period as 5 remarked closing parenthesis by Hildebrand .. d open parenthesis n closing parenthesis = d openDue parenthesis\quad to n\quad plus 5040t h i closing s \quad parenthesisconnection i period\ oquad perioda l s o \quad Conjectures \quad C 1 −− C 3 \quad were \quad c o n s i d e r e d \quad ex − tremely difficult , \quad if not hopeless . \quad ( Problem \quad 1 1 . 3 is believed to be of the Independently of open parenthesis 1 1 period 5 closing(2002)) parenthesis. Heath hyphen Brown open parenthesis 1982same closing depth parenthesis\quad foundas \quad a conditionalthe \quad proofTwin of \quad Prime \quad Conjecture , \quad as \quad remarked by Hildebrand C 2 under EH periodIt was Finally a great he succeeded surprise open when parenthesis C . Spiro Heath ( 1 98 hyphen 1 ) was Brown able open to prove parenthesis 1 984 closing\ begin parenthesis{ a l i g n ∗} closing parenthesis in showing C 1 (unconditionally 2002 ) using ) the . ideas of Spiro comma .. in combination with significant d(n) = d(n + 5040) i.o. (11.5) \endnew{ a ideas l i g n of∗} his own period His method led also to C 2 comma but C 3 remained open period C 3 was proved just \ centerline { It was a greatIndependently surprise when of ( C 1 1 . . Spiro 5 ) Heath ( 1 - 98 Brown 1 ) was ( 1982 able ) found to prove a } recently by J periodconditional hyphen proof C period of .. Schlage hyphen Puchta open parenthesis 2003 slash 2005 closing parenthesis periodC His 2 method under involved EH . Finally both he succeeded ( Heath - Brown ( 1 984 ) ) in \ begintheoretical{ a l i g andn ∗} computational methods period \ tag ∗{$ ( 1showing 1 . C 1 5 unconditionally ) $} d ( using n the )=d ideas of Spiro ( , n+5040 in combination ) i An important featurewith significant of all these results new ideas were comma of his as own pointed . out by Heath hyphen .Brown o open . parenthesis 1 982 closing parenthesis in connection with the conditional solution of C 2 : quoteleft \end{ a l i g n ∗} His method led also to C 2 , but C 3 remained open . C 3 was proved It should just recently by J . - C . Schlage - Puchta ( 2003 / 2005 ) . His method be noted at this point that in solving Capital Omega open parenthesis n closing parenthesis = Capital Omega \ hspace ∗{\ f i l linvolved} Independently both theoretical of ( 1 1 and . computational5 ) Heath − Brown methods ( 1982 . ) found a conditional proof of open parenthesis n plusAn 1 closing important parenthesis feature we of shall all thesenot have results were , as pointed out by Heath specified Capital- Omega Brown open ( 1 parenthesis 982 ) in connection n closing parenthesis with the comma conditional or even solutionthe parity of of CapitalC 2 : ‘ Omega open\noindent parenthesisC 2 n underclosing parenthesis EH . Finally period he Thus succeeded we avoid the ( parity Heath problem− Brown comma ( 1 984 ) ) in showing C 1 unconditionallyIt should using the ideas of Spiro , \quad in combination with significant rather than solve it period quoteright Ω(n) = Ω(n + 1) new ideas ofbe his noted own at . this point that in solving we shall not have Our Theorem 9specified yields in aΩ( rathern), or quick even way the a parity new solution of Ω(n of). ConjecturesThus we avoid the parity problem C 1 endash C 3 with the additional advantage that we can solve them even if the His method led, also rather to than C 2 solve , but it . C ’ 3 remained open . C 3 was proved just common consecutiveOur value Theorem of f open 9 parenthesis yields in a n rather closing parenthesisquick way = a f new open solution parenthesis of Conjec-n plus 1 closing parenthesisrecently open by parenthesis J . − C. f =\quad d commaSchlage Capital− OmegaPuchta or omega ( 2003 closing / 2005 parenthesis ) . His is specified method period involved both theoretical andtures computational C 1 – C 3 with methods the additional . advantage that we can solve them More precisely weeven can if prove the common consecutive value of f(n) = f(n + 1)(f = d, Ω or ω) is Theorem 10 openspecified parenthesis . GGPY 200 ? ? closing parenthesis period For any A greater equal 3 we have Anomega important open parenthesis feature n closing of all parenthesis these = results omega open were parenthesis , as pointed n plus 1 closing out byparenthesis Heath =− A comma Brown ( 1 982 ) in connection withMore the precisely conditional we can prove solution of C 2 : ‘ It should i period o periodTheorem 10 ( GGPY 200??). For any A ≥ 3 we have ω(n) = ω(n+1) = A, Theorem 1 1 .. open parenthesis GGPY 200 ? ? closing parenthesis period .. For any A greater equal 4 we \noindent be notedi . o . at this point that in solving $ \Omega ( n ) = \Omega have Capital Omega open parenthesis n closing parenthesis200??) =. Capital OmegaA open≥ 4 parenthesis nΩ( plusn) =1 closing ( n + 1Theorem )$ weshallnothave 1 1 ( GGPY For any we have parenthesis = Ω(n + 1) = A, i . o . A comma i period o period 200??). 24 | A d(n) = d(n+1) = A, \noindent s p e cTheorem i f i e d $ 1\Omega 2 ( GGPY( n )For any ,$ oreventheparityofwe have $ \Omega Theorem 1 2 openi . parenthesiso . GGPY 200 ? ? closing parenthesis period For any 24 bar A we have d open parenthesis( n ) n closing . $ parenthesis Thuswe = avoid d open the parenthesis parity n plus problem 1 closing , parenthesis = A comma rather than solveConjectures it . ’ C 1 – C 3 are also interesting if the shift 1 is replaced by i period o period2 or Conjectures C 1by endash a general C 3 are shift also interestingb ∈ Z+, that if the is , shift the 1 problem is replaced of by whether 2 or Ourby Theorem a general shift9 yields b in Z to in the a power rather of plus quick comma way that a new is comma solution the problem of Conjectures of whether CEquation: 1 −− C 3 open with parenthesis the additional 1 1 period 6 advantage closing parenthesis that we .. f can open solveparenthesis them n closing even parenthesisif the = f opencommonconsecutivevalueof parenthesis n plus b closing parenthesis $f (f n(n) = )f(n + =b) f ( n + 1 (11).6) ( f = d , \Omega $ or $ \omega ) $ is specified . holds infinitelyholds often if infinitely f = d comma often Capital if f Omega= d, Ω or or omegaω. period \ centerline {More precisely we can prove }

\noindent Theorem10(GGPY $200 ? ? ) . $ Forany $A \geq 3 $ we have $ \omega ( n ) = \omega ( n + 1 ) = A , $

\noindent i . o .

\noindent Theorem 1 1 \quad (GGPY $200 ? ? ) .$ \quad For any $ A \geq 4 $ we have $ \Omega ( n ) = \Omega ( n + 1 ) = $ $ A , $ i . o .

\noindent Theorem12(GGPY $200 ? ? ) . $ Forany $24 \mid A $ wehave$d ( n ) = d ( n + 1 ) =A ,$

\noindent i . o .

\ hspace ∗{\ f i l l } Conjectures C 1 −− C 3 are also interesting if the shift 1 is replaced by 2 or

\noindent by a general shift $ b \ in Z ˆ{ + } , $ that is , the problem of whether

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 1 . 6 ) $} f(n)=f(n+b) \end{ a l i g n ∗}

\noindent holds infinitely often if $ f = d , \Omega $ or $ \omega . $ Landau quoteright s problems on primes .. 385 \ hspaceThese∗{\ resultsf i l were l }Landau proved by’ s C problems period Pinner on open primes parenthesis\quad 1385 997 closing parenthesis for every value b using an Theseingenious results extension were of proved Heath hyphen by C Brown . Pinner quoteright ( 1 997 s method ) for for every f = d and value Capital $ Omegab $ using periodan Y periodingenious Butt hyphen extension of Heath − Brown ’ smethod for $ f = d$ and $ \Omega . $ Y . Butt − kewitz open parenthesis 2003 closing parenthesis extended PuchtaLandau quoteright ’ s problems s result on for primes f = omega385 and for infinitelykewitz many ( 2003 ) extended Puchta ’ s result for $ f = \omega $ and for infinitely many integer shift sThese $b results . $ were proved by C . Pinner ( 1 997 ) for every value b integer shift s busing period an ingenious extension of Heath - Brown ’ s method for f = d and Our methods yield a full extension of the results for f = omega and Capital Omega with Our methods yieldΩ. Y .a Butt full - extension kewitz ( 2003 of ) the extended results Puchta for ’ $s result f = for \fomega= ω and$ for and $ \Omega $ specified commoninfinitely values of many f open integer parenthesis shift n s closingb. parenthesis = f open parenthesis n plus b closing parenthesiswith = A and a partial extension specifiedcommonvaluesofOur methods yield $f a full ( extension n ) of = the f results ( for nf += ω and bΩ )with = A$ for d open parenthesisspecified n closing common parenthesis values = of df open(n) = parenthesisf(n + b) = nA plusand b closinga partial parenthesis extension = A period andTheorem a partial 13 open extension parenthesis GGPY 200 ? ? closing parenthesis period If b in Z to the power of plus comma A greater equal 4 comma then Capital Omega open parenthesis n closing parenthesis = Capital Omega open parenthesis\ begin { a l n i g plus n ∗} b closing parenthesis = A commaford(n) = d(n + b) = A. ford(n)=d(n+b)=A. i period o period + \end{ a l i g n ∗} Theorem 13 ( GGPY 200??). If b ∈ Z ,A ≥ 4, then Ω(n) = Ω(n + b) = A, Theorem 14 openi .parenthesis o . GGPY 200 ? ? closing parenthesis period If b in Z to the power of plus comma A greater equal A open parenthesis b closing parenthesis comma+ then omega open parenthesis n closing \noindent Theorem13(GGPYTheorem 14 ( GGPY $200200??) ?. If ?b )∈ Z ,A .$≥ A If(b), then $b ω\(nin) = ω(Zn + ˆ{b)+ = } , parenthesis = omegaA, openi . parenthesis o . nTheorem plus b closing parenthesis 1 5 = ( GGPY 200??). If b ∈ A A\ commageq i period4 , o $ period then $ \Omega ( n ) = \Omega ( n + b ) Z+, bequivalence − negationslash 15( mod 30), 48 | A, then =Theorem A , .. $ 1 5 .. open parenthesis GGPY 200 ? ? closing parenthesis period .. If b in Z to the power of plus comma b equivalence-negationslash 1 5 open parenthesis mod 30 closing parenthesis comma 48 bar A comma .. then\noindent i . o . d(n) = d(n + b) = A, i.o. d open parenthesis n closing parenthesis = d open parenthesis n plus b closing parenthesis = A comma i \noindent Theorem14(GGPYWe will show below $200 the simple ? ? deduction ) .$ of Theorem If $b 10\ fromin TheoremZ ˆ{ + } , period o period 9 in the most important case A = 3. We consider the admissible system A We\geq will showA below ( the b simple ) deduction ,$then$ of Theorem\omega 10 from Theorem( n 9 ) = \omega ( n +in b the most ) important = $ case A = 3 period We consider the admissible system Equation: open parenthesis 1 1 period 7 closing18m + parenthesis 1, 24m + .. 1, 18 m27m plus+ 1 1 comma 24 m plus 1 comma(11.7) 27 m \noindent $ A , $ i . o . plus 1 and the relations Theoremand the relations\quad 1 5 \quad (GGPY $200 ? ? ) .$ \quad I f $ b \ in Z ˆ{ + } ,3 openbparenthesis equivalence 1 8− mnegationslash plus 1 closing parenthesis 1 5 = 2 ($ open parenthesismod $30 2 7 m ) plus , 1 closing 48 parenthesis\mid A plus, $ 1 comma\quad Equation:then open parenthesis 1 13(18 periodm + 8 1) closing = 2(27 parenthesism + 1) + 1, .. 4 open parenthesis 1 8 m plus 1 closing parenthesis = 3 open parenthesis 24 m4(18 plusm 1 closing+ 1) = 3(24parenthesism + 1) + plus 1, 1 comma 9 open parenthesis(11.8) 24 m plus\ begin 1 closing{ a l i g parenthesis n ∗} = 8 open parenthesis 2 7 m plus 1 closing parenthesis plus 1 period 9(24m + 1) = 8(27m + 1) + 1. d(n)=d(n+b)=A,i.o.Since by Theorem 9 at least two of the forms in open parenthesis 1 1 period 7 closing parenthesis will be simulta\end{ a hyphen l i g n ∗} Since by Theorem 9 at least two of the forms in ( 1 1 . 7 ) will be neously E sub 2simulta hyphen numbers - neously not divisibleE − numbers by 2 and not 3 for divisible infinitely many by 2 values and 3 m for comma infinitely We will show below the simple deduction2 of Theorem 10 from Theorem 9 we obtain by openmany parenthesis values 1m, 1we period obtain 8 closing by ( parenthesis 1 1 . 8 ) a sequence sequence x subxi → i right ∞ with arrow infinity with inEquation: the most open important parenthesis case 1 1 period $ A 9 closing = 3 parenthesis . $ We .. consider omega open the parenthesis admissible x sub systemi closing parenthesis = omega open parenthesis x sub i plus 1 closing parenthesis = 2 plus 1 = 3 period \ beginThe case{ a l i of g n a∗} general .. A greater 3 .. can beω(x deducedi) = ω(x ..i + in 1) .. = a 2.. + similar 1 = 3. way from (11.9) \ tag ∗{$ ( 1 1 . 7 ) $} 18m+1 , 24m+1 , 27m Theorem 9 with someThe additional case of a ideas general period A > 3 can be deduced in a similar +1 2 1 period .. Theway exceptional from Theorem set in Goldbach 9 with quoteright some additional s Problem ideas . \endHardy{ a l i andg n ∗} Littlewood1 ..2 open. parenthesis The exceptional 1 924 closing set parenthesis in Goldbach .. examined ’ s Problem the problem whether one can Hardy and Littlewood ( 1 924 ) examined the problem whether \noindentbound fromand aboveone the the can relations number E open parenthesis X closing parenthesis of Goldbach exceptional even numbers below X commabound which cannotfrom above be expressed the number as a sumE( ofX) twoof Goldbach primes comma exceptional i period e even period numbers \ begin { a l i g n ∗} Equation: openbelow parenthesisX, which 1 2 period cannot 1 closing be expressed parenthesis as.. E a open sum parenthesis of two primes X closing , i .parenthesis e . = bar E3(18m+1)=2(27m+1)+1, bar = vextendsingle-vextendsingle open brace n less or equal X comma 2 bar n comma n negationslash-equal\\\ tag ∗{$ ( 1 1 . 8 ) $} 4(18m+1)=3(24m+1 p plus p to the power of prime closing brace vextendsingle-vextendsingle period 0 )They + attacked 1 , the\\ problem9(24m+1)=8(27m+1) with theE( celebratedX) =| E |= circle|{n ≤ methodX, 2 comma| n, n invented6= p + p }|. (12.1) +by 1 Hardy . comma ..They Littlewood attacked and theRamanujan problem period with .. the They celebrated could not circle prove any method result , invented \endunconditionally{ a l i g n ∗} by period Hardy However , Littlewood comma they showedand Ramanujan . They could not prove any Equation: openresult parenthesis unconditionally 1 2 period 2 closing . However parenthesis , they .. E showed open parenthesis X closing parenthesis ll X toSince the power by Theorem of 1 slash 92 plus at epsilon least two of the forms in ( 1 1 . 7 ) will be simulta − neously $ E { 2 } − $ numbers not divisible by 2 and 3 for infinitely many values $ m , $ E(X) X1/2+ε (12.2) we obtain by ( 1 1 . 8 ) a sequence $ x { i }\rightarrow \ infty $ with

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 1 . 9 ) $}\omega ( x { i } ) = \omega ( x { i } + 1 ) = 2 + 1 = 3 . \end{ a l i g n ∗}

The case of a general \quad $ A > 3 $ \quad can be deduced \quad in \quad a \quad similar way from Theorem 9 with some additional ideas .

\ centerline {1 2 . \quad The exceptional set in Goldbach ’ s Problem }

\ hspace ∗{\ f i l l }Hardy and Littlewood \quad ( 1 924 ) \quad examined the problem whether one can

\noindent bound from above the number $ E ( X ) $ of Goldbach exceptional even numbers below $X , $ which cannot be expressed as a sum of two primes , i . e .

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 2 . 1 ) $} E ( X ) = \mid E \mid = \arrowvert \{ n \ leq X , 2 \mid n , n \not= p + p ˆ{\prime }\} \arrowvert . \end{ a l i g n ∗}

They attacked the problem with the celebrated circle method , invented by Hardy , \quad Littlewood and Ramanujan . \quad They could not prove any result unconditionally . However , they showed

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 2 . 2 ) $} E(X) \ l l X ˆ{ 1 / 2 + \ varepsilon } \end{ a l i g n ∗} 3 86 J´anos Pintz under the assumption of GRH . This result is even today the best conditional one , apart from the improvement of Goldston ( 1 989 / 1992 ) who replaced Xε

with log3 X.

The ﬁrst unconditional estimate of type E(X) = o(X) was made possible more than a decade later by the method of I . M . Vinogradov ( 1 937 ) , which yielded the celebrated Vinogradov ’ s three prime theorem . This theorem states that every suﬃciently large odd number can be written as a sum of three primes . Thus , using Vinogradov ’ s method , van der Corput ( 1937 ) , Estermann ( 1 938 ) and Cˇ udakov ( 1 938 ) simultaneously and independently proved the unconditional estimate (12.3) E(X) AX( log X)−A for any A > 0. This shows that Goldbach ’ s Conjecture i s true in the statistical sense that almost all even numbers are Goldbach numbers . The above result was the best known for 35 years when Vaughan ( 1 972 ) improved it to

√ E(X) X exp(−c Xlog). (12.4) Just three years later Montgomery and Vaughan ( 1 975 ) succeeded in showing the very deep estimate

1−δ E(X) ≤ X for X > X0(δ), (12.5) with an unspeciﬁed , but explicitly calculable , δ > 0. Several attempts were made to show ( 1 2 . 5 ) with a reasonable ( not too small ) value of δ. These investigations led to the values δ = 0.5 ( Chen , Liu ( 1 989 ) ) , (12.6) δ = 0.79 ( H . Z . Li ( 1 999 ) ) , δ = 0.86 ( H . Z . Li ( 2000 a ) ) . Finally , the author could show unconditionally ( see Section 1 8 for some

details)

Theorem 16 (J.P.) . There exists a fix ϑ < 2/3 such that E(X) Xϑ. If we consider the analogous problem ( cf . ( 1 2 . 6 ) ) for the difference of primes , then the above - mentioned results are all transferable to this case as well , thereby furnishing an approximation to the Generalized Twin Prime Conjecture . Thus , defining analogously to E(X) in ( 1 2 . 1 ) 3 86 .. Jaacutenos Pintz \noindentunder the3 assumption 86 \quad of GRHJ \ ’{ a period} nos This Pintz result is even today the best conditional underone comma the assumption apart from the of improvement GRH . This of Goldston result open is evenparenthesis today 1 989 the slash best 1992 conditional closing parenthesis who replaced X to the power of epsilon \noindentwith log toone the power , apart of 3 Xfrom period the improvement of Goldston ( 1 989 / 1992 ) who replaced $ XThe ˆ{\ firstvarepsilon unconditional}$ estimate of type E open parenthesis X closing parenthesis = o open parenthesis X closing parenthesis was made possible \ begin { a l i g n ∗} more than a decade later by the methodE0 of(X I) period = {n ≤ MX, period2 | n, n Vinogradov6= p − p0}, open parenthesis 1(12 937.7) closing parenthesiswith \ commalog ˆ{ which3 } X. \endyielded{ a l i g the n ∗} celebratedwe can Vinogradov show that quoteright all but sX ..2 three/3 even prime integers theorem below periodX .. Thiscan be.. theorem written as states that everythe sufficiently large odd number can be written as a sum of \ hspacethree primes∗{\ f i periodl ldifference}The first of two unconditional primes . Theorem estimate 1 of7 (J.P type $E.).E0( (X) X X )ϑ with = o (Thus X comma ) $ .. wasmadea using fixed Vinogradov constant possible quoterightϑ < 2/3. s method comma van der Corput .. open parenthesis 1937 closing parenthesis comma .. Estermann \noindentopen parenthesismore than 1 938 a closing decade parenthesis later and by Ccaronthe method udakov of open I parenthesis . M . Vinogradov 1 938 closing ( 1 parenthesis 937 ) , which simultaneouslyyielded the and celebrated independently Vinogradov proved the ’ s \quad three prime theorem . \quad This \quad theorem statesunconditional that estimate every sufficiently large odd number can be written as a sum of threeopen parenthesis primes . 1 2 period 3 closing parenthesis E open parenthesis X closing parenthesis ll A X open parenthesis log X closing parenthesis to the power of minus A .. for any .. A greater 0 period ThusThis , shows\quad thatusing .. Goldbach Vinogradov quoteright ’ s s Conjecture method , i s van true der in the Corput statistical\quad sense ( 1937 ) , \quad Estermann (that 1 938 almost ) and all even $ numbers\check{ areC} Goldbach$ udakov numbers ( 1 period 938 ) The simultaneously above result was and independently proved the unconditionalthe best known for estimate 35 years when Vaughan open parenthesis 1 972 closing parenthesis improved it to Equation: open parenthesis 1 2 period 4 closing parenthesis .. E open parenthesis X closing parenthesis ll X exponent\noindent parenleftbig$(12.3)E(X) sub minus c radicalbig-line of X sub log parenrightbig\ l l periodA X ( $ l o g $ X ) ˆJust{ − threeA years}$ later\quad Montgomeryf o r any and\quad Vaughan$ A .. open> parenthesis0 . $ 1 975 closing parenthesis .. succeeded in showing the very deep estimate ThisEquation: shows open that parenthesis\quad Goldbach 1 2 period 5 ’ closing s Conjecture parenthesis i .. s E opentrue parenthesis in the statistical X closing parenthesis sense less orthat equal almostX to the power all even of 1 minus numbers delta for are X greater Goldbach X sub numbers 0 open parenthesis . The above delta closing result parenthesis was comma with an unspecified comma but explicitly calculable comma delta greater 0 period Several attempts were \noindentmade to showthe open best parenthesis known for 1 2 period 35 years 5 closing when parenthesis Vaughan with ( 1 a reasonable972 ) improved open parenthesis it to not too small closing parenthesis value of delta period These \ begininvestigations{ a l i g n ∗} led to the values \ tagdelta∗{$ = ( 0 period 1 5 2 open . parenthesis 4 ) $ Chen} E(X) comma Liu open parenthesis\ l l X 1 989\exp closing( parenthesis{ − } c closing\ sqrt { X } {\ log } parenthesis). comma \endopen{ a l parenthesis i g n ∗} 1 2 period 6 closing parenthesis delta = 0 period 79 open parenthesis H period Z period Li open parenthesis 1 999 closing parenthesis closing parenthesis comma Justdelta three = 0 period years 86 open later parenthesis Montgomery H period and Z period Vaughan Li open\quad parenthesis( 1 975 2000 ) a\ closingquad parenthesissucceeded closing in parenthesisshowing period the very deep estimate Finally comma the author could show unconditionally open parenthesis see Section 1 8 for some \ begindetails{ a closing l i g n ∗} parenthesis \ tagTheorem∗{$ ( 16 open 1 parenthesis 2 . 5 J period ) $} P periodE(X) closing parenthesis\ leq periodX .. ˆ{ There1 exists− a \ fixdelta vartheta} lessf o r X2 slash> 3 suchX that{ 0 E} open( parenthesis\ delta X closing), parenthesis ll X to the power of vartheta period \endIf{ wea l consider i g n ∗} the analogous problem .. open parenthesis cf period .. open parenthesis 1 2 period 6 closing parenthesis closing parenthesis .. for the difference of \noindentprimes commawith then an the unspecified above hyphen , mentioned but explicitly results are all calculable transferable to $ this , case\ delta > 0 . $as well Several comma attempts thereby furnishing were an approximation to the Generalized Twin Prime Conjecture period Thus comma defining analogously to E open parenthesis X closing parenthesis in open parenthesis\noindent 1made 2 period to 1 closingshow ( parenthesis 1 2 . 5 ) with a reasonable ( not too small ) value of $ \ delta . $Equation: These open parenthesis 1 2 period 7 closing parenthesis .. E to the power of prime open parenthesis X closing parenthesis = open brace n less or equal X comma 2 bar n comma n negationslash-equal p minus p to the\noindent power of primeinvestigations closing brace comma led to the values we can show that all but X to the power of 2 slash 3 even integers below X can be written as the \noindentdifference of$ two\ delta primes period= 0 . 5 ($ Chen,Liu(1989)), $Theorem ( 1 1 7 2 open . parenthesis 6 ) J period\ delta P period= closing 0 parenthesis . 79 period ($ H.Z.Li(1999)), E to the power of prime open parenthesis X closing parenthesis ll X to the power of vartheta with a fixed constant vartheta less 2 slash 3 period \ centerline { $ \ delta = 0 . 86 ($ H.Z.Li(2000a)). }

\ hspace ∗{\ f i l l } Finally , the author could show unconditionally ( see Section 1 8 for some

\ begin { a l i g n ∗} d e t a i l s ) \end{ a l i g n ∗}

\noindent Theorem 16 ( J . P . ) . \quad There exists a fix $ \vartheta < 2 / 3$ suchthat $E ( X ) \ l l X ˆ{\vartheta } . $

If we consider the analogous problem \quad ( c f . \quad ( 1 2 . 6 ) ) \quad for the difference of primes , then the above − mentioned results are all transferable to this case as well , thereby furnishing an approximation to the Generalized Twin Prime Conjecture . Thus , defining analogously to $E ( X ) $ in ( 1 2 . 1 )

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 2 . 7 ) $} E ˆ{\prime } ( X ) = \{ n \ leq X , 2 \mid n , n \not= p − p ˆ{\prime }\} , \end{ a l i g n ∗}

\noindent we can show that all but $ X ˆ{ 2 / 3 }$ even integers below $ X $ can be written as the

\noindent difference of two primes . Theorem17(J.P $. ) . Eˆ{\prime } (X) \ l l X ˆ{\vartheta }$ with a fixed constant $ \vartheta < 2 / 3 . $ Landau quoteright s problems on primes .. 387 \ hspaceNaturally∗{\ commaf i l l }Landau we are not ’ .. s able problems to prove on the primes stronger generalization\quad 387 of the twin prime conjecture open parenthesis 2 period 1 5 closing parenthesis for any single even number N since thisNaturally would , we are not \quad able to prove the stronger generalization of the twinimply prime the Bounded conjecture Gap Conjecture ( 2 . 1period 5 ) for any single even number $ N $ since this would imply the Bounded Gap Conjecture . 13 period .. The Ternary Goldbach Conjecture and Descartes quoterightLandau ’ s problems on primes 387 Conjecture period Primes as a basis \ centerline {13 . Naturally\quad The , we Ternary are not Goldbach able to Conjecture prove the stronger and Descartes generalization ’ } of The name Ternarythe Goldbach twin prime Conjecture conjecture refers ( to 2 the . 1 conjecture 5 ) for any that single even number N since every odd integer larger than 5 can be written as a sum of three primes period \ centerline { Conjecturethis would . imply Primes the as Bounded a basis Gap} Conjecture . This conjecture appeared13 . first The actually Ternary at Waring Goldbach open parenthesis Conjecture 1 770 closing and parenthesis Descartes period ’ The first comma albeit The name Ternary Goldbach ConjectureConjecture refers . Primes to the as conjecture a basis that conditional commaThe result name concerning Ternary this wasGoldbach achieved Conjecture in the above refers hyphen to mentioned the conjecture pi hyphen that everyoneering odd work integer of Hardy larger and Littlewood than 5 open can parenthesis be written 1 923 as closing a sum parenthesis of three period primes They showed . that This conjectureevery appeared odd integer first larger actually than 5 at can Waring be written ( 1 as 770 a sum) . The of three first primes , albeit if there . This conjecture appeared first actually at Waring ( 1 770 ) . The first conditionali s a theta less , 3 slash result 4 such concerning that no Dirichlet this L was hyphen achieved function vanishes in the in above the halfplane− mentioned pi − oneering work, albeitof Hardy conditional and Littlewood , result concerning ( 1 923 ) this . They was showedachieved that in the if above there - Re s greater thetamentioned open parenthesis pi - oneering a weaker work form of of GRH Hardy closing and parenthesis Littlewood comma ( 1 923then ) every . They sufficiently large odd num hyphen \noindent i sshowed a $ \theta that if there< 3 / 4$ such that no Dirichlet $L − $ function vanishes in the halfplane ber can be writteni s as a aθ sum < of three3/4 primessuch thatperiod no As Dirichletan approximationL− function to the vanishes in the Goldbach Conjecture we may consider the problem of whether the set P of \noindent Rehalfplane $ s > \theta ( $ a weaker form of GRH ) , then every sufficiently large odd num − primes open parenthesisRe s > θ extended( a weaker with form the element of GRH 0 closing ) , then parenthesis every sufficiently forms a basis large or an odd asymptotic num - basis berof finite can order be written period The as existence a sum of of a number three Sprimes to the power . As of an * such approximation that every integer to larger the Goldbach Conjectureber can be we written may consider as a sum the of three problem primes of . whether As an approximation the set $ P to $ the of than 1 can be writtenGoldbach as the Conjecture sum of at most we mayS to the consider power of the * primes problem was of first whether proved by the set P primesSchnirelman ( extended open parenthesis with the 1 930 element comma 1933 0 ) closing forms parenthesis a basis period or an Let asymptotic us denote by Sbasis the smallest of finite orderof primes . The ( existence extended with of a the number element $ 0 S ) ˆforms{ ∗ } a$ basis such or thatan asymptotic every integer larger number S to the power of * with ∗ than 1 can bebasis written of finite as the order sum . of The at existence most $ of S aˆ{ number ∗ }$ primesS such was that first every proved by the above property period Similarly let S sub 1 denote the smallest number with the ∗ Schnirelman (integer 1 930 larger , 1933 than ) . 1 Letcan be us written denote as by the $ sum S $ of atthe most smallestS primes number was $ S ˆ{ ∗ }$ property that allfirst sufficiently proved large by numbers Schnirelman can be ( expressed 1 930 , as 1933 the sum ) . Let us denote by S the with ∗ of S sub 1 primessmallest period As number we mentionedS with in Section the above 2 comma property the conjecture . Similarly formulated let byS1 denote theDescartes above prior property to Goldbach . Similarly is equivalent let to S = $ 3 S period{ 1 So}$ we may denote restate the it as smallest number with the property thatthe all smallest sufficiently number with large the numbers property can that be all expressed sufficiently as large the numbers sum Descartes quoterightcan be Conjecture expressed period as the S = sum S sub of 1 =S1 3primes period . As we mentioned in Section 2 o fApproaching $ S { 1 the}$ Goldbach primes Conjecture . As we from mentioned this direction in we Section can try to 2 , the conjecture formulated by Descartes prior, the to conjecture Goldbach formulated is equivalent by Descartes to $ S prior = to Goldbach3 . $ is So equivalent we may restate it as give upper boundsto forS = S 3 and. So S we sub may 1 period restate The it original as work of Schnirelman relied on two basic results comma proved by Brun quoteright s sieve and combinatorial methods comma \noindent DescartesDescartes ’ Conjecture ’ Conjecture $ ..S S= S1 == 3. S { 1 } = 3 . $ respectively periodApproaching the Goldbach Conjecture from this direction we can try open parenthesisto i closing give upper parenthesis bounds The for numberS and G openS1. The parenthesis original x closing work ofparenthesis Schnirelman of Goldbach numbersApproaching below x the satisfies Goldbach G open parenthesis Conjecture x closing from parenthesis this direction greater equal we cancx try to give upper boundsrelied for on two $ S basic $ and results $ , S proved{ 1 } by Brun. $ The ’ s sieve original and combinatorial work of Schnirelman relied for x greater equalmethods 4 with ,an respectively absolute constant . c greater 0 period onopen two parenthesis basic results ii closing parenthesis , proved Every by Brun set A subset ’ s sieve Z to the and power combinatorial of plus with positive methods lower asymptotic , respectively . ( i ) The number G(x) of Goldbach numbers below x satisfies G(x) ≥ cx density forms an for x ≥ 4 with an absolute constant c > 0. asymptotic basis of( finite ii ) Every order period set A ⊂ Z+ with positive lower asymptotic density forms ( iThis ) The combinatorial number approach $G led ( to x the estimates ) $ of Goldbach numbers below $ x $ satisfies $ G ( xan ) asymptotic\geq cx basis $ of finite order . S sub 1 less or equal 2208 RomanovThis combinatorial open parenthesis approach 1935 closing led parenthesis to the estimates comma f oopen r $ parenthesis x \geq 1 3 period4 $ 1 withclosing an parenthesis absolute S sub constant 1 less or equal $ c 71 ..> Heilbronn0 endash . $ Landau endash

Scherk open parenthesis 1 936 closing parenthesisS1 comma≤ 2208 Romanov(1935), ( iiS sub ) 1Every less or set equal67 $A .. Ricci\subset open parenthesisZ ˆ{ 1+ 936}$ comma with 1 positive937 closing parenthesis lower asymptotic period density forms an asymptoticThe breakthrough basis(13 came.1) ofS in1 finite 1≤ 93771 whenHeilbronn order I period . M – period Landau Vinogradov – Scherk open ( 1 parenthesis 936 ) , 1 937 closing parenthesis invented S1 ≤ 67 Ricci ( 1 936 , 1 937 ) . \ centerlinehis .. method{ This to estimateThe combinatorial breakthrough .. trigonometric approach came .. sums in .. led 1 over 937 to primes when the .. estimates andI . Mproved . Vinogradov ..} his ( 1 937 celebrated theorem) invented comma according his method to which every to estimate suﬃciently large trigonometric odd number sums over \ [Scan be{ written1 }\ asprimesleq the sum2208 andof three proved Romanov primes comma his ( celebrated which 1935 implies theorem ) , \ ,] according to which every Theorem open parenthesissuﬃciently I largeperiod odd M period number Vinogradov can be closing written parenthesis as the sum period of S three sub 1 primes less or equal 4 period , which implies \noindent $ (Theorem 1 3( .I . M 1 . Vinogradov ) S { 1).S}\1 ≤leq4. 71 $ \quad Heilbronn −− Landau −− Scherk ( 1 936 ) ,

\ centerline { $ S { 1 }\ leq 67 $ \quad Ricci ( 1 936 , 1 937 ) . }

The breakthrough came in 1 937 when I . M . Vinogradov ( 1 937 ) invented h i s \quad method to estimate \quad trigonometric \quad sums \quad over primes \quad and proved \quad h i s celebrated theorem , according to which every sufficiently large odd number can be written as the sum of three primes , which implies

\noindent Theorem ( I .M. Vinogradov $ ) . S { 1 }\ leq 4 . $ 3 88 .. Jaacutenos Pintz \noindentAlthough his3 88 result\quad is nearlyJ \ ’ optimal{ a} nos comma Pintz it gave no clue for a good estimate of S period The estimates for S were first reached by the elementary method of AlthoughSchnirelman his period result Later is results nearly used optimal a combination , it of gave elementary no clue and analytic for a good estimate ofmethods $ S involving . $ The in many estimates cases heavy for computations $ S $ wereas well first period Subsequent reached by the elementary method of Schnirelman . Later results used a combination of elementary and analytic improvements for3 88 S wereJános as followsPintz : methodsLine 1 S involvingless 2 times 1 in 0 to many the power cases of heavy 10 Scaron computations anin open parenthesis as well 1964 . closing Subsequent parenthesis comma improvements forAlthough $ S $ his were result as is follows nearly optimal : , it gave no clue for a good es- Line 2 S less or equaltimate 6 times of 1 0S. toThe the power estimates of 9 Klimov for S openwere parenthesis first reached 1 969 byclosing the parenthesis elementary comma S less or equal 1method 59 .. Deshouillers of Schnirelman open parenthesis . Later 1 results 972 slash used 73 closing a combination parenthesis of comma elementary \ [ \Sbegin less or{ a equal l i g n e1 d 1} 5S .. Klimov< 2 comma\cdot Pilt quoteright1 0 ai ˆ{ comma10 }\ Scaroncheck eptickaj{S} a openanin parenthesis ( 1964 1972 ), \\ and analytic methods involving in many cases heavy computations as well closing parenthesis. comma Subsequent improvements for S were as follows : SS less\ orleq equal6 61 ..\ Klimovcdot open1 parenthesis 0 ˆ{ 9 1} 978Klimov closing parenthesis ( 1 comma 969 ) , \end{ a l i g n e d }\ ] S less or equal 55 .. Klimov open parenthesisS 1 < 9752 · closing1010 Sanin parenthesisˇ (1964) comma, S less or equal 27 Vaughan open parenthesisS 1977≤ 6 closing· 109 Klimov parenthesis(1969) comma, \ centerlineS less or equal{ $ 26 S .. Deshouillers\ leq 1 open 59 parenthesis $ \quad 1 975Deshouillers slash 76 closing ( parenthesis 1 972 / comma 73 ) , } S less or equal 25 .. Zhang comma DingS ≤ 159 open parenthesisDeshouillers 1 983 ( 1 closing 972 / parenthesis 73 ) , comma \noindentS less or equal$ S 19S ≤ ..\115 Rieselleq Klimov comma1 Vaughan1 , Pilt 5 $’ open ai\quad, parenthesisSˇ eptickajKlimov 1983 a ( , 1972closing Pilt ) parenthesis ’ , aiS ≤ 61 $ , commaKlimov\check ({ 1S} $ eptickajS less or aequal ( 1972 7978 .. Ramareacute ) ) , ,S ≤ 55 openKlimov parenthesis ( 1 975 1 995 ) , closing parenthesis period $The S best\ leq known conditional61 $ \quad resultsKlimov are the (following 1 978 period ) , $Theorem S \ leq open parenthesis55 $ \quad KanieckiKlimov open parenthesis (S 1≤ 97527 1)V 995aughan , closing(1977) parenthesis, closing parenthesis period .. RH implies S less or equal 6 period S ≤ 26 Deshouillers ( 1 975 / 76 ) , \ [STheorem\ leq .... open27 parenthesis Vaughan Deshouillers ( 1977 comma ).... Effinger , \ ] comma .... te Riele comma .... Zinoview .... S ≤ 25 Zhang , Ding ( 1 983 ) , open parenthesis 1997 closing parenthesis comma .... Saouter S ≤ 19 Riesel , Vaughan ( 1983 ) , open parenthesis 1 998 closing parenthesis closing parenthesis period .. The assumption of GRH implies the \ centerline { $ S \ leq 26 $ \quadS ≤ 7 DeshouillersRamaré( 1 995 ( 1 ) 975. / 76 ) , } validity of the Ternary Gold hyphenThe best known conditional results are the following . bach Conjecture for every odd integer N greater 5 comma and consequently the estimate \ centerline { $Theorem S \ leq ( Kaniecki25 $ \quad ( 1 995Zhang ) ) . , DingRH (implies 1 983 )S ≤ , 6}. S less or equal 4Theorem period ( Deshouillers , Effinger , te Riele , Zinoview ( 1997 ) , It .. is unclear yet .. whether it i s easier to deal with the Descartes Con hyphen \ centerline { $Saouter S \ leq 19 $ \quad Riesel , Vaughan ( 1983 ) , } j ecture than with( 1 the 998 open ) ) . parenthesisThe assumption Binary closing of parenthesis GRH implies Goldbach the validity Conjecture of the period Ternary Earlier methods for \ centerline { $Gold S -\ leq bach Conjecture7 $ \quad forRamar every\ ’{ odde} integer( 1 995N )> . 5}, and consequently estimation of thethe exceptional estimate set would yield the same estimates for the ex hyphen ceptional sets of the two different problems period However comma the method leading \ centerlineto Theorem{ 1The 6 open best parenthesis known for conditional a brief discussion results see Section are the1 8 comma following for more . details} see Pintz open parenthesis 2006 closing parenthesis closingS parenthesis≤ 4. yields a better bound for this case period Specifically\noindent commaTheorem we can ( prove Kaniecki ( 1 995 ) ) . \quad RH implies $ S \ leq 6 . $ the following period \noindent TheoremIt\ h is f i unclear l l ( Deshouillers yet whether , it\ ih s f easier i l l E tof f i dealn g e r with , \ theh f i Descartesl l te R i e Con l e , \ h f i l l Zinoview \ h f i l l ( 1997 ) , \ h f i l l Saouter Theorem 18 period- j ecture .. All but than O open with parenthesis the ( Binary X to ) the Goldbach power of Conjecture3 slash 5 log to . Earlier the power methods of 10 X closing parenthesis integersfor below estimation X can be written of the as exceptional set would yield the same estimates for \noindentthe sum of( at 1 most 998 three ) ) primes . \quad commaThe .. where assumption the last prime of GRH open parenthesis implies the if it exists validity closing ofparenthesis the Ternary Gold − bach Conjecturethe forex - every ceptional odd setsinteger of the $ two N different> 5 problems , $ and . consequently However , the the estimate can be method leading to Theorem 1 6 ( for a brief discussion see Section 1 8 , chosen as 2 commafor more 3 or 5 details period see Pintz ( 2006 ) ) yields a better bound for this case . \ beginWe remark{ a l i g n that∗} while the methods of proving Theorems 1 6 and 18 are S \ leq 4Specifically . , we can prove the following . similar comma neither one implies the other period openO(X parenthesis3/5 log10 X) However comma the proofX of Theorem \end{ a l i g n ∗} Theorem 18 . All but integers below can be 1 8 i s written as the sum of at most three primes , where the last prime ( if easier comma comparableit exists ) to can an beestimate chosen of type as E2 open , 3 or parenthesis5 . X closing parenthesis ll X to the power ofI t 4\ slashquad 5 logis to unclear the power yet of c\ Xquad openwhether parenthesis it cf period i s easier open parenthesis to deal 1 2 with period the 1 closing Descartes parenthesis Con − j ecture than withWe remark the ( thatBinary while ) Goldbach the methods Conjecture of proving . Theorems Earlier 1 methods 6 and 18 for are closing parenthesissimilar , neither one implies the other . ( However , the proof of Theorem estimationfor Goldbach of quoteright the exceptional s Problem period set closing would parenthesis yield the same estimates for the ex − ceptional sets1 8 of i s the two different problems . However , the method leading E(X) X4/5 logc X( to Theorem 1easier 6 ( for , comparable a brief discussion to an estimate see of Section type 1 8 , for more detailscf . ( 1 2 see Pintz ( 2006. ) 1 ) ) )yields for Goldbach a better ’ s bound Problem for . ) this case . Specifically , we can prove the following .

\noindent Theorem 18 . \quad Allbut $O ( Xˆ{ 3 / 5 }\ log ˆ{ 10 } X ) $ integers below $ X $ can be written as the sum of at most three primes , \quad where the last prime ( if it exists ) can be chosen as 2 , 3 or 5 .

We remark that while the methods of proving Theorems 1 6 and 18 are similar , neither one implies the other . ( However , the proof of Theorem 1 8 i s

\noindent easier , comparable to an estimate of type $E ( X ) \ l l X ˆ{ 4 / 5 }\ log ˆ{ c } X ($ cf.(12.1)) for Goldbach ’ s Problem . ) Landau quoteright s problems on primes .. 389 \ hspace14 period∗{\ ..f i Gaps l l }Landau between consecutive’ s problems Goldbach on primes numbers \quad 389 Denoting the consecutive Goldbach numbers by 4 = g 1 less g 2 less period period period .. we \ centerlinemay try to give{14 upper . \quad boundsGaps for the between occurring consecutive maximal gaps Goldbach numbers } Equation: open parenthesis 14 period 1 closing parenthesis .. A open parenthesis X closing parenthesis = maximum\ hspace ∗{\ g subf i kl l less} Denoting or equal X the to the consecutive power of open Goldbach parenthesis numbers g k plus 1 byminus $ g 4 k closing = parenthesis g 1 < g 2 < . . . $ \quad we Landau ’ s problems on primes 389 in this sequence period14 Goldbach . Gaps quoteright between s Conjecture consecutive is naturally Goldbach equivalent numbers to A open parenthesis X closing parenthesis = 2 \noindent may try toDenoting give upper the consecutive bounds for Goldbach the occurring numbers maximal by 4 = g1 gaps< g2 < ... for X greater equalwe 4 period .. Thus comma .. the problem of upper estimation of A open parenthesis X closing parenthesismay .. represents try to give upper bounds for the occurring maximal gaps \ begina new{ a approximation l i g n ∗} to Goldbach quoteright s Conjecture period .. In other words we may \ tagask∗{ :$ .. ( for which 14 functions . 1 .. f ) open $} parenthesisA ( X X closing ) parenthesis= \max ..{ cang .. we{ k guarantee}\ leq .. at ..X least}ˆ{ ..( g k + 1 } − g k ) (gk+1 one .. Goldbach A(X) = max −gk) (14.1) \end{ a l i g n ∗} g ≤X number in an interval of type open parenthesis X commak X plus f open parenthesis X closing parenthesis closing parenthesis period .. This problem differs from \noindent in thisin this sequence sequence . . Goldbach Goldbach ’ s ’ s Conjecture Conjecture is is naturally equivalent equivalent to to $ A the other approximationsA(X) = 2 infor the followingX ≥ respect4. Thus period , The the sharpest problem results of for upper estimation (the X problems ) = of Sections 2 $ 1 2 endash 1 3 and 1 5 endash 1 8 all use the circle method comma which f o r $ X \geqof A(X4) represents . $ \quad aThus new approximation , \quad the problem to Goldbach of upper ’ s Conjecture estimation . of i s specifically designedIn other to words treat additive we may problems ask : period for In which contrast functions to this commaf(X the) can we $ Abest estimates( X for ) A$ open\quad parenthesisr e p r e s X e n closing t s parenthesis can be derived from information concerning the a new approximationguarantee to Goldbach at least ’ s one Conjecture Goldbach . \quad numberIn in other an interval words of we type may distribution of primes(X,X + periodf(X)). This problem differs from the other approximations in the askThe : following\quad propositionfor which i s functions contained in\ thequad special$ case f vartheta ( X sub ) 1 $ = 7\ slashquad 1 2can plus\quad epsilonwe comma guarantee \quad at \quad l e a s t \quad one \quad Goldbach numberinanintervaloftypefollowing respect . The $( sharpest X results , X for + the fproblems ( X of Sections ) ) 1 2 .$ – \quad This problem differs from vartheta sub 21 = 3 1 and slash 1 65 –plus 1 8 epsilon all use in the the circle work method of Montgomery , which and i s Vaugham specifically open designed parenthesis to 1 975 closingthe other parenthesis approximations period in the following respect . The sharpest results for the problemstreat of Sections additive problems 1 2 −− 1 . 3 In and contrast 1 5 −− to this1 8 , all the usebest the estimates circle for methodA(X) , which Proposition periodcan .. be Let derived us suppose from we information have four positive concerning constants the vartheta distribution sub 1 comma of primes vartheta . sub 2 commai s specifically c sub 1 and designed to treat additive problems . In contrast to this , the The following proposition i s contained in the special case ϑ1 = 7/12 + ε, bestc sub estimates 2 less c sub 1 for vartheta $ A sub 1 ( .. with X the ) following $ can properties be derived : from information concerning the distributionϑ of2 = primes 1/6 + ε in . the work of Montgomery and Vaugham ( 1 975 ) . open parenthesisProposition a closing parenthesis . .. everyLet interval us suppose of type we open have square four bracket positive X minus constants Y comma X closing square bracket .. with X to the power of vartheta 1 less Y less X slash 2 .. contains at The followingϑ proposition1, ϑ2, c1 and c i2 < s c contained1ϑ1 with the in following the special properties case : $ \vartheta { 1 } = least c sub 1 Y slash( a log ) X primesevery for interval any X greater of type X sub[X 0 comma− Y,X] with Xϑ1 < Y < X/2 7open / parenthesis 1 2 b + closing\ varepsilon parenthesis .. for, .. $ all .. but .. c sub 2 X slash log X .. integer .. values .. n in contains at least c1Y/ log X primes for any X > X0, open$ square\vartheta bracket{ X comma2 } = 2 X closing 1 / square 6 bracket + \ ..varepsilon the .. interval$ in the work of Montgomery and Vaugham ( 1 975 ) . ( b ) for all but c2X/ log X integer values open square bracket n minus X to the power of vartheta 2 comma n closing square bracket .. contains a prime \noindent Propositionn ∈ [X, .2X\quad] theLet intervalus suppose we have four positive constants $ \vartheta { 1 } for any X greater X sub 0ϑ period .. Then [n − X 2, n] contains a prime for any X > X0. Then , Equation:\vartheta open parenthesis{ 2 } , 14 period c { 21 closing}$ and parenthesis .. A open parenthesis X closing parenthesis ll X to the$ c power{ 2 of} vartheta< c sub{ 1 vartheta1 }\vartheta sub 2 period { 1 }$ \quad with the following properties : In this way comma the combination of any pairA of(X estimates) Xϑ1 fromϑ2 . open parenthesis 3 period(14 1.2) 3 closing parenthesis( a ) \quad .. andevery interval of type $ [ X − Y , X ] $ \quad with $ X ˆ{\vartheta } 1 open< parenthesisY < 3XIn period this / 1 way 6 closing2 $ , the\ parenthesisquad combinationc o n implies t a i n s of a at any bound pair for ofA open estimates parenthesis from x (closing 3 . 1 parenthesis 3 ) l e a s t $ c { 1 } Y /$ log $X$ primes for any $X > X { 0 } , $ period Combining theand result ( 3 . vartheta 1 6 ) implies sub 1 = a 2 bound 1 slash 40for ofA Baker(x). Combining endash the result ϑ1 = 21/40 Harman endashof Pintz Baker open – parenthesis Harman 200– Pintz 1 closing ( 200 parenthesis 1 ) with with the the estimate estimate varthetaϑ2 = 1/20 sub + 2ε =of 1 slash 20\ hspace plus epsilon∗{\ f ofi l lChCh}( period .b Jia ) \ Jia (quad and open takingf parenthesis o r \quad into anda account l l taking\quad thatbut the\quad first - mentioned$ c { 2 } workX gives / $ l o g $ X $ \quad i n t e g e r \quad values \quad $ n \ in [ X , 2 X ] $ \quad the \quad i n t e r v a l into account thatactually the first some hyphen exponent mentionedϑ work1 slightly gives actually less than some 2 1 exponent / 40 ) we vartheta obtain sub 1 slightly less thanTheorem 2 1 slash 40( closing Baker parenthesis , Harman we , obtain Jia , Pintz ) . All intervals of type \noindent $ [ n −21 X ˆ{\vartheta } 2 , n ] $ \quad contains a prime for any Theorem open parenthesis[X,X + X 800 Baker] contain comma at Harman least one comma Goldbach Jia comma number Pintz , closing that is parenthesis , period .. All intervals$ X > of typeX open{ 0 square} . bracket $ \quad X commaThen X plus X 2 1 divided by 800 sub closing square bracket contain at least one Goldbach number comma that is comma gn + 1 − gn g21/800 ⇔ A(X) X21/800. (14.3) \ beginEquation:{ a l i g open n ∗} parenthesis 14 period 3 closing parenthesisn .. g n plus 1 minus g n ll g sub n to the power of \ tag ∗{$ ( 14 . 2 ) $} A(X) \ l l X ˆ{\vartheta { 1 }\vartheta { 2 }} 2 1 slash 800 Leftrightarrow A open parenthesisThe first X closing conditional parenthesis estimate ll X to the , power of 2 1 slash 800 period . The first conditional estimate comma \endEquation:{ a l i g n ∗} open parenthesis 14 period 4 closing parenthesis .. g n plus 1 minus g n ll log to the power of 3 g n Leftrightarrow A open parenthesisgn X+ closing1 − gn parenthesislog3 gn ll⇔ logA( toX) the powerlog3 ofX 3 XonRH on RH (14.4) Inwas this proved way by , Linnik the combination open parenthesis of 1952 any closing pair parenthesis of estimates comma fromwhile the ( 3 best . i1 s 3the ) following\quad and : (Theorem 3 . 16 .. ) open implieswas parenthesis proved abound by Kaacutetai Linnik for ( 1952 .. $A open ) , parenthesis while ( x thebest ) 1 967 i . s closing $ the Combining following parenthesis : theTheorem closing result parenthesis $ \vartheta { 1 } 2 =period 2 .. RH 1 implies /( Kátai g 40$ n plus 1 (ofBaker minus 1 967 g ) n ) ll.−− log toRH the implies power of 2gn g+1 n Leftrightarrow−gn log A opengn parenthesis⇔ A(X) X closing parenthesisHarman −− ll Pintz ( 200 1 ) with the estimate $ \vartheta { 2 } = 1 / 20 + \ varepsilon $ of Ch . Jia ( and taking log to the power of 2 X period 2 into account that the first − mentioned worklog X. gives actually some exponent $ \vartheta { 1 }$ slightly less than 2 1 / 40 ) we obtain

\noindent Theorem ( Baker , Harman , Jia , Pintz ) . \quad All intervals of type $ [ X , X + X \ f r a c { 2 1 }{ 800 } { ] }$ contain at least one Goldbach number , that is ,

\ begin { a l i g n ∗} \ tag ∗{$ ( 14 . 3 ) $} g n + 1 − g n \ l l g ˆ{ 2 1 / 800 } { n } \Leftrightarrow A(X) \ l l X ˆ{ 2 1 / 800 } . \end{ a l i g n ∗}

\ centerline {The first conditional estimate , }

\ begin { a l i g n ∗} \ tag ∗{$ ( 14 . 4 ) $} g n + 1 − g n \ l l \ log ˆ{ 3 } g n \Leftrightarrow A(X) \ l l \ log ˆ{ 3 } X on RH \end{ a l i g n ∗}

\noindent was proved by Linnik ( 1952 ) , while the best i s the following : Theorem \quad (K\ ’{ a} t a i \quad ( 1 967 ) ) . \quad RHimplies $g n + 1 − g n \ l l \ log ˆ{ 2 } g n \Leftrightarrow A(X) \ l l $

\ begin { a l i g n ∗} \ log ˆ{ 2 } X. \end{ a l i g n ∗} 390 .. Jaacutenos Pintz \noindent1 5 period390 .. Goldbach\quad exceptionalJ \ ’{ a} nos numbers Pintz in short intervals The results of Section 1 2 raise the problem of whether we can prove the \ centerlineanalogue of{ E1 open 5 . parenthesis\quad Goldbach X closing exceptional parenthesis = o numbers open parenthesis in short X closing intervals parenthesis} for short intervals comma that is comma \ hspace ∗{\ f i l l }The results of Section 1 2 raise the problem of whether we can prove the Equation: open390 parenthesisJános Pintz 1 5 period 1 closing parenthesis .. E open parenthesis X comma Y closing parenthesis : = E open parenthesis X plus Y closing parenthesis minus E open parenthesis X closing parenthesis \noindent analogueof1 5 . $E Goldbach ( X exceptional ) = o numbers ( X in )$ short for intervals short intervals ,that is , = o open parenthesis YThe closing results parenthesis of Section comma 1 2 raise the problem of whether we can prove for some function Y = Y open parenthesis X closing parenthesis period The above relation means that almost \ begin { a l i g n ∗}the all even analogue of E(X) = o(X) for short intervals , that is , \ tagintegers∗{$ ( are Goldbach 1 5 numbers . 1 in ) a short$} E(X,Y):=E(X+Y interval of type open square bracket X comma X plus Y open parenthesis) − E X closing ( Xparenthesis ) = closing o square ( bracket Y ) period , \endThe{ a first l i g n result∗} of such type was provedE by(X,Y Ramachandra) := E(X + openY ) − parenthesisE(X) = o(Y 1) 973, closing parenthesis(15.1) with an interval of type \noindent forfor some some function function $YY = Y (X =). The Y above ( Xrelation ) means . $ that The almost above all relation even means that almost all even Equation: openintegers parenthesis are 1 Goldbach5 period 2 closing numbers parenthesis in a short .. Y =interval Y open of parenthesis type [X,X X closing+ Y (X parenthesis)]. =integers X to the power are of Goldbach vartheta sub numbers 3 comma in vartheta a short sub 3 interval = 3 divided of by type 5 plus epsilon $ [ period X , X + Y ( X )The ] first. $ result of such type was proved by Ramachandra ( 1 973 ) with The result of Ramachandraan interval was of improvedtype to vartheta sub 3 = 1 slash 2 plus epsilon in Perelli comma Pintz open parenthesis 1 992 closing parenthesis period .. Soon afterwards comma .. simultaneously and indepen- dentlyThe first Perelli endashresult Pintz of such type was proved by Ramachandra ( 1 973 ) with an 3 intervalopen parenthesis of type 1 993 closing parenthesisY and= Y Mikawa(X) = openXϑ3 , parenthesis ϑ = + 1ε. 992 closing parenthesis(15 proved.2) the 3 5 significantly stronger estimates vartheta sub 3 = \ begin { a l i g n ∗} 7 slash 36 plus epsilonThe result and vartheta of Ramachandra sub 3 = 7 slash was 48 improved plus epsilon to commaϑ3 = 1/ respectively2+ε in Perelli period , Pintz A new ( feature of\ tag Mikawa∗{$ quoteright( 11 992 s5 result ) . Soon 2 ) afterwards $} Y = , Y simultaneously ( X ) and = independently X ˆ{\vartheta Perelli { 3 }} , was\vartheta that comma– similarly{ Pintz3 } ( to= 1 the 993\ f work r a) c and{ of3 Iwaniec Mikawa}{ 5 and} ( Jutila+ 1 992 open\ varepsilon ) proved parenthesis the 1979significantly. closing parenthesis stronger comma \end{ a l i g n ∗} it was based estimates ϑ3 = 7/36 + ε and ϑ3 = 7/48 + ε, respectively . A new feature of on a combinationMikawa of analytic ’ s result and sieve was methods that period, similarly Further to refinements the work of this Iwaniec and Jutila \noindentmethod yieldedThe( theresult 1979 sharper ) ,of it results Ramachandra was based on was a combination improved to of analytic $ \vartheta and sieve{ methods3 } = . 1 / 2vartheta + \ varepsilon sub 3Further = 7 slash$ refinements 78 in plus Perelli epsilon of .. this , Ch Pintz period Jia open parenthesis 1 995 b comma 1 995 c closing parenthesis( 1 992 comma ) . \methodquad Soon yielded afterwards the sharper , \ resultsquad simultaneously and independently Perelli −− Pintz ( 1 993 ) and Mikawa ( 1 992 ) proved the significantly stronger estimates $ \vartheta { 3 } open parenthesisϑ3 1= 5 7 period/78 + ε 3 closingCh . Jia parenthesis ( 1 995 vartheta b , 1 995 sub c )3 ,=(15 7 slash.3) ϑ 83 1= plus 7/81 epsilon + ε H.Z. .. H period Z =period $ Li open parenthesisLi ( 1 995 1 995 ) , closing parenthesis comma $ 7 / 36 + \ varepsilon $ and $ \vartheta { 3 } = 7 / 48 + vartheta sub 3 = 1 1 slashϑ3 1= 60 11 plus/160 epsilon + ε Baker .. Baker , Harman comma Harman , Pintz comma ( 1995 Pintz / 97 open ) . parenthesis 1995 slash\ varepsilon 97 closing parenthesisCurrently, $ respectively period , the best known . A new estimates feature are of the Mikawa following ’ s . result wasCurrently that comma , similarlyTheorem the best toknown(the Ch estimates . work Jia 1 of 996 are Iwaniec the b ) following. and period JutilaAlmost ( 1979 all even ) , integers it was are based onTheorem a combination open parenthesisGoldbach of analytic numbers Ch period andJia 1 sieve996 b closing methods parenthesis . Further period refinements.... Almost all even of integers this are ϑ Goldbach numbersin every interval of type [X,X + X 3], for ϑ3 = 7/18 + ε. More precisely \noindentin every intervalmethodwe of have type yielded open square the sharper bracket X results comma X plus X to the power of vartheta 3 closing square bracket comma for vartheta sub 3 = 7 slash 1 8 plus epsilon period More precisely \noindentwe have $ \vartheta { 3 } = 7 / 78 + \ varepsilon $ \quad Ch . Jia ( 1995b , 1995 c ) , E(X,Xϑ3 ) AXϑ3 log−A X foranyA > 0. (15.4) $Equation: ( 1 open 5 parenthesis . 3 1 ) 5 period\vartheta 4 closing parenthesis{ 3 } = .. E open7 /parenthesis 8 1 X comma + \ Xvarepsilon to the power $ of\quad varthetaH.Z. sub 3 closingTheorem Li(1995) parenthesis( Kaczorowski , ll A X to the – power Perelli of vartheta – Pintz sub ( 1 3 993 log to) ) the. powerUnder of minus the A X for any A greater 0 periodGRH we have \ centerlineTheorem open{ $ parenthesis\vartheta Kaczorowski{ 3 } endash= Perelli 1 1 endash / Pintz 1 open 60 parenthesis + \ varepsilon 1 993 closing parenthesis$ \quad Baker , Harman , Pintz ( 1995 / 97 ) . } closing parenthesis period .. Under the GRH we6+ haveε 6+ε \noindentEquation:Currently open parenthesis , the 1 5 best periodE(X, known 5log closing estimatesX) parenthesis = o(log areX ..) E theforanyε open following parenthesis > 0. . X comma(15 log.5) to the power of 6 plus epsilon X closing parenthesis = o open parenthesis log to the power of 6 plus epsilon X closing \noindent TheoremWe ( conclude Ch . Jia this 1 996 section b ) with . \ h a f further i l l Almost problem all , even which integers is a combina- are Goldbach numbers parenthesis for anytion epsilon of greater the approaches 0 period of Section 1 2 and the present section . Specifically We conclude this section with a further problem comma whichY = isY ( aX combination) \noindent inevery, we can interval ask for the oftype shortest $[ interval X , Xfor + which Xˆ we{\ canvartheta guarantee} 3 ] of the approachesbeyond of Section ( 1 1 5 2 . and 1 ) the an present estimate section of type period Specifically comma we can , $ask ffor o r the $ shortest\vartheta interval Y{ =3 Y} open= parenthesis 7 / X 1closing 8 parenthesis + \ varepsilon for which we can. guarantee $ More beyond precisely open parenthesis 1 5 period 1 closing parenthesis an estimate of type 1−δ \noindentEquation:we open have parenthesis 1 5 period 6 closingE(X,Y parenthesis) Y .. E, open parenthesis X comma(15. Y6) closing parenthesis ll Y to the power of 1 minus delta comma \ begin { a l i g n ∗}with a given absolute constant δ > 0. The strongest known result of this with a given absolutekind was constant achieved delta greater recently 0 period by A The . Languasco strongest known ( 2004 result ) . of this \ tagkind∗{$ was ( achieved 1 5recently . by 4 A period ) $} LanguascoE ( open X parenthesis , X ˆ{\ 2004vartheta closing parenthesis{ 3 }} period) \ l l A X ˆ{\varthetaTheorem{ 3 }}\( Languascolog ˆ{ − ) .A } XThe f o r estimate any A( 1> 5 .0 6 ) . is Theorem .. open parenthesis Languasco7/24+ closingε parenthesis period .. The .. estimate open parenthesis 1 5 \end{ a l i g n ∗} true for Y = X if ε > 0 is arbitrary , with a suitably chosen period 6 closing parenthesisδ = δ(ε) > ..0 is. true for Y = X to the power of 7 slash 24 plus epsilon .. if epsilon greater 0 is arbitrary comma .. with a suitably chosen delta = delta open parenthesis epsilon closing parenthesis\noindent greaterTheorem 0 period ( Kaczorowski −− P e r e l l i −− Pintz ( 1 993 ) ) . \quad Under the GRH we have \ begin { a l i g n ∗} \ tag ∗{$ ( 1 5 . 5 ) $} E(X, \ log ˆ{ 6 + \ varepsilon } X ) = o ( \ log ˆ{ 6 + \ varepsilon } X ) f o r any \ varepsilon > 0 . \end{ a l i g n ∗}

We conclude this section with a further problem , which is a combination of the approaches of Section 1 2 and the present section . Specifically , we can ask for the shortest interval $Y = Y ( X ) $ for which we can guarantee beyond ( 1 5 . 1 ) an estimate of type

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 5 . 6 ) $} E(X,Y) \ l l Y ˆ{ 1 − \ delta } , \end{ a l i g n ∗}

\noindent with a given absolute constant $ \ delta > 0 . $ The strongest known result of this kind was achieved recently by A . Languasco ( 2004 ) .

\noindent Theorem \quad ( Languasco ) . \quad The \quad estimate ( 1 5 . 6 ) \quad i s true f o r $ Y = X ˆ{ 7 / 24 + \ varepsilon }$ \quad i f $ \ varepsilon > 0 $ is arbitrary , \quad with a suitably chosen $ \ delta = \ delta ( \ varepsilon ) > 0 . $ Landau quoteright s problems on primes .. 39 1 \ hspace1 6 period∗{\ f .. i Thel l }Landau Goldbach ’ endash s problems Linnik Problem on primes \quad 39 1 As another appro to the power of x-i mation to Goldbach quoteright s Problem comma Linnik open parenthesis 1\ centerline 95 1 comma ..{1 1 953 6 . closing\quad parenthesisThe Goldbach −− Linnik Problem } proved that every even integer can be expressed as a sum of two primes and As another $ appro ˆ{ x−i }$ mation to Goldbach ’ s Problem , Linnik ( 1 95 1 , \quad 1 953 ) K powers of two period In his original work K was an unspecifiedLandau large ’number s problems period on primes 39 1 provedOne can that try to every show Linnik even quoteright integer s can theorem be with expressed explicitly as given a sum values of of twoK comma primes and $ K $ powers of two .1 In 6 . his originalThe Goldbach work –$ KLinnik $ was Problem an unspecified large number . at least for even numbersAs another N greaterappro N subx−i 0mation period The to Goldbachbest possible ’ estimate s Problem K = , 0 Linnik i s ( 1 95 Oneclearly can equivalent try to to show Goldbach Linnik quoteright ’ s theorem s Conjecture with for explicitly N greater N sub given 0 period values of $ K , $ at least for1 even , 1numbers 953 ) proved $ N that> everyN { even0 } integer. $ The can bebest expressed possible as aestimate sum $ K Linnik quoterightof s two .. proof primes was .. and significantlyK powers .. simplified of two by . .. In Gallagher his original .. open work parenthesisK was 1 975 an closing =parenthesis 0 $ period i s .. Later clearly equivalentunspecified to Goldbach large number ’ s . Conjecture One can try for to show $ N Linnik> N ’ s theorem{ 0 } . with $ explicit estimatesexplicitly for K were given based values on Gallagher of K, quoterightat least for s work even : numbers N > N0. The best K = 54 0 .. Liupossible endash Liu estimate endash WangK = 0openi s parenthesis clearly equivalent 1 998 b closing to Goldbach parenthesis ’ s comma Conjecture LinnikEquation: ’ s open\quad parenthesisproof was 1 6 period\quad 1significantly closing parenthesis\ ..quad K tosimplified the power of K by =\ subquad = 25Gallagher 2 250 to \quad ( 1 975 ) . \quad Later explicit estimatesfor N > for N0. $ K $ were based on Gallagher ’ s work : the power of 0 H sub TLinnik period ’ period s proof Z sub was period tosignificantly the power of Z simplified period Li open by parenthesis Gallagher 2000 b ( closing parenthesis Wang sub open parenthesis 1999 closing parenthesis comma to the power of comma \ centerline { $1 K 975 = ) . 54 Later 0 $ explicit\quad estimatesLiu −− Liu for −−K wereWang based ( 1 998 on Gallagher b ) , } ’ s K = 1 906 .. Hwork period : Z period Li open parenthesis 2001 closing parenthesis period The conditional estimates commaK proved= 540 underLiu GRH – Liu comma – Wang were the( 1 following 998 b ) ,: \ beginK ={ 770a l i ..g n J∗} period Y period Liu comma M period C period Liu comma T period Z period Wang open parenthesis\ tag ∗{$ ( 1998 1 a closing 6 parenthesis . 1 comma) $} K ˆ{ K } = { = } 25 { 2 } 250 ˆ{ 0 } H { T . } . Z ˆ{ Z. } { . } KLi ( 20000 bZ. ) { Wang }, ˆ{ , } { ( 1999 ) open parenthesis 1 6 period 2 closingK parenthesis== 252250 K =H 200T..Z ... JLi period(2000b Y)W period ang(1999) Liu, comma M period(16.1) C period Liu, } comma T period Z period Wang open parenthesis 1999 closing parenthesis comma \end{ a l i g n ∗} K = 1 60 .. Tianze Wang open parenthesisK = 1 999 1906 closingH . parenthesis Z . Li ( 2001 period ) . These results wereThe improved conditional by D period estimates R period , proved Heath under hyphen GRH Brown , comma were the J period following hyphen : C period \ centerline { $ K = 1 906 $ \quad H.Z. Li ( 2001 ) . } Puchta open parenthesis 2002K = closing770 J parenthesis . Y . Liu , M . C . Liu , T . Z . Wang ( 1998 a ) , to K = 13 open(16 parenthesis.2) K =200 and KJ = . 7 Y on . Liu GRH , closingM . C parenthesis . Liu , T . and Z . simultaneously Wang ( 1999 and ) , independently \ centerline {The conditional estimates , proved under GRH , were the following : } by K = 160 Tianze Wang ( 1 999 ) . I period Ruzsa and the authorThese to results were improved by D . R . Heath - Brown , J . - C . \ centerlineTheorem 19{ open$Puchta K parenthesis = ( 2002 770 Pintz ) $ endash\quad RuzsaJ .Y. open Liuparenthesis ,M.C. 2003 comma Liu ,T. 200 ? closing Z .Wang( parenthesis 1998 a ) , } closing parenthesisto periodK = .. 13( All sufficientlyand K = large 7 on even GRH inte ) andhyphen simultaneously and independently \noindentgers can be$( expressedby 1 as a 6 sum . of two 2 primes ) and K= at most 200$ eight powers\quad of J .Y. Liu ,M.C. Liu ,T. Z . Wang( 1999 ) , two period .. UnderI . Ruzsa GRH the and same the result author is true to with at most seven powers of two period \ centerline { $ K = 1 60 $ \quad Tianze Wang ( 1 999 ) . } The proof of TheoremTheorem 1 9 relies 19 on( the Pintz Structural – Ruzsa Theorem(2003, of200?)) Section. 1All 8 sufficiently large even L inte - gers can be expressed as a sum of two primes and at most eight \ hspaceand on∗{\ a muchf i l lpowers more}These effective of results twotreatment . were of Under improvedthe exponential GRH by the Dsum . same sum R . exp result Heath open is− parenthesis trueBrown with , 2 at J to .mostthe− powerC . ofPuchta ( 2002 ) nu alpha closing parenthesisseven powers of two . \noindentnu = 1 to $KThe = proof 13 of Theorem ( $ and 1 9 $Krelies on = the 7 Structural $ onGRH) Theorem and simultaneously of Section and independently by than those applied1 8 in open parenthesis 1 6 period 1 closing parenthesis endash open parenthesis 1 6 period 2 closing\noindent parenthesisI . Ruzsaperiod and the author to As a natural refinement of the result of Linnik we may askL for an asymp hyphen \noindent Theorem 19 ( Pintz −− Ruzsa $( 2003 , 200 ? ) ) .$ \quad All sufficiently large even inte − totic formula for the number R sub k to the power of prime open parenthesis N closingP parenthesis of representationgers can be of an expressedand even on integere a much as as a more sum of effective two primes treatment and of at the most exponential eight powers sum of exp ν twothe .sum\quad of twoUnder(2 primesα) GRH and k the powers same of two result period is This true problem with i s atopen most for every seven powers of two . value of k period Recently comma however comma in collaboration with A period Languasco and A period \ hspaceZaccagnini∗{\ f comma i l l }The we were proof able of to Theorem show the following 1 9 relies periodν = 1 on the Structural Theorem of Section 1 8 Theorem 20 openthan parenthesis those applied Languasco in endash ( 1 6 .Pintz 1 ) endash– ( 1 6 Zaccagnini . 2 ) . open parenthesis 2007 closing parenthesis\ [L closing\ ] parenthesisAs period a natural .. Let refinement k be any positive of the result of Linnik we may ask for an asymp integer comma X .. sufficiently large period .. Then comma0 .. a f-t er the eventual deletion of at most - totic formula for the number Rk(N) of representation of an even integere C open parenthesisas thek closing sum parenthesis of two primes X to the and powerk powers of 3 slash of 5two open . Thisparenthesis problem log X i closings open parenthesis for \noindent and on a much more effective treatment of the exponential sum $ \sum $ to the power of openevery parenthesis value 10 of closingk. Recently parenthesis , however .. even integers , in collaboration below X comma with we A can . Languasco give an asymptotic forexp hyphen $ ( 2 ând{\nu A .}\alpha ) $ mula for R subZaccagnini k to the power , we of prime were open able parenthesis to show the N closing following parenthesis . for the remaining even values \ [ \nu = 1 \ ] of N less X period Theorem 20 ( Languasco – Pintz – Zaccagnini ( 2007 ) ) . Let k be any positive integer ,X sufficiently large . Then , a f − t er the eventual deletion of at most \noindent than those applied in ( 1 6 . 1 ) −− ( 1 6 . 2 ) . C(k)X3/5( log X)(10) even integers below X, we can give an asymptotic for - mula for R0 (N) for the remaining even values of N < X. As a natural refinement of thek result of Linnik we may ask for an asymp − totic formula for the number $ R ˆ{\prime } { k } ( N ) $ of representation of an even integere as the sum of two primes and $ k $ powers of two . This problem i s open for every value of $ k . $ Recently , however , in collaboration with A . Languasco and A .

\noindent Zaccagnini , we were able to show the following .

\noindent Theorem 20 ( Languasco −− Pintz −− Zaccagnini ( 2007 ) ) . \quad Let $ k $ be any positive integer $ , X$ \quad sufficiently large . \quad Then , \quad a $ f−t $ er the eventual deletion of at most

\noindent $ C ( k ) X ˆ{ 3 / 5 } ( $ l o g $ X ) ˆ{ ( 10 ) }$ \quad even integers below $ X , $ we can give an asymptotic for − mula f o r $ R ˆ{\prime } { k } ( N ) $ for the remaining even values of $N < X . $ 392 J´anos Pintz 17. F − u rther approximations to Goldbach ’ s Conjecture H . Mikawa ( 1 993 ) studied the moments of the diﬀerences of Goldbach

∞ numbersG = {gk}k=1, X ∗ α ∗ 0 Mα(X) = (gn+1 − gn) , gk = min(gk, X), α ≥ 0, 0 = 1. (17.1) gn≤X

300 He proved that M3(X) X log X and

α−1 Mα(X) = 2 X + o(X) for 0 < α < 3. (17.2) The above moments are sensible both for the total size E(X) of the ex - ceptional set ( cf . ( 1 2 . 1 ) ) and for the concentration of Goldbach exceptional numbers . As a sharper form of ( 1 7 . 2 ) we may formulate the α−1 1−δ(α) Conjecture A .Mα(X) = 2 X + O(X ) holds for any α ≥ 0 and

correspondingδ(α) > 0. To see the diﬃculty of Conjecture A we remark that ( i ) for α = 0 the assertion i s equivalent to the deep theorem E(X) X1−δ of Montgomery and Vaughan ( cf . ( 1 2 . 5 ) ) . ( ii ) The truth of Conjecture A i s equivalent to ε Conjecture B .gn + 1 − gn ε gn for any ε > 0. Using the Structural Theorem of Section 18 and the theorem of Baker – Harman – Jia – Pintz ( cf . ( 14 . 3 ) ) about the gaps between consecutive Gold - bach numbers we can deduce 341 Theorem 2 1 (J.P.) . Conjecture A is true for any α < 21 = 16.238... The following result refers to the concentration of Goldbach exceptional numbers . Theorem 22 (J.P.) . One can discard a set E0 of at most O(X3/5 log10 X) Goldbach exceptional numbers m ∈ [X/2,X] so that the remaining set will contain at most C Goldbach exceptional numbers in any interval of type [Y,Y + Y 1/3] ⊆ [X/2,X], where C is an explicitly calculable absolute con - stant . The above result clearly implies that E(X) X2/3. In Section 16 we have seen that starting from any even number and subtracting from it a number of the form

K X 2νi ,K ≤ 8, (17.3) i = 1 392 .. Jaacutenos Pintz \noindent1 7 period392 F-u rther\quad approximationsJ \ ’{ a} nos to Pintz Goldbach quoteright s Conjecture H period Mikawa open parenthesis 1 993 closing parenthesis .. studied the moments of the differences of Goldbach\ centerline { $ 1 7 . F−u $ rther approximations to Goldbach ’ s Conjecture } numbers G = open brace g k closing brace sub k = 1 to the power of comma to the power of infinity Equation: \ hspace ∗{\ f i l l }H . Mikawa ( 1 993 ) \quad studied the moments of the differences of Goldbach open parenthesis 1( 7 an period empty 1 closing sum meansparenthesis 0 ) we.. M arrive sub alpha at a open Goldbach parenthesis number X closing . The parenthesis numbers = sum open parenthesis g sub n plus 1 to the power of * minus g n closing parenthesis to the power of alpha comma \ begin { a l i g n ∗}of g sub k to the powertype of * ( = 1minimum 7 . 3 ) form open a parenthesis very thin g set k comma , having X closing at most parenthesis ( log X comma)K elements alpha greater equalnumbers 0 comma G 0 to =the power\{ ofg 0 = 1 k period\} gˆ n{\ toinfty the power} { of lessk or = equal 1 X ˆ{ , }}\\\ tag ∗{$ ( 1 7 . 1 )below $} M {\X. Onealpha can} prove( theX analogue ) = of\sum this result( g , too ˆ{ ,∗ that } { startingn + 1 } He proved thatfrom M sub any 3 open even parenthesis X closing parenthesis ll X log to the power of 300 X and − Equation:g n open ) ˆ parenthesis{\alpha 1} 7 period, g 2 ˆ{ closing ∗ } parenthesis{ k } = .. M\min sub alpha( open g parenthesis k , X X closing ) parenthesis, \alpha = 2 to\geq the power0 of alpha , 0 minus ˆ{ 0 1 X} plus= o open 1 parenthesis . \\ g X n closing ˆ{\ leq parenthesisX } for 0 less alpha less\end 3{ perioda l i g n ∗} The above moments are sensible both for the total size E open parenthesis X closing parenthesis of the ex hyphen\ centerline {He proved that $M { 3 } (X) \ l l X \ log ˆ{ 300 } X $ andceptional} set open parenthesis cf period open parenthesis 1 2 period 1 closing parenthesis closing parenthesis and for the concentration of Goldbach exceptional \ beginnumbers{ a l iperiod g n ∗} As a sharper form of open parenthesis 1 7 period 2 closing parenthesis we may formulate the \ tagConjecture∗{$ ( A 1 period 7 M sub . alpha 2 open ) $} parenthesisM {\alpha X closing} parenthesis( X )= 2 to = the 2 power ˆ{\ ofalpha alpha minus− 1 X1 plus} X O open + parenthesis o ( X X to the ) power f o r of 1 0 minus< delta\alpha open parenthesis< 3 alpha . closing parenthesis closing parenthesis\end{ a l i g ....n ∗} holds for any alpha greater equal 0 .... and corresponding delta open parenthesis alpha closing parenthesis greater 0 period TheTo above see the moments difficulty of are Conjecture sensible A we both remark for that the total size $ E ( X ) $ of the ex − ceptionalopen parenthesis set (i closing cf . parenthesis ( 1 2 . 1 for ) alpha ) and = 0 for the assertion the concentration i s equivalent to of the Goldbach deep theorem exceptional E open parenthesisnumbers X . closing As a parenthesis sharper form ll of ( 1 7 . 2 ) we may formulate the X to the power of 1 minus delta of Montgomery and Vaughan open parenthesis cf period open parenthesis 1 2\noindent period 5 closingConjecture parenthesis A closing $ . parenthesis M {\alpha period } ( X ) = 2 ˆ{\alpha − 1 } Xopen + parenthesis O ( ii X closing ˆ{ 1 parenthesis− \ delta The truth( of Conjecture\alpha A i) s equivalent} ) $ to\ h f i l l holds for any $ \Conjecturealpha B\geq period g0 n $ plus\ 1h minus f i l l gand n ll sub epsilon g sub n to the power of epsilon for any epsilon greater 0 period \ beginUsing{ thea l i g Structural n ∗} Theorem of Section 18 and the theorem of Baker endash correspondingHarman endash Jia endash\ delta Pintz( open\ parenthesisalpha cf) period> ..0 open parenthesis . 14 period 3 closing parenthesis closing\end{ a parenthesis l i g n ∗} .. about the gaps between consecutive Gold hyphen bach numbers we can deduce \ centerlineTheorem 2 1{To .. open see parenthesis the difficulty J period P of period Conjecture closing parenthesis A we remark period .. that Conjecture} A is true for any alpha less 341 divided by 2 1 = 1 6 period 238 period period period \ hspaceThe following∗{\ f i lresult l }( i refers ) f oto r the $ concentration\alpha = of Goldbach 0 $ the exceptional assertion i s equivalent to the deep theorem $ Enumbers ( period X ) \ l l $ Theorem 22 open parenthesis J period P period closing parenthesis period .... One can discard a set E to the\noindent power of prime$ X of ˆ{ at1 most− O open \ delta parenthesis}$ ofX to Montgomery the power of 3and slash Vaughan 5 log to the ( cf power . ( of 1 10 2 X . closing 5 ) ) . parenthesis \ centerlineGoldbach exceptional{( ii ) numbers The truth m in open of Conjecture square bracket A X i slash s equivalent 2 comma X closing to } square bracket .. so that the remaining set will \noindentcontain atConjectureB most C .. Goldbach $. .. exceptional g n numbers + 1 in any− intervalg of n type\ l l {\ varepsilon } g ˆ{\ varepsilon } { n }$ f o ropen any square $ \ bracketvarepsilon Y comma> Y plus0 Y to . the $ power of 1 slash 3 closing square bracket subset equal open square bracket X slash 2 comma X closing square bracket comma .. where C is an explicitly calculable absolute conUsing hyphen the Structural Theorem of Section 18 and the theorem of Baker −− Harmanstant period−− Jia −− Pintz ( c f . \quad ( 14 . 3 ) ) \quad about the gaps between consecutive Gold − bachThe above numbers result we clearly can implies deduce that E open parenthesis X closing parenthesis ll X to the power of 2 slash 3 period \noindentIn .. SectionTheorem .. 16 .. we 2 1have\quad .. seen(J.P.). that .. starting from\quad any evenConjecture number and A is true for any $ \alpha < subtracting\ f r a c { 341 from}{ it a2 number 1 } of=1 the form 6 . 238 . . .$ K Equation: open parenthesis 1 7 period 3 closing parenthesis .. sum 2 to the power of nu sub i comma K lessThe or following equal 8 comma result i = 1 refers to the concentration of Goldbach exceptional numbersopen parenthesis . an empty sum means 0 closing parenthesis we arrive at a Goldbach number period The numbers of \noindenttype openTheorem parenthesis 22 1 7( period J . P 3 . closing ) . \ parenthesish f i l l One form can a very discard thin set a comma set $ having E ˆ{\ atprime most open}$ parenthesisofatmost log X $O closing ( parenthesis Xˆ{ 3 to the / power 5 }\ of K elementslog ˆ{ below10 } XX period ) $ One can prove the analogue of this result comma too comma that starting from any even \noindent Goldbach exceptional numbers $ m \ in [ X / 2 , X ] $ \quad so that the remaining set will contain at most $ C $ \quad Goldbach \quad exceptional numbers in any interval of type

\noindent $ [ Y , Y + Y ˆ{ 1 / 3 } ] \subseteq [ X / 2 , X ] , $ \quad where $ C $ is an explicitly calculable absolute con − stant .

\ centerline {The above result clearly implies that $E ( X ) \ l l X ˆ{ 2 / 3 } . $ }

In \quad Sec tion \quad 16 \quad we have \quad seen that \quad starting from any even number and subtracting from it a number of the form

\ begin { a l i g n ∗} K \\\ tag ∗{$ ( 1 7 . 3 ) $}\sum 2 ˆ{\nu { i }} ,K \ leq 8 , \\ i = 1 \end{ a l i g n ∗}

\noindent ( an empty sum means 0 ) we arrive at a Goldbach number . The numbers of

\noindent type ( 1 7 . 3 ) form a very thin set , having at most ( log $X ) ˆ{ K }$ elements below $X . $ One can prove the analogue of this result , too , that starting from any even Landau quoteright s problems on primes .. 393 \ hspacenumber∗{\ andf i adding l l }Landau to it a ’number s problems of type open on primesparenthesis\quad 1 7 period393 3 closing parenthesis we can reach a Goldbach \noindentnumber periodnumber The andresult adding below shows to itthat a this number can be of achieved type for ( any 1 7 N . less 3 or ) equalwe can X reach a Goldbach with a set having just two suitably chosen elements below X period \noindentTheorem 23number period .... . The Let X result greater below X sub 0 shows period ....that Then this we have can integers be achieved a comma for b less any or equal $ N X \ leq X $ Landau ’ s problems on primes 393 such that for number and adding to it a number of type ( 1 7 . 3 ) we can reach a every N less or equal X at least one of N comma N plus a comma N plus b is a Goldbach number \noindent withGoldbach a set having just two suitably chosen elements below $ X . $ From the abovenumber result we . can The further result deduce below the shows existence that of this an arbi can hyphen be achieved for any N ≤ X trarily thin universalwith aset set S with having just two suitably chosen elements below X. \noindentEquation:Theorem open parenthesis 23 . 1\ h 7 f period i l l Let 4 closing $ X parenthesis> X ..{ Z to0 the} power. $ of\ h plus f i l = l GThen minus we S periodhave integers $ a , b Theorem\ leq X $23 such. Let thatX >for X0. Then we have integers a, b ≤ X such that Theorem 24 periodfor Let f open parenthesis x closing parenthesis be an arbitrary increasing function with limint x right arrow infinityevery to theN power≤ X ofat f least open parenthesis one of N,N x closing+ a, N parenthesis+ b is a Goldbach = number \noindentinfinity periodevery .. Then $ N we can\ leq find a sX$ et S such atleastoneof that $N , N + a , N + b $ is a GoldbachFrom the number above result we can further deduce the existence of an arbi - Equation: opentrarily parenthesis thin 1 universal 7 period 5 setclosingS with parenthesis .. bar n less or equal X semicolon n in S bar less or equal f open parenthesis X closing parenthesis for X greater X sub 0 comma G minus S = Z to the power of plusFrom comma the above result we can further deduce the existence of an arbi − + trarilythat is comma thin every universal integer can set be written $ S $ as with the differenceZ = G − of S a. Goldbach number (17.4) and an element of S period Theorem 24 . Let f(x) be an arbitrary increasing function with \ begin18 period{ a l i g.. n Explicit∗} formulas in the additive theory of primes limf( x) = ∞. Then we can find a s et S such that \ tagIn∗{ ..$ this ( .. section 1x 7 ..→∞ we . .. sketch 4 .. ) one $} ..Z of ˆ the{ + ..} basic= .. ideas G ..− behindS. the .. new .. ap hyphen \endproximation{ a l i g n ∗} of Goldbach quoteright s Conjecture period In Section 2 we cited the fact that + Hilbert expressed the hope that the| n ≤ RiemannX; n ∈ S endash |≤ f(X von)forX Mangoldt > X0, PrimeG − Number S = Z , (17.5) \noindentFormula openTheorem parenthesis 24 . 3 period Let 1$ closing f ( parenthesis x ) might $ be help an with arbitrary the solution increasing of the Goldbach function comma with Twin$ \lim Prime{ x that\rightarrow is , every integer\ infty can} beˆ{ writtenf ( as} thex difference ) = $of a Goldbach number $and\ infty .. Generalized.and $ .. an\ Twinquad element ..Then Prime of we.. ConjecturesS can. find period a s ..et We $ .. S will $ .. sketchsuch ..that below .. a .. two hyphen dimensional analogue18 .of open Explicit parenthesis formulas 3 period 1 in closing the parenthesis additive comma theory which of primes plays a basic role in the\ begin proof{ ofa l iall g n ∗} In this section we sketch one of the basic ideas \ tagTheorems∗{$ ( 16 1 endashbehind 7 24 .theopen 5 parenthesis new ) $}\ ap inmid some - proximation casesn directly\ leq of commaX Goldbach ;in some n ’ s cases Conjecture\ in throughS Theorem .\mid In 1\ 6leq commaf ( X )Section f o r 2 X we> citedX the{ fact0 } that,G Hilbert− expressedS = Z the ˆ{ hope+ } that, the \endfor{ examplea l i g n ∗} closingRiemann parenthesis – von period Mangoldt Prime Number Formula ( 3 . 1 ) might help with Let X be any largethe number solution comma of the P less Goldbach or equal , square Twin root Prime of X a and parameter Generalized to be chosen later Twin period \noindentWe will applythat thePrime is circle , everymethod Conjectures integerwith major . can and be Weminor written arcs will defined as sketch theby difference below of a a two Goldbach - number andEquation: an element opendimensional parenthesis of $ S 1 8 analogue period . $ 1 closing of ( 3 parenthesis . 1 ) , which .. M plays = union a basicof q less role or equal in the p open proof parenthesis of a sub comma to theall power of union of a sub = 1 from q closing parenthesis = 1 to q bracketleftbigg a divided by\ centerline q minus P divided{18Theorems . by\quad qX sub 16Explicit comma – 24 ( ina divided formulassome cases by q in plus directly the P divided additive , in some by qX cases theory to the through power of primes of Theorem bracketrightbigg} comma m = open square1 6 , bracket 0 comma 1 closing square bracket backslash M period In \quad t h i s \quad s e c t i o n \quad we \quad sketch \quad one \quad o f the \quad b a s i c \quad i d e a s \quad behind the \quad new \quad ap − We consider asfor usual example ) . √ proximation1 of GoldbachLet X be any ’ s large Conjecture number .,P In≤ SectionX a parameter 2 we cited to be the chosen fact later that . HilbertEquation: expressed openWe parenthesis will the apply hope 1 8 theperiod that circle 2 closing the method Riemann parenthesis with−− major ..von R open andMangoldt parenthesis minor Prime arcs m defined closing Number parenthesis by = toFormula the power ( of 3 = . 0 1 from ) might plus integral help = with R sub the 1 open solution parenthesis of them closing Goldbach parenthesis , Twin plus Prime R sub 2 open and \quad Generalized \quad Twin \quad Prime \quad Conjectures . \quad We \quad w i l l \quad sketch \quad below \quad a \quad two − parenthesis m closing parenthesis comma to p 1 to the power of sum plus] sub p 2 to the power of = m integral dimensional analogue of ( 3 .[ 1S )q ,)=1 whicha P playsa aP basic role in the proof of all log p 1 times log p 2 = integral toM the= power(a ofa S toq the[ − power of+ 2 open, parenthesism = [0, 1] \ alphaM. closing parenthesis(18.1) e , =1 q qX q qX open parenthesis minus m alpha closingq parenthesis≤p d alpha M, m \noindent Theorems 16 −− 24 ( in some cases directly , in some cases through Theorem 1 6 , where We consider as usual Equation: open parenthesis 1 8 period 3 closing parenthesis .. S open parenthesis alpha closing parenthesis 1 =\noindent sum log pefor open exampleparenthesis ) p .alpha closing parenthesis comma X sub 1 = X to the power of 1 minus epsilon sub 0 comma L = log X period X sub 1 less p less or equal X R 2 Let $X$ be any large number $= ,+ = PR1(m)+\ leqR2(m), \ sqrtS ({α)eX(−mα}$)dα a parameter to be chosen later . R(m) = 0 P (18.2) + =mR R We will apply the circle method withp1p major2 log andp1·log minorp2= arcs defined by M m \ begin { a l i g n ∗} \ tag ∗{$ ( 1where 8 . 1 ) $} M = \bigcup { q \ leq p } ( a ˆ{\bigcup { a }} { , } { = 1 }ˆ{ q ) = 1 } { q } [ \ f r a c { a }{ q } − \ f r a c { P }{ qX } { , }\ f r a c { a }{ q } + \ f r a c { P }{ qX }ˆ{ ] } , mX = [ 0 , 1 ] \setminus M. S(α) = log pe(pα),X = X1−ε0 ,L = log X. (18.3) \end{ a l i g n ∗} 1 X1 < p ≤ X \ centerline {We consider as usual }

\ centerline {1 }

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 8 . 2 ) $} R ( m ) = ˆ{ = } 0 ˆ{ + \ int = R { 1 } ( m ) + R { 2 } ( m ) , } { p 1 ˆ{\sum { + }} { p } 2 ˆ{ = m }{\ int }\ log p 1 \cdot \ log p 2 = \ int }ˆ{ S ˆ{ 2 } ( \alpha ) e ( − m \alpha ) d \alpha }\\ M m \end{ a l i g n ∗}

\noindent where

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 8 . 3 ) $} S( \alpha ) = \sum \ log pe ( p \alpha ),X { 1 } = X ˆ{ 1 − \ varepsilon { 0 }} , L = \ log X. \\ X { 1 } < p \ leq X \end{ a l i g n ∗} 394 J´anos Pintz The contribution of the minor arcs can be estimated well on average by Parseval ’ s identity and Vinogradov ’ s method or Vaughan ’ s method . This

yieldsforP ≤ X2/5 X X L10X | E2 |= |{m; 2 | m, m ∈ [ X],R2(m) > √ }| √ (18.4) 2 , L P .

In such a way , many problems are reduced to the behaviour of R1(m). In the classical treatment M, that is ,P, i s chosen in such a way that an asymptotic evaluation of R1(m) as

Y 1 ) R (m) ∼ S(m) := C (1 + p (18.5) 1 0 −1 p|m would be possible , due to the uniform distribution of primes in the arith - metic progressions with moduli q ≤ P. This requires by Siegel ’ s theorem ( cf . Siegel ( 1 936 ) ) the strong upper bound

P logA X (18.6) for any arbitrarily large but ﬁxed A > 0. This yields a relatively weak bound in ( 1 8 . 4 ) for | E2 | . The idea of Montgomery and Vaughan ( 1 975 ) was to evaluate exactly the eﬀect of the possible ( essentially unique ) Siegel zero and to show some weaker positive lower bound for R1(m) in place of ( 18 . 5 ) by using a deep theorem of Gallagher ( 1 970 ) about the statistically good distribution of primes in arithmetic progressions modulo

q ≤ P := Xδ (18.7) with a small but ﬁxed positive value of δ. As a generalization of this idea we evaluate exactly the eﬀect of all ‘ gen - eralized exceptional zeros ’ % = β + iγ of all L− functions modulo

q ≤ P := X2/5 (18.8) with

H β ≥ 1 − | γ |≤ T, (18.9) log X , where H and T are large parameters . We denote the set of the above zeros by E(H,T ). In practice we may choose H and T as large absolute constants . The evaluation gives rise to ‘ generalized singular series ’ S(χi, χj, m) sat - i sfying 394 .. Jaacutenos Pintz \noindentThe contribution394 \quad of theJ minor\ ’{ a arcs} nos can Pintz be estimated well on average by Parseval quoteright s identity and Vinogradov quoteright s method or Vaughan quoteright s method period This\ hspace ∗{\ f i l l }The contribution of the minor arcs can be estimated well on average by yields for P less or equal X to the power of 2 slash 5 Equation: open parenthesis 1 8 period 4 closing parenthesis\noindent .. bar EParseval sub 2 bar = ’ vextendsingle-vextendsingle-vextendsingle-vextendsingle s identity and Vinogradov ’ s method or Vaughan braceleftbigg ’ ms semicolonmethod . 2 This bar m comma m in bracketleftbigg X divided by 2 sub comma X bracketrightbigg comma R sub 2 open paren- \ begin { a l i g n ∗} thesis m closing parenthesis greater X divided| S(χi, by χj square, m) |≤ rootS(χ of0, χL0 bracerightbigg, m) = S(m). vextendsingle-vextendsingle-(18.10) vextendsingle-vextendsingley i e l d s f o r P \ llleq L to theX power ˆ{ 2 of 10 / X divided 5 }\\\ bytag square∗{$ root ( of 1 P sub 8 period . 4 ) $}\mid E { 2 }\mid = \arrowvert \{ m ; 2 \mid m , m \ in [ \ f r a c { X }{ 2 } { , } In such a way commaThe evaluation many problems of R are1(m reduced) depends to the on behaviour the generalized of R sub 1 open exceptional parenthesis zeros m closing X],Rparenthesis period , { 2 } ( m ) > \ f r a c { X }{\ sqrt { L }}\}\arrowvert \ l l \ f rIn a c the{ L classical ˆ{ 10whose treatment} X number}{\ Msqrt comma i{ sP bounded that}} is{ comma. by} Ce P4 commaH by a i density s chosen theoremin such a way of Jutila that an ( 1 977 \endasymptotic{ a l i g n ∗} evaluation) . The of Rresulting sub 1 open explicit parenthesis formula m closing forms parenthesis the basis as for many further results Equation: open. parenthesis 1 8 period 5 closing parenthesis .. R sub 1 open parenthesis m closing parenthesis thicksimIn such S aopen way parenthesis , many m problems closing parenthesis are reduced : = C sub to 0 the product behaviour p bar m parenleftbigg of $ R { 1 plus1 } p 1( divided m by) minus . $ 1 to the power of parenrightbigg Inwould the be classical possible comma treatment due to the $M uniform , distribution $ that of is primes $ , in the P arith , $hyphen i s chosen in such away that an asymptoticmetic progressions evaluation with moduli of q $ less R or{ equal1 } P period( m This ) requires $ as by Siegel quoteright s theorem open parenthesis cf period Siegel open parenthesis 1 936 closing parenthesis closing parenthesis the strong upper\ begin bound{ a l i g n ∗} \ tagEquation:∗{$ ( open 1 parenthesis 8 . 1 5 8 period ) $ 6} closingR { parenthesis1 } ( .. m P ll ) log to\sim the powerS of A ( X m ) : =for C any{ ..0 arbitrarily}\prod large{ butp .. fixed\mid .. Am greater} ( 0 period 1 .. + This p yields\ f r .. a c a{ relatively1 }{ − weak1 }ˆ{ ) } \endbound{ a l i ing n open∗} parenthesis 1 8 period 4 closing parenthesis for bar E sub 2 bar period The idea of Montgomery and Vaughan open parenthesis 1 975 closing parenthesis .. was to evaluate exactly \noindentthe effect ofwould the possible be possible open parenthesis , due essentially to the uniqueuniform closing distribution parenthesis Siegel of zeroprimes and to in show the some arith − meticweaker progressions positive lower bound with for moduli R sub 1 open $ q parenthesis\ leq mP closing . $ parenthesis This requires in place of by open Siegel parenthesis ’ s theorem 18( period cf . 5 Siegel closing parenthesis ( 1 936 by ) )using the a deep strong upper bound theorem of Gallagher .. open parenthesis 1 970 closing parenthesis .. about .. the statistically good distribution of\ begin { a l i g n ∗} \ tagprimes∗{$ in ( arithmetic 1 8 progressions . 6 modulo ) $} P \ l l \ log ˆ{ A } X \endEquation:{ a l i g n ∗} open parenthesis 1 8 period 7 closing parenthesis .. q less or equal P : = X to the power of delta with a small but fixed positive value of delta period \noindentAs a generalizationf o r any of\ thisquad ideaarbitrarily we evaluate exactly large the but effect\quad of all quoteleftf i x e d \ genquad hyphen$ A > 0 . $ \quaderalizedThis exceptional y i e l d s zeros\quad quoterighta relatively rho = beta weak plus i gamma of all L hyphen functions modulo boundEquation: in (open 18 parenthesis . 4 ) for 1 8 period $ \mid 8 closingE parenthesis{ 2 }\ ..mid q less or. equal $ P : = X to the power of 2 slash 5 Thewith idea of Montgomery and Vaughan ( 1 975 ) \quad was to evaluate exactly theEquation: effect open of parenthesis the possible 1 8 period ( essentially 9 closing parenthesis unique .. beta ) Siegel greater zeroequal 1 and minus to H show divided some by log Xweaker sub comma positive bar gamma lower bar less bound or equal for T comma $ R { 1 } ( m ) $ in place of ( 18 . 5 ) by using adeep theoremwhere H andof Gallagher T are large parameters\quad ( period 1 970 We ) denote\quad theabout set of\ thequad abovethe zeros statistically good distribution of primesby E open in parenthesis arithmetic H comma progressions T closing parenthesis modulo period In practice we may choose H and T as large absolute constants period \ beginThe evaluation{ a l i g n ∗} gives rise to quoteleft generalized singular series quoteright S open parenthesis chi i comma chi\ tag sub∗{ j$ comma ( 1 m closing 8 parenthesis . 7 ) sat $} hyphenq \ leq P : = X ˆ{\ delta } \endi sfying{ a l i g n ∗} Equation: open parenthesis 1 8 period 1 0 closing parenthesis .. bar S open parenthesis chi i comma chi sub\noindent j commawith m closing a small parenthesis but bar fixed less or positive equal S open value parenthesis of $ \ chidelta 0 comma. chi $ 0 comma m closing parenthesis = S open parenthesis m closing parenthesis period AsThe a generalization .... evaluation .... of of R sub this 1 open idea parenthesis we evaluate m closing exactly parenthesis the .... depends effect .... of on all the .... ‘ gen generalized− ....eralized exceptional exceptional zeros comma zeros ’ $ \varrho = \beta + i \gamma $ o f a l l $ L − $whose functions number i s modulo bounded by Ce to the power of 4 H by a density theorem of Jutila open parenthesis 1 977 closing parenthesis period \ beginThe resulting{ a l i g n ∗} explicit formula forms the basis for many further results period \ tag ∗{$ ( 1 8 . 8 ) $} q \ leq P : = X ˆ{ 2 / 5 } \end{ a l i g n ∗}

\noindent with

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 8 . 9 ) $}\beta \geq 1 − \ f r a c { H }{\ log X } { , } \mid \gamma \mid \ leq T, \end{ a l i g n ∗}

\noindent where $ H $ and $ T $ are large parameters . We denote the set of the above zeros by $E ( H , T ) .$ Inpracticewemaychoose $H$ and $T$ as large absolute constants . The evaluation gives rise to ‘ generalized singular series ’ $ S ( \ chi i , \ chi { j } , m ) $ s at − i s f y i n g

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 8 . 1 0 ) $}\mid S( \ chi i , \ chi { j } , m ) \mid \ leq S( \ chi 0 , \ chi 0 , m ) = S ( m ). \end{ a l i g n ∗}

\noindent The \ h f i l l e v a l u a t i o n \ h f i l l o f $ R { 1 } ( m ) $ \ h f i l l depends \ h f i l l on the \ h f i l l generalized \ h f i l l exceptional zeros ,

\noindent whose number i s bounded by $ Ce ˆ{ 4 H }$ by a density theorem of Jutila ( 1 977 ) . The resulting explicit formula forms the basis for many further results . Landau quoteright s problems on primes .. 395 LandauTheorem ’ s .. problems 25 .. open on parenthesis primes J\ periodquad 395 P period closing parenthesis period .. Under the .. above .. conditionsTheorem we\quad have for25 even\quad n in (J.P.). \quad Under the \quad above \quad conditions we have for even $ nopen\ squarein $ bracket X slash 2 comma X closing square bracket .. the explicit formula : Equation: open parenthesis 1 8 period 1 1 closing parenthesis .. R sub 1 open parenthesis m closing parenthesis \noindent $ [ X / 2 , X ] $ \quad the explicit formula : = rho i comma rho subLandau j in E ’ open s problems parenthesis on primes H comma395 T closingTheorem parenthesis 25 from O open(J.P.) parenthesis. Xe to the power of minus cH closing parenthesis plus O open parenthesis X to the power of minus 1 slash 3 H closing \ begin { a l i g n ∗}Under the above conditions we have for even n ∈ parenthesis plus O[ toX/ S2 plus,X] tothe the powerexplicit of open formula parenthesis : to the power of m closing parenthesis plus sum sum S\ tag open∗{ parenthesis$ ( 1 chi 8 i comma . chi 1 sub 1 j comma ) $} mR closing{ 1 } parenthesis( m Capital ) Gamma= \varrho open parenthesisi , rho i\varrho { j } closing\ in parenthesisE ( H Capital , Gamma T ) open ˆ{ parenthesisO ( Xe rho ˆ sub{ − j closingcH } parenthesis) + divided O (by Capital X ˆ{ Gamma− 1 + open/ parenthesis 3 H } parenleftbigg) + O rho} { i XS divided{ + by}ˆ square{ ( } rootˆ{ m of T to ) the} power+ \ ofsum plus parenrightbiggm%i\sum%j −1 S( rho comma\ chi i , \ chi { j } , m ) }\ fO r a(Xe c {\−cHGamma)+O(X−1/3H )+( O \Γ(varrho%i)Γ(%j) i ) \Gamma ( R1(m) = %i, %j ∈ E(H,T ) (18.11) to the power of j closing parenthesis to the powerS( ofm m)+ toPP the powerS(χi,χ of,m rho) i to the+) power of plus rho sub j minus \varrho { j } ) }{\Gamma ( { ( +}\ f r a c {\varrhoj Γ((i %iX√{ X%j}}{\) sqrt { T }}ˆ{+{ ) }} 1 T , \varrhowhere the{ generalized, }ˆ{ j singular} ) } sˆ{ eriesm Sˆ{\ openvarrho parenthesisi chi ˆ{ i comma+ }\ chivarrho sub j comma{ j m} closing − 1 parenthesis}} \end{ a l i g n ∗} satisfy where the generalized singular s eries S(χi, χj, m) satisfy Equation: open parenthesis 1 8 period 1 2 closing parenthesis .. bar S open parenthesis chi i comma chi sub j\ commanoindent m closingwhere parenthesis the generalized bar less or equal singular epsilon s eries $ S ( \ chi i , \ chi { j } ,unless m the )$ fo l lowingsatisfy three conditions all hold open| S( parenthesisχi, χj, m) |≤ condε chi is the conductor of chi(18 comma.12) C open parenthesis epsilon closing parenthesis .. a constant depending on epsilon closing parenthesis \ beginopen{ parenthesisa l i g n ∗}unless 1 8 period the fo 1 l 3 lowing closing three parenthesis conditions .. cond all chi hold i bar ( Ccond openχ parenthesisis the conductor epsilon closing parenthesis\ tag ∗{$ ( m comma 1of 8 condχ, . chi sub 1 j bar 2 C open) $}\ parenthesismid S( epsilon closing\ chi parenthesisi , m comma\ chi ..{ condj } chi, i m ) \mid C(\εleq) a constant\ varepsilon depending on ε) ( 1 8 . 1 3 ) cond χi | C(ε)m, cond to the power of hline chi sub j less epsilon to the power of−3 minus 3 period \endIntroducing{ a l i g n ∗} theχ notationj | C(ε)m, cond χi χj < ε . Equation: open parenthesis 1 8 period 14 closingIntroducing parenthesis the .. notationE sub 0 open parenthesis X closing parenthesis =\noindent braceleftbiggunless m semicolon the 2fo bar l m lowing comma m three in bracketleftbigg conditions X divided all hold by 2 sub( cond comma $ X\ bracketrightbiggchi $ is the conductor of $ \ chi , $ comma m element-slash G bracerightbigg comma X E0(X) = {m; 2 | m, m ∈ [ X], melement − slashG}, (18.14) one can deduce from Theorem 24 results about the quoteleft2 , structure quoteright of a set E sub 1 open parenthesis\noindent X closing$ C parenthesis ( \ varepsilon ) $ \quad a constant depending on $ \ varepsilon ) $containing the evenone canm quoteright deduce froms with Theorem R sub 1 open 24 resultsparenthesis about m closing the ‘ parenthesis structure less’ of X a divided set by ( 1 8 . 1 3 ) \quad cond $ \ chi i \mid C( \ varepsilon ) m , $ square root of L subE1( periodX) cond $ \ chi { j }\mid C( \ varepsilon X ) m , $ \quad cond $ \ chi Theorem 26 opencontaining parenthesis the J period even Pm period’ s with commaR1 Weak(m) < Structural√ Theorem closing parenthesis period i ˆ{\ r u l e {3em}{0.4 pt }}\ chi { j } < \ varepsilonL . ˆ{ − 3 } . $ .. There are positiveTheorem abso hyphen 26 ( J . P . , Weak Structural Theorem ) . There are lute constants c sub 0 comma K and a set E sub 1 open parenthesis X closing parenthesis .. with the properties \ centerline { Introducingpositive abso the - lute notation constants} c0,K and a set E1(X) with the properties : : ( 1 8 . 1 5 ) open parenthesis 1 8 period 1 5 closing parenthesis \ beginLine{ 1a K l i gLine n ∗} 2 E sub 0 open parenthesis X closing parenthesis subset equal E sub 1 open parenthesis X closing\ tag ∗{ parenthesis$ ( 1 cup 8 E sub . 2 open 14 parenthesis ) $} E X{ closing0 } parenthesis( X ) comma = bar\{ E subm 2 open ; parenthesis 2 K \mid X mclosing , parenthesis m \ in bar ll[ L to\ f the r a c power{ X }{ of 102 X} to{ the, } powerX of ]3 slash , 5 comma m element E[ sub 1− opens l a s parenthesis h G \} X E (X) ⊆ E (X) ∪ E (X), | E (X) | L10X3/5, E (X) ⊆ A , d > Xc0 , closing, parenthesis subset0 equal1 union2 of A sub d2 sub nu comma d sub nu1 greater X todν the powerν of c sub 0 comma\end{ a Line l i g n 3∗} nu = 1 ν = 1 where A sub d denotes the multiples of an integer d period \noindent one can deduce from Theorem 24 results about the ‘ structure ’ of a set Theorem 26 clearlywhere impliesAd Edenotes open parenthesis the multiples X closing of parenthesis an integer ll Xd. to the power of 1 minus delta period However$ E { comma1 } ( in a muchXTheorem )more $ 26 clearly implies E(X) X1−δ. However , in a much sophisticated waymore we can show the much stronger \noindentTheorem 27containing opensophisticated parenthesis the J even periodway we P $m$ period can show comma ’ s the with Strong much Structural$R stronger{ 1 Theorem} ( closing m )parenthesis< \ f period r a c { X }{\ sqrt { L }} { . }$ .... Theorem 26 is trueTheorem with 27 ( J . P . , Strong Structural Theorem ) . Theorem 26 is \noindents ome c subTheorem 0 greatertrue with 26 1 slash ( J 3 . period P . , Weak Structural Theorem ) . \quad There are positive abso − lute constants $ c { 0 } , K$ andaset $E { 1 } ( X ) $ \quad with the properties : This result immediatelys ome yieldsc0 > 1 Theorem/3. 1 6 and plays a crucial role in (the 1 8 proofs . 1 of 5 some ) This other resultresults amongimmediately Theorems yields 1 7 endash Theorem 24 period 1 6 and plays a crucial role in Finally commathe we just proofs briefly of mention some other that an results analogous among explicit Theorems formula 1and 7 – 24 . \ [ \resultsbegin analogous{ a l i g n e d to}Finally TheoremsK \\ , we 1 9just endash briefly 2 7 mentioncan be proved that for an the analogous representation explicit formula and Eof even{ 0 integers} (X)results as differences analogous\ ofsubseteq two to primes Theorems periodE { 11 9} – 2(X) 7 can be proved\cup for theE representa-{ 2 } (X ), \mid tionE { of2 even} integers(X) as differences\mid \ofl l twoL primes ˆ{ 10 .} X ˆ{ 3 / 5 } , E { 1 } (X) \subseteq \bigcup A { d {\nu }} , d {\nu } > X ˆ{ c { 0 }} , \\ \nu = 1 \end{ a l i g n e d }\ ]

\noindent where $ A { d }$ denotes the multiples of an integer $ d . $

\ hspace ∗{\ f i l l }Theorem 26 clearly implies $E ( X ) \ l l X ˆ{ 1 − \ delta } . $ However , in a much more

\noindent sophisticated way we can show the much stronger

\noindent Theorem 27 ( J . P . , Strong Structural Theorem ) . \ h f i l l Theorem 26 is true with

\noindent s ome $ c { 0 } > 1 / 3 . $

This result immediately yields Theorem 1 6 and plays a crucial role in the proofs of some other results among Theorems 1 7 −− 24 .

Finally , we just briefly mention that an analogous explicit formula and results analogous to Theorems 1 9 −− 2 7 can be proved for the representation of even integers as differences of two primes . 396 .. Jaacutenos Pintz \noindent19 period ..396 Approximations\quad J \ ’{ toa} Landaunos Pintz quoteright s first problem In this last section we will briefly summarize the most important results \ centerlinein connection{19 with . the\quad problemApproximations of whether the polynomial to Landau n to ’ the s power first of problem 2 plus 1 represents} infinitely many primes period \ hspace ∗{\ f i l l } In this last section we will briefly summarize the most important results Let f be a polynomial396 János with integerPintz coefficients irreducible over the ra hyphen t ionals and without a fixed prime divisor period Let p open parenthesis f closing parenthesis be the minimal \noindent in connection19 . with Approximations the problem of to whether Landau the ’ s polynomial first problem $ n ˆ{ 2 } + number In this last section we will briefly summarize the most important 1 $such represents that f represents integers with at most p open parenthesis f closing parenthesis prime factors infinitely infinitely manyresults primes . often comma andinlet connection deg f be the with degree the of problem f period of The whether first result the polynomial n2 +1 represents Equation: openinfinitely parenthesis many 1 9 period primes 1 closing . parenthesis .. p open parenthesis f closing parenthesis less orLet equal $ 4 f degree $ be f minus a polynomial 1 comma with integer coefficients irreducible over the ra − t ionals and withoutLet f be a fixed a polynomial prime divisorwith integer . Let coefficients $ p (irreducible f ) over$ be the the ra minimal number was proved more- than t ionals 80 years and ago without by H period a fixed Rademacher prime open divisor parenthesis . Let 1924p(f) closingbe the parenthesis minimal period Later results number \noindentwere such that $ f $ represents integers with at most $ p ( f ) $ prime factors infinitely often , and letsuch deg that $f frepresents $ be the integers degree with of at $ most fp .(f $) prime The firstfactors result infinitely p open parenthesisoften f closing , and parenthesis let deg f lessbe theor equal degree 3 degree of f. f minusThe first 1 Ricci result open parenthesis 1 936 closing parenthesis comma \ beginopen{ parenthesisa l i g n ∗} 1 9 period 2 closing parenthesis p open parenthesis f closing parenthesis less or equal deg \ tag ∗{$ ( 1 9 . 1 ) $} p ( f ) \ leq 4 \deg f − 1 , f plus c log open parenthesis deg f closing parenthesisp(f) ..≤ Kuhn4 deg f open− 1, parenthesis 1 953 comma 1(19 954.1) closing parenthesis\end{ a l i g comman ∗} p open parenthesiswas f proved closing parenthesis more than less 80 or years equal ago degree by f H plus . Rademacher 1 Buhscarontab ( 1924 open )parenthesis . Later 1 967 closing\noindent parenthesiswasresults provedperiod were more than 80 years ago by H . Rademacher ( 1924 ) . Later results wereIf deg f = 2 comma then first Kuhn .. open parenthesis 1 953 closing parenthesis .. proved p open parenthesis f closing parenthesis less or equal 3 comma ..p while(f) ≤ the3 deg bestf − 1 Ricci(1936), \ [result p is( at present f ) \ leq 3 \deg f − 1 Ricci ( 1 936 ) , \ ] (19.2) p(f) ≤ deg f + c log ( deg f) Kuhn ( 1 953 , 1 954 ) , Theorem open parenthesis Iwaniec open parenthesis 1978 closing parenthesis closing parenthesis period .. If deg f = 2 and f open parenthesis 0 closing parenthesis .. is odd comma then p open parenthesis f closing p(f) ≤ deg f + 1 Buhstabˇ (1967). parenthesis\noindent less$(19.2)p(f) or equal 2 period \ leq $ deg $f + c$ log(deg $ fCorollary ) $ period\quad n toKuhnIf the deg power ( 1f 953 of= 2 plus , 2, 1then 1 954 = P first sub) , 2 Kuhn infinitely ( o 1 f-t 953 en period ) proved p(f) ≤ 3, Another approachwhile i s to the ask best Capital Omega hyphen type estimates about the largest prime \ [divisor p ( P open f parenthesisresult ) \ isleq at f openpresent\deg parenthesisf n + closing 1 parenthesis Buh \check closing{ s } parenthesistab ( of f 1open 967 parenthesis ) n. closing\ ] parenthesisTheorem period For the( Iwaniec special case ( 1978 of f open) ) . parenthesisIf deg n closingf = 2 parenthesisand f(0) = nis to odd the power , of 2 2 plus 1 comma suchthen estimatesp(f) ≤ 2. Corollary . n + 1 = P2 infinitely o f − t en . were achieved by HooleyAnother open parenthesis approach 1967 i closing s to ask parenthesisΩ− type estimates about the largest \ hspaceEquation:∗{\ openf i l lprime} parenthesisIf deg 1$f 9 period = 3 closing 2 ,$ parenthesis then .. firstKuhn P open parenthesis\quad n to( the1 953 power ) of\quad 2 plusproved 1 closing$ p parenthesis ( fdivisor greater) \ leq nP to(f the(n))3 powerof f ,(n of$). 1For\ periodquad the 1 specialwhile i period the case o period best of f semicolon(n) = n2 +1, such estimates finally the sharpestwere known achieved estimate by Hooleyi s ( 1967 ) \noindentTheorem openresult parenthesis is at Deshouillers present comma Iwaniec open parenthesis 1 982 closing parenthesis closing parenthesis period P open parenthesis n to the power of 2 plus 1 closing parenthesis greater n to the power of 1 period\noindent 202468Theorem period period ( Iwaniec period .. i ( period 1978 o ) periodP )(n .2 +\quad 1) > n1Ifdeg.1 i.o.; $f = 2$ and (19.3) $f ( 0It) is easy$ \ toquad see thatisodd,then Landau quoteright $p s first ( conjecture f ) would\ leq follow if2 we could . $ Corollaryshow that $finally . nˆ the{ sharpest2 } + known 1 = estimate P { i2 s }$ infinitely o $ f−t $ en . 2 1.202468... open parenthesisTheorem 1 9 period 4( closing Deshouillers parenthesis , Iwaniec braceleftbig(1982)) square.P root(n of+ p 1) to> the n power of bracerightbigi . o less\ hspace c divided∗{\ byf i l square l.}Another root ofapproach p .. i period i o periods to askcomma $ \Omega − $ type estimates about the largest prime with a suitable constantIt c is period easy to see that Landau ’ s first conjecture would follow if we \noindent divisorcould $P ( f ( n ) )$ of $f ( n ) .$ Forthespecialcaseof This was shown in the weaker form open√ } brace square root of p closing brace less or equal cp to the power of $ f ( n ) = n ˆ{ 2 } +√ 1c , $ such estimates minus alpha with show that (19.4) { p < p i . o . , with a suitable constant c. √ alpha = 1 divided by 1 5This minus was epsilon shown .. I in period the weaker M period form Vinogradov{ p} ≤ opencp−α parenthesiswith 1 940 closing parenthesis\noindent commawere achieved by Hooley1 ( 1967 ) α = 15 − ε I . M . Vinogradov ( 1 940 ) , open parenthesis 1 9 period 51 closing parenthesis alpha = 1 divided by 1 0 minus epsilon .. I period M period (19.5) α = 10 − ε I . M . Vinogradov ( 1 976 , Ch . 4 ) , Vinogradov\ begin { a l openi g n ∗} parenthesis 1 976 comma Ch period 4 closing parenthesis comma \ tag ∗{$ ( 1 9 . 3 ) $} P ( n ˆ{ 2 } + 1 ) > n ˆ{ 1 . 1 } alpha = 0 period 1 63 period period periodα Kaufman= 0.163... open Kaufman parenthesis(1979) 1979. closing parenthesis period iThe . best o known . result ; is \end{ a l i g n ∗} The best known result is

\noindent finally the sharpest known estimate i s

\noindent Theorem( Deshouillers ,Iwaniec $( 1 982 ) ) . P ( nˆ{ 2 } + 1 ) > n ˆ{ 1 . 202468 . . . }$ \quad i . o .

\ hspace ∗{\ f i l l } It is easy to see that Landau ’ s first conjecture would follow if we could

\noindent show that $ ( 1 9 . 4 ) \{\ sqrt { p }ˆ{\}} < \ f r a c { c }{\ sqrt { p }}$ \quad i . o . , with a suitable constant $ c . $

\ centerline { This was shown in the weaker form $ \{\ sqrt { p }\}\ leq cp ˆ{ − \alpha }$ with }

\ centerline { $ \alpha = \ f r a c { 1 }{ 1 5 } − \ varepsilon $ \quad I . M . Vinogradov ( 1 940 ) , }

\noindent $ ( 1 9 . 5 ) \alpha = \ f r a c { 1 }{ 1 0 } − \ varepsilon $ \quad I .M. Vinogradov ( 1 976 , Ch . 4 ) ,

\ [ \alpha = 0 . 1 63 . . . Kaufman ( 1979 ) . \ ]

\noindent The best known result is Landau quoteright s problems on primes .. 397 \ hspaceTheorem∗{\ openf i l l parenthesis}Landau Balog ’ s problems open parenthesis on primes 1983 closing\quad parenthesis397 comma Harman open parenthesis 1 983 closing parenthesis closing parenthesis period open brace square root of p closing brace ll epsilon p to the power\noindent of minusTheorem( 1 divided by Balog 4 plus epsilon( 1983 i period ) ,Harman o period $( 1 983 ) ) . \{\ sqrt { p } \}\Finallyl l we mention\ varepsilon that Hardyp and ˆ{ Littlewood − \ f r a c open{ 1 parenthesis}{ 4 } + 1 923\ closingvarepsilon parenthesis}$ expressed i . o . a num hyphen Landau ’ s problems on primes 397 Finally we mention that Hardy and Littlewood ( 1 923 ) expressed1 a num − ber of conjectures in their landmark paper about additive problems involv√ hyphen − +ε ber of conjecturesTheorem in their( Balog landmark ( 1983 ) paper, Harman about(1983)) additive. { p} problems εp 4 i involv . o . − ing primes period ..Finally Some of we them mention deal with that prime Hardy values and of polynomials Littlewood semicolon ( 1 923 one ) expressed of a ingthem primes i s exactly . \ thequad quantitativeSome of form them of the deal first with conjecture prime of Landau values period of polynomials ; one of them i s exactlynum the- ber quantitative of conjectures form in their of the landmark first paper conjecture about additiveof Landau prob- . Conjecture openlems parenthesis involv - Hardy ing endashprimes Littlewood . Some open of parenthesis them deal 1923 with closing prime parenthesis values of closing parenthesis period .. The number of primes n to the power of 2 plus 1 less or equal x \noindent Conjecturepolynomials ( Hardy ; one−− ofLittlewood them i s exactly ( 1923 the ) quantitative ) . \quad The form number of the first of primes is asymptoticallyconjecture equal to of Landau . $ nEquation: ˆ{ 2 } open+ parenthesis 1 \ leq 1 9 periodx $ 6 closing parenthesis .. product p greater 2 parenleftbigg 1 minus p is asymptoticallyConjecture equal to( Hardy – Littlewood ( 1923 ) ) . The number of primes 1 divided by minusn 12 + parenleftbigg 1 ≤ x is asymptotically minus 1 divided equal by p to to the power of parenrightbigg parenrightbigg square root of X divided by log X sub period \ beginReferences{ a l i g n ∗} \ tag ∗{$ ( 1 9 . 6 ) $}\prod { p > 2√ } ( 1 − p \ f r a c { 1 }{ − open square bracket 1 closing square bracket RY period ..1 J− period1 ) ..X Backlund comma Udieresis ber .. die .. (1 − p ( ) (19.6) Differenzen1 } ( \ zwischenf r a c { − .. den1 ..}{ Zahlenp }ˆ comma{ ) } .. die) zu\ f.. r a den c {\ ..sqrt ersten{ X n }}{\ log X } { . } −1 p log X . \endPrimzahlen{ a l i g n ∗} teilerfremd sind period Commentationesp>2 in honorem E period L period Lindelodieresisf period Annales Acad period References \ centerlineSci period Fenn{ References period 32 open} parenthesis 1 929 closing parenthesis comma Nr period 2 comma 1 endash 9 [ 1 ] R . J . 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Bombieri , On the large sieve . Mathematika 1 2 ( 1 965 ) , 20 1 – 225 . [ 10 ] E . Bombieri Archivopen square Math bracket. 40 ( 6 1 closing 983 ) square , 434 bracket−− 440 M period . } B period Barban comma The density of zeros of , H . Davenport , Small differences between prime numbers . Proc . Roy . Soc . Ser . A 293 ( Dirichlet L hyphen s eries and the problem of the sum of ¨ [ 6 ] M . B .1 Barban 966 ) , 1 – , 18 The . [ 1 density 1 ] A . of Brauer zeros , of H . Dirichlet Zeitz , U $ber L eine− zahlentheoretische$ s eries and the problem of the sum of primes and nearBehauptung primes period von Legendre Mat period. Sber Sb period . Berliner 61 open Math parenthesis. Ges . 29 ( 1 930 963 ) closing , 1 1 6 – parenthesis 1 25 . [ 1 2 ] comma 418primes endash and 425 open near parenthesis primes .Russian Mat . closing Sb . parenthesis 61 ( 1 963 period ) , 418 −− 425 ( Russian ) . H . Brocard , L ’ intermédiairedes math . 4 ( 1897 ) , p . 1 59 . [ 13 ] V . Brun , Le crib le d ’ open square bracket 7 closing square bracket P period T period Bateman comma R period A period Horn Eratosthéneet le théor e` me de Goldbach . Videnselsk . Skr . 1 ( 1 920 ) , Nr . 3 . comma[ 7 ] A P heuristic . T . Bateman asymptotic , formula R . A concerning . Horn the, A distribution heuristic asymptotic formula concerning the distribution [ 14 ] A . A . Buhˇstab , New improvements in the sieve of Eratosthenes . Mat . Sb . 4 ( 1 938 ofof prime prime numbers numbers period . Math Math . period Comp Comp . 1 6period ( 1 1 962 6 open ) , parenthesis 363 −− 367 1 962 . closing parenthesis comma ) , 357 – 387 ( Russian ) . [ 1 5 ] A . A . Buhˇstab , Sur la décomposition des nombres pairs en 363 endash 367 periodsomme de deux composantes dont chacune est forméed ’ un nombre bornéde facteurs premiers . [ 8open ] J square . Bertrand bracket 8 closing ,M\ ’ square{ e} moire bracket sur J period le nombre Bertrand de comma valeurs Meacutemoire que peut sur le prendre nombre de une valeurs fonction quand on y permute lesDoklady lettres Akad qu. Nauk ’ . elle SSSR enferme29 ( 1 940 . ) , J 544 $ – 548 . .\ [acute 1 6 ] A .{E A} . Buh$ˇstab cole, New Roy results . Polytechnique 1 8 ( 1845 ) , 1 23 −− 140 . que peut prendre unein the fonction investigation quand of on Goldbach – Euler ’ s pro blem and the problem of twin prime numbers . y permute les lettres qu quoteright elle enferme period J period Eacute cole Roy period Polytechnique 1 8 \noindent [ 9Doklady ] E . Akad Bombieri . Nauk . SSSR , On1 62 the( 1 large 965 ) , 735 sieve – 738 ( Rus . Mathematika - sian ) . [ 17 ] A 1 . A 2 .Buh ( 1ˇstab 965, A ) , 20 1 −− 225 . open parenthesis 1845combinatorial closing parenthesis strengthening comma of the Eratosthenian 1 23 endash sieve 140 method period . Usp . Mat . 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Sb . 4 ( 1 938 ) , 357 −− math387 period ( Russian 4 open parenthesis ) . 1897 closing parenthesis comma p period 1 59 period [open 15 square ]A.A.Buh bracket 13 closing\v{ s } squaretab , bracket Sur l V a period d\ ’{ e Brun} composition comma Le crib des le d nombres quoteright pairs Eratostheacutene en somme de deux composantes etdont le theacuteor chacune egrave est me form de\ Goldbach’{ e}e d period ’ un Videnselsk nombre born period\ ’{ Skre} periodde facteurs 1 open parenthesis premiers 1 920 . Doklady closing Akad . Nauk . parenthesisSSSR 29 comma ( 1 940 ) , 544 −− 548 . [Nr 16 period ]A.A.Buh 3 period \v{ s } tab , New results in the investigation of Goldbach −− Euler ’ s pro blem and the problemopen square of bracket twin prime 14 closing numbers square bracket . Doklady A period Akad A period . Nauk Buhscarontab . SSSR 1 comma 62 ( 1 New 965 improvements ) , 735 −− 738 ( Rus − ins thei a n sieve ) . of Eratosthenes period Mat period Sb period 4 open parenthesis 1 938 closing parenthesis comma 357[ endash17 ]A.A. Buh\v{ s } tab , A combinatorial strengthening of the Eratosthenian sieve method . Usp . Mat387 . open Nauk parenthesis 22 ( 1 Russian 967 ) closing , no parenthesis . 3 , 1 99 period−− 226 ( Russian ) . [open 18 ] square Y . bracket Buttkewitz 1 5 closing , Master square bracket ’ s Thesis A period . A Freiburg period Buhscarontab Univ . , comma 2003 Sur . la deacutecom- position[ 19 des ]F. nombres Carlson pairs en $somme , de\ddot deux{U composantes} $ ber die Nullstellen der Dirichletschen Reihen und der Riemannscher $ \dontzeta chacune− $ est formeacutee d quoteright un nombre borneacute de facteurs premiers period Doklady Akad periodFunktion Nauk period . Arkiv f . Math . Astr . Fys . 1 5 ( 1 920 ) , No . 20 . 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Jaacutenos Pintz \noindentopen square398 bracket\quad 20 closingJ \ ’{ a} squarenos Pintz bracket P period L period Ccaron ebyscaronev comma Meacutemoire sur les nombres premiers period Meacutemoire des seuvants eacutetrangers de [ 20l quoteright ] P . L Acad $ period . \check Sci period{C} St$ period eby\ Peacutetersbourgv{ s }ev , M\ ’{ 7e} openmoire parenthesis sur les 1850 nombres closing parenthesis premiers . M\ ’{ e} moire des seuvants \ ’{ e} t r a n g e r s de commal ’ Acad 1 7 endash . Sci 33 .period St . P\ ’{ e} tersbourg 7 ( 1850 ) , 1 7 −− 33 . open square bracket[ 44 21 ] P closing . Erd squareohungarumlaut bracket Jings Run, Some Chen pro blemscomma on On number the theoryrepresentation , in : Analytic of a andlarge even integer[ 2 1 as ] the Jing sum Run of a Chen prime and , On the representation of a large even integer as the sum of a prime and the product ofelementary at most number two theory primes ( Marseille . Kexue , 1 983 Tongbao ) . Publ . Math 1 7 . (Orsay 1 966 , 86 - 1 ) ( 1 , 983 385 ) , 53−− – 57386 . ( Chinese ) . the product of at most[ 45 ] P two . Erd primesohungarumlaut period Kexues Tongbao , L . Mirsky 1 7 open, The parenthesisdistribution of 1 values 966 closing of the divisor parenthesis comma 385 endashfunction 386 opend( parenthesisn). Proc . London Chinese Math closing . Soc parenthesis . ( 3 ) 2 ( 1 period952 ) , 257 – 271 . [ 22open ] square Jingbracket Run Chen 22 closing , On square the representation bracket Jing Run Chen of a comma large On even the representation integer as of the a large sum even of a prime and the product of at[ 46 ] mostT . two Estermann primes, Eine . neue Sci Darstellung . Sinica und 1 neue 6 Anwendungen( 1 973 ) der , Viggo 1 57 Brunschen−− 1 76 . integer as the sumMethode of a prime. J and . Reine Angew . 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Zinoviev comma A complete Vinogradov 3 hyphen primes theorem under the Riemann hypothesis period Electron period Res period Announc period Amer[ 39 period ] L . Math E . period Dickson , A new extension of Dirichlet ’ s theorem on prime numbers . Messenger of MathSoc period . ( 2 3 )open , 33parenthesis ( 1904 1 997 ) closing, 155 parenthesis−− 1 6 1 comma . 99 endash 104 period open square bracket 39 closing square bracket L period E period Dickson comma A new extension of Dirichlet quoteright[ 40 ] P s .theorem G . L on . prime Dirichlet numbers , period Beweis Messenger des Satzes of , dass jede unbegrenzte arithmetische Progression , derenMath perioderstes open Glied parenthesis und Differenz 2 closing parenthesis ganze Zahlen comma ohne 33 open gemeinscha parenthesis 1 $ 904 f− closingt $ lichenparenthesis Factor sind , un − commaendlich 1 55 vieleendash 1 Primzahlen 6 1 period enth \”{a} lt . Abhandl . Kgl . Preu \ ss . Akad . Wiss . ( 1837 ) , 45 −− 8 1 . [ Werke , Iopen , 313 square−− bracket342 . 40 G closing . Reimer square , bracket Berlin P period , 1889 G period ; French L period translation Dirichlet comma : J Beweis . math des Satzes . pures appl . 4 , 1839 , 393 −− comma dass jede unbegrenzte arithmetische Progression comma \ centerlinederen erstes{ Glied422 und. ] Differenz} ganze Zahlen ohne gemeinscha f-t lichen Factor sind comma un hyphen endlich viele Primzahlen enthadieresislt period Abhandl period Kgl period Preu germandbls period Akad period[ 41 Wiss] P period. D . open T . parenthesis A . Elliott 1837 closing , H . parenthesis Halberstam comma , A 45 conjecture endash 8 1 period in prime open square number bracket theory . Symposia WerkeMathematica comma 4 INDAM , Rome , 59 −− 72 , Academic Press , London , 1 968 / 69 . 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GoldstonProceedings of , the On Amalfi Hardy Conference and on Littlewood Analytic Number ’ Theory s contribution ( Maiori , 1 989 to ) . 1 the 1 5 – Goldbach 1 conjecture , in : open square bracket55 , Univ 52 closing . Salerno square , Salerno bracket , 1 992 D . period [ 53 ] D A . period A . Goldston Goldston , Y comma . Motohashi On Hardy , J . Pintz and ,Littlewood C . quoterightProceedings s contribution of the to Amalfi the Goldbach Conference conjecture on comma Analytic in : Number Theory ( Maiori , 1 989 ) . 1 1 5 −− 1 55 , Univ . SalernoY.Y , Salernoı ld ı r ,ı m 1, 992Small . gaps between primes exist . Proc . Japan Acad . 82 A ( 2006 ) Proceedings of, the 6 1Amalfi – 65 . [ Conference 54 ] D . A . Goldstonon Analytic , J Number. Pintz , C Theory . Y ı openld ı parenthesisr ı m , Primes Maiori in Tuples comma. 1 989 closing[ 53 parenthesis ] D . Aperiod . Goldston 1 1 5 endash ,Y 1 . 55 Motohashi comma , J . Pintz ,C . Y . 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E 57 . closingLittlewood square, Some bracket pro Dblems period of ‘ PartitioA period Numerorum Goldston ’ comma , III : On S period the expression W period of Grahama comma J period Pintz comma C period Y period Y i ld i r i m comma Small gaps between products \noindent [ 59number ] A as . a Granville sum of primes ,. Acta Unexpected Math . 44 ( irregularities 1 923 ) , 1 – 70 . [ 66 ] inG the . H . distribution Hardy , J . E . of prime numbers , in : Proceed − of two primes periodLittlewood Proc period, Some London problems Math of ’ Partitio period Numerorum Soc period ’ , open V : A parenthesis further contribution 200 ? closing to the study parenthesis commaings to of appear the Internationalcomma preprint at Congress arXiv : math of slash Mathematicians 60961 5 ( Z\”{u} rich , 1 994 ) . Vol . 1 , 2 , 388 −− 399 , Birkh \”{a} userof Goldbach , Basel ’ s problem , 1 995. Proc . . London Math . Soc . 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Littlewood parenthesisˇ , comma Some 1 problems 2 endash 28 of period ’ Partitio Numerorum ’ , V : A further contributionSumme to the von study höchstens of 71 Goldbach Primzahlen darstellen ’ s problem.C asopis . pˇest. Proc Math . London . Fys . 65 Math( 1 936 . ) , Soc 1 1 . ( 2 ) 22 ( 1 924 ) , open square bracket7 – 141 6 . 1 [ closing E . square bracket .. G period Greaves comma Sieves in Number Theory period Springer46 −− comma56 . 200 1 period open square bracket 62 closing square bracket J period Hadamard comma Sur les zeacuteros de la fonction zeta\noindent open parenthesis[ 67 ] s\ closingquad G parenthesis . Harman de Riemann , Primes period in Comptesshort intervals Rendus Acad . period Math Sci . periodZeitschr . 1 80 ( 1 982 ) , 335 −− 348 . [Paris 68 ] 1\ 22quad openG parenthesis . Harman 1896 , Onclosing the parenthesis distribution comma 1470 of endash $\ sqrt 1473{ p period}$ modulo one . Mathematika 30 ( 1 983 ) , 104 −− 1 1 6 . open square bracket 63 closing square bracket J period Hadamard comma Sur la fonction zeta open parenthesis s\noindent closing parenthesis[ 69 ] period D . ComptesR . Heath Rendus− Brown Acad period , A parity Sci period problem Paris 1 23 from open sieve parenthesis theory 1896 . closing Mathematika 29 ( 1 982 ) , 1 −− 6 . parenthesis[ 70 ] D comma . R .p period Heath 93− periodBrown , Prime twins and Siegel zeros . Proc . London Math . Soc . ( 3 ) 47 (open 1 983 square ) , bracket 1 93 64−− closing224 .square bracket J period Hadamard comma Sur la distribution des zeacuteros de[ la 71 fonction ] D . zeta\quad openR. parenthesis\quad s closingHeath parenthesis− Brown et , s\quad es conseacutequancesThe divisor arith $ f hyphen−u $ nction at consecutive integers . \quad Mathematika 31 (meacutetiques 1 984 ) , 141period−− Bull149 period . Soc period Math period France 24 open parenthesis 1896 closing parenthesis comma[ 72 1 ] 99 D endash . R . 220 Heath period− Brown , Almost − prime $ k − $ tuples . Mathematika 44 ( 1 997 ) , 245 −− 266 . [open 73 ]square D . bracket R . Heath 65 closing− Brown square bracket , H . .. Iwaniec G period H , period On the Hardy difference comma J period between E period consecutive Littlewood prime numbers . commaInvent Some . Math pro blems . 55 of quoteleft ( 1 979 Partitio ) , 49 Numerorum−− 69 . quoteright comma III : On the [expression 74 ] D . of R a . number Heath as− aBrown sum of , primes J . − periodC . Acta Puchta Math , period Integers 44 open represented parenthesis 1 as 923 a closing sum of primes and powers parenthesisof two . comma Asian 1 endash J . Math 70 period . 6 ( 2002 ) , no . 3 , 535 −− 565 . [open 75 ]square H . bracket Heilbronn 66 closing , E square . Landau bracket .., GP period . Scherk H period , AlleHardy grocomma\ ss J perioden ganzen E period Zahlen Littlewood lassen sich als Summe commavon h Some\”{o} problemschstens of 71quoteright Primzahlen Partitio darstellen Numerorum quoteright $ . comma\check V{C :} A$ further a s o p i s p\v{e} st . Math . Fys . 65 ( 1 936 ) , 117 −− 141 . [ E . contribution to the study of Goldbach quoteright s problem period Proc period London Math period Soc period open parenthesis 2 closing parenthesis 22 open parenthesis 1 924 closing parenthesis comma 46 endash 56 period open square bracket 67 closing square bracket .. G period Harman comma Primes in short intervals period Math period Zeitschr period 1 80 open parenthesis 1 982 closing parenthesis comma 335 endash 348 period open square bracket 68 closing square bracket .. G period Harman comma On the distribution of square root of p modulo one period Mathematika 30 open parenthesis 1 983 closing parenthesis comma 104 endash 1 1 6 period open square bracket 69 closing square bracket D period R period Heath hyphen Brown comma A parity problem from sieve theory period Mathematika 29 open parenthesis 1 982 closing parenthesis comma 1 endash 6 period open square bracket 70 closing square bracket D period R period Heath hyphen Brown comma Prime twins and Siegel zeros period Proc period London Math period Soc period open parenthesis 3 closing parenthesis 47 open parenthesis 1 983 closing parenthesis comma 1 93 endash 224 period open square bracket 71 closing square bracket D period .. R period .. Heath hyphen Brown comma .. The divisor f-u nction at consecutive integers period .. Mathematika 31 open parenthesis 1 984 closing parenthesis comma 141 endash 149 period open square bracket 72 closing square bracket D period R period Heath hyphen Brown comma Almost hyphen prime k hyphen tuples period Mathematika 44 open parenthesis 1 997 closing parenthesis comma 245 endash 266 period open square bracket 73 closing square bracket D period R period Heath hyphen Brown comma H period Iwaniec comma On the difference between consecutive prime numbers period Invent period Math period 55 open parenthesis 1 979 closing parenthesis comma 49 endash 69 period open square bracket 74 closing square bracket D period R period Heath hyphen Brown comma J period hyphen C period Puchta comma Integers represented as a sum of primes and powers of two period Asian J period Math period 6 open parenthesis 2002 closing parenthesis comma no period 3 comma 535 endash 565 period open square bracket 75 closing square bracket H period Heilbronn comma E period Landau comma P period Scherk comma Alle gro germandbls en ganzen Zahlen lassen sich als Summe von hodieresischstens 71 Primzahlen darstellen period Ccaron asopis pecaronst period Math period Fys period 65 open parenthesis 1 936 closing parenthesis comma 1 1 7 endash 141 period open square bracket E period 400 János Pintz Landau , Collected Works , 9 , 351 – 375 , Thales Verlag ; The Collected Papers of Hans Arnold Heilbronn , 1 97 – 2 1 1 , J . 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[ 92 ] Chaohua Jia , Almost all short intervals containing prime numbers . Acta Arith . 76 ( 1 996 a ) , 2 1 – 84 . [ 93 ] Chaohua Jia , On the exceptional s et of Goldbach numbers in a short interval . Acta Arith . 77 ( 1 996 b ) , no . 3 , 207 – 287 . [ 94 ] Chaohua Jia , Ming - Chit Liu , On the largest prime factor of integers . Acta Arith . 95 ( 2000 ) , No . 1 , 17 – 48 . [ 95 ] M . Jutila , On numbers with a large prime factor , I . J . Indian Math . Soc . ( N . S . ) 37 ( 1 973 ) , 43 – 53 . [ 96 ] M . Jutila , On Linnik ’ s constant . Math . Scand . 41 ( 1 977 ) , 45 – 62 . [ 97 ] L . Kaniecki , On Sˇ nirelman ’ s constant under the Riemann hypothesis . Acta Arith . 72 ( 1 995 ) , 361 – 374 . [ 98 ] J . Kaczorowski , A . Perelli , J . Pintz , A note on the exceptional s et for Goldbach ’ s problem in short intervals . Monatsh . Math . 1 1 6 , no . 3 - 4 ( 1 995 ) , 275 – 282 . Corrigendum : ibid . 1 1 9 ( 1 995 ) , 2 1 5 – 2 1 6 . [ 99 ] ´ , A remark on a paper of Ju . V . Linnik . Magyar Tud . Akad . Mat . Fiz . I.Katai √ Oszt . Közl. 1 7 ( 1 967 ) , 99 – 100 . [ 100 ] R . M . Kaufman , The distribution of { p}. Mat . Zametki 26 ( 1 979 ) , 497 – 504 ( Russian ) . [ 101 ] N . I . Klimov , On the computation of Sˇ nirelman ’ s constant . Volˇzsk ij Mat . Sbornik 7 ( 1 969 ) , 32 – 40 ( Russian ) . [ 102 ] N. I . Klimov , A refinement of the estimate of the absolute constant in the Goldbach – Sˇ nirelman 400 .. Jaacutenos Pintz \noindentLandau comma400 \ Collectedquad J \ Works’{ a} nos comma Pintz 9 comma 351 endash 375 comma Thales Verlag semicolon The Collected Papers of Hans Arnold LandauHeilbronn , Collected comma 1 97 Works endash 2 , 1 9 1 comma, 351 −− J period375 Wiley , Thales 1 988 period Verlag closing ; The square Collected bracket Papers of Hans Arnold Heilbronnopen square , bracket 1 97 −− 76 closing211 square , J bracket. Wiley1988 A period J . period ] Hildebrand comma Erd ohungarumlaut s quoteright problems on consecutive integers comma Paul Erd ohungarumlaut s and his Mathe hyphen [ 76 ] A . J .problem Hildebrand , in : Number , Erd theory : $ Collection ohungarumlaut of Studies in Additive $ s ’ Number problems Theory on. Nauˇcn. consecutive Trudy integers , Paul Erd matics I periodKuibyˇsev. Bolyai Society Gos . Ped Mathematical . Inst . 1 58 Studies( 1 975 ) 1 , 141 comma – 30 ( Russian Budapest ) . [ comma103 ] N . 2002I . Klimov comma, A 305 new endash 31$ ohungarumlaut7 period $ s and his Mathe − ˇ matics I . Bolyaiestimate Society of the absolute Mathematical constant in the Goldbach Studies – S 1nirelman 1 , Budapest pro blem ., Izv 2002 . VUZ ,no 305 . 1 ( 1−− 31 7 . open square bracket978 ) , 77 25 –closing 35 ( Russian square ) . bracket D period Hilbert comma Gesammelte Abhandlungen period Vol period 3 comma 290 endash 329 comma Springer comma Berlin comma 1 935 period \ centerlineopen square{ bracket[ 77 ]78 D closing . Hilbert square bracket , Gesammelte .. G period Abhandlungen Hoheisel comma Primzahlpro . Vol . 3 bleme , 290 in der−− Analysis329 , Springer , Berlin , 1 935 . } period SBer period Preuss period Akad period Wiss period comma Berlin comma 1 930 comma [ 78580 ] endash\quad 588G period . Hoheisel , Primzahlpro bleme in der Analysis . SBer . Preuss . Akad . Wiss . , Berlin , 1 930 , 580open−− square588 bracket . 79 closing square bracket .. C period Hooley comma .. On the greatest prime factor of a quadratic polynomial period Acta Math period .. 1 1 7 [ 79open ] parenthesis\quad C 1. 967 Hooley closing , parenthesis\quad On comma the greatest 281 endash prime299 period factor of a quadratic polynomial . Acta Math . \quad 1 1 7 (open 1 967 square ) , bracket 281 −− 80 closing299 . square bracket M period N period Huxley comma On the differences of primes in arithmetical progressions period Acta Arith period 1 5 [ 80open ] parenthesis M . N . Huxley1 968 slash , 69 On closing the differencesparenthesis comma of 367 primes endash in 392 arithmetical period progressions . Acta Arith . 1 5 (open 1 968 square / 69 bracket ) , 81 367 closing−− 392 square . bracket M period N period Huxley comma On the difference between consecutive primes period Invent math period 1 5 open parenthesis 1 972 closing parenthesis comma [ 811 64 ] endash M . N 1 70 . Huxleyperiod , On the difference between consecutive primes . Invent math . 1 5 ( 1 972 ) , 1open 64 −− square1 70 bracket . 82 closing square bracket M period N period Huxley comma Small differences between consecutive primes period Mathematika 2 0 open parenthesis 1 973 closing parenthesis comma [ 82229 ] endash M . N 232 . period Huxley , Small differences between consecutive primes . Mathematika 2 0 ( 1 973 ) , 229open−− square232 bracket . 83 closing square bracket M period N period Huxley comma Small differences between consecutive primes II period Mathematika 24 open parenthesis 1 977 closing parenthesis comma [ 83142 ] endash M . N 1 52 . period Huxley , Small differences between consecutive primes II . Mathematika 24 ( 1 977 ) , 142open−− square1 52 bracket . 84 closing square bracket M period N period Huxley comma An application of the Fouvry endash Iwaniec theorem period Acta Arith period 43 open parenthesis 1 984 closing parenthesis comma [ 84441 ] endash M . N 443 . period Huxley , An application of the Fouvry −− Iwaniec theorem . Acta Arith . 43 ( 1 984 ) , 441open−− square443 bracket . 85 closing square bracket A period E period Ingham comma On the difference between consecutive primes period Quart period J period Math period Oxford ser period [ 858 open ] A parenthesis . E . Ingham 1 937 closing , On parenthesis the difference comma 255 between endash 266 consecutive period primes . Quart . J . Math . Oxford ser . 8open ( 1 square 937 ) bracket , 255 86−− closing266 . square bracket H period Iwaniec comma Almost primes represented by quadratic polynomials period Invent period math period 47 open parenthesis 1 978 closing parenthesis comma [ 861 71 ] endash H . Iwaniec 188 period , Almost primes represented by quadratic polynomials . Invent . math . 47 ( 1 978 ) , 1open 71 −− square188 bracket . 87 closing square bracket H period Iwaniec comma M period Jutila comma Primes in short intervals period Arkiv Mat period 1 7 open parenthesis 1 979 closing parenthesis comma 1 67 endash 1 76 period\ centerline { [ 87 ] H . Iwaniec ,M . Jutila , Primes in short intervals . Arkiv Mat . 1 7 ( 1 979 ) , 1 67 −− 1 76 . } open square bracket 88 closing square bracket H period Iwaniec comma J period Pintz comma Primes in short\ centerline intervals period{ [ 88 Monatsh ] H . Iwaniecperiod Math ,period J . Pintz 98 open , parenthesis Primes in 1 984 short closing intervals parenthesis . comma Monatsh 1 1 5 . Math . 98 ( 1 984 ) , 1 1 5 −− 143 . } endash 143 period [ 89open ] square\quad bracketChaohua 89 closing Jia square, Difference bracket .. Chaohuabetween Jia consecutive comma Difference primes between . Sci consecutive . China primes Ser . A 38 ( 1 995 a ) , 1 1 63 −− period1 186 Sci . period China Ser period A 38 open parenthesis 1 995 a closing parenthesis comma 1 1 63 endash 1 186 period [ 90open ] square\quad bracketChaohua 90 closing Jia ,square Goldbach bracket numbers .. Chaohua in Jia a comma short Goldbach interval numbers , I in. Sciencea short interval in China 38 ( 1 995 b ) , 385 −− comma406 . I period Science in China 38 open parenthesis 1 995 b closing parenthesis comma 385 endash 406 period [ 9open 1 ] square\quad bracketChaohua 9 1 closing Jia square , Goldbach bracket ..numbers Chaohua in Jia a comma short Goldbach interval numbers , II in .a short Science interval in China 38 ( 1 995 c ) , comma5 13 II−− period523 Science . in China 38 open parenthesis 1 995 c closing parenthesis comma 5 13 endash 523 period [ 92open ] square\quad bracketChaohua 92 closing Jia square, \quad bracketAlmost .. Chaohua all short Jia comma intervals .. Almost containing all short intervals prime containing numbers . \quad Acta Arith . \quad 76 prime( 1 numbers 996 a ) period , 2 .. 1 Acta−− 84 Arith . period .. 76 open parenthesis 1 996 a closing parenthesis comma 2 1 endash 84 period [ 93open ] square\quad bracketChaohua 93 closing Jia , square On the bracket exceptional .. Chaohua Jia s et comma of Goldbach On the exceptional numbers s et in of a Goldbach short interval . Acta Arith . numbers77 ( 1996b) in a short interval , no period . 3 , Acta 207 Arith−− period287 . 77 open parenthesis 1 996 b closing parenthesis comma no period 3 comma 207 endash 287 period [ 94open ] square\quad bracketChaohua 94 closing Jia ,square Ming bracket− Chit .. Chaohua Liu , On Jia comma the largest Ming hyphen prime Chit factor Liu comma of integersOn the . Acta Arith . 95 largest( 2000 prime ) factor ,No of . integers 1 , 17 period−− 48 Acta . Arith period 95 open parenthesis 2000 closing parenthesis comma No period 1 comma 17 endash 48 period [ 95open ]M. square Jutila bracket 95 , closing On numbers square bracket with a M largeperiod Jutila prime comma factor On numbers , I . J with . aIndian large prime Math factor . Soc . ( N . S . ) 37 ( 1 973 ) , comma43 −− I period53 . J period Indian Math period Soc period open parenthesis N period S period closing parenthesis 37 open parenthesis 1 973 closing parenthesis comma \ centerline43 endash 53{ [ period 96 ] M . Jutila , On Linnik ’ s constant . Math . Scand . 41 ( 1 977 ) , 45 −− 62 . } open square bracket 96 closing square bracket M period Jutila comma On Linnik quoteright s constant period Math[ 97 period ] L . Scand Kaniecki period 41 , open\quad parenthesisOn $ \ 1check 977 closing{S} $ parenthesis nirelman comma ’ s 45 constant endash 62 periodunder the Riemann hypothesis . Acta Arith . 72 (open 1 995 square ) , bracket 361 −− 97 closing374 . square bracket L period Kaniecki comma .. On Scaron nirelman quoteright s constant under the Riemann hypothesis period Acta Arith period 72 [ 98open ] parenthesis J . Kaczorowski 1 995 closing , A parenthesis . Perelli comma , J 361 . endash Pintz 374 , A period note on the exceptional s et for Goldbach ’ s problemopen square in bracket short 98 intervals closing square . Monatshbracket J period . Math Kaczorowski . 1 1 6 comma , no A . period 3 − 4 Perelli ( 1 comma 995 ) J , period 275 −− 282 . Corrigendum : Pintzibid comma . 119(1995) A note on the exceptional ,215 s et−− for2 Goldbach 1 6 . quoteright s problem in short intervals period Monatsh period Math period 1 1 6 comma no period 3 hyphen 4 open parenthesis[ 99 ] I 1 . 995 K\ closing’{ a} tai parenthesis , A remark comma on 275 a endash paper 282 of period Ju . Corrigendum V . Linnik : . Magyar Tud . Akad . Mat . Fiz . Oszt . K\”{o} z l . 17(1967ibid period 1 1 ) 9 open ,99 parenthesis−− 100 . 1 995 closing parenthesis comma 2 1 5 endash 2 1 6 period [open 100 square ] R . bracket M . Kaufman 99 closingsquare , The bracket distribution I period Kaacutetai of $ \{\ commasqrt A remark{ p }\} on a paper. of $ Ju Mat period . Zametki 26 ( 1 979 ) , 497 −− 504 ( Russian ) . V[ period 101 Linnik ] N . period I . Klimov Magyar Tud , On period the Akad computation period Mat ofperiod $ Fiz\check period{S Oszt} $ period nirelman Kodieresiszl ’ s period constant . Vol\v{z} sk $ ij1 7 $ open Mat parenthesis . Sbornik 1 967 7 closing ( 1 parenthesis 969 ) , comma 99 endash 100 period 32open−− square40 ( bracket Russian 100 ) closing . square bracket R period M period Kaufman comma The distribution of open brace[ 102 square ] N root . I of . p closingKlimov brace , A period refinement Mat period of Zametki the estimate 26 open parenthesis of the absolute 1 979 closing constant parenthesis in the Goldbach −− comma$ \check 497 endash{S} $ 504 nirelman open parenthesis problem Russian ,in closing : Number parenthesis theory period : Collection of Studies in Additive Number Theory . Nauopen\v{ squarec}n . bracket Trudy 101 Kuiby closing\v{ squares }ev .bracket Gos .N Ped period . I Inst period . Klimov 1 58 ( comma 1 975 On ) the , 14computation−− 30 ( of Russian ) . Scaron[ 103 nirelman ] N . quoteright I . Klimov s constant , A periodnew estimate Volzcaronsk of ij Mat the period absolute Sbornik constant 7 open parenthesis in the 1 Goldbach 969 closing −− parenthesis$ \check{ commaS} $ nirelman pro blem . Izv32 endash . VUZno 40 open . 1 parenthesis ( 1 978 Russian ) , 25 closing−− 35 parenthesis ( Russian period ) . open square bracket 102 closing square bracket N period I period Klimov comma A refinement of the estimate of the absolute constant in the Goldbach endash Scaron nirelman problem comma in : Number theory : Collection of Studies in Additive Number Theory period Nauccaronn period Trudy Kuibyscaronev period Gos period Ped period Inst period 1 58 open parenthesis 1 975 closing parenthesis comma 14 endash 30 open parenthesis Russian closing parenthesis period open square bracket 103 closing square bracket N period I period Klimov comma A new estimate of the absolute constant in the Goldbach endash Scaron nirelman pro blem period Izv period VUZ no period 1 open parenthesis 1 978 closing parenthesis comma 25 endash 35 open parenthesis Russian closing parenthesis period Landau quoteright s problems on primes .. 40 1 Landauopen square ’ s problems bracket 104 on closing primes square\quad bracket40 N 1 period I period Klimov comma G period Z period Pilt quoteright[ 104 ]N. ai comma I . T Klimov period A ,G.Z.period Scaron Pilt eptickaja ’ ai comma ,T.A An estimate $ . for\ thecheck absolute{S} $ constant eptickaja , An estimate for the absolute constant in the Goldbach endash Scaron nirelman problem period Issled period po teorii ccaronisel comma Saratov No period\ hspace 4 open∗{\ f parenthesis i l l } in the 1 972 Goldbach closing parenthesis−− $ \ commacheck{ 35S} endash$ nirelman 5 1 problem . Issled . po teorii \v{c} isel , Saratov No . 4 ( 1 972 ) , 35 −− 5 1 open parenthesis Russian closing parenthesis period ˇ \ [( RussianLandau ) . ’ s\ problems] on primes 40 1 [ 104 ] N . I . Klimov , G . Z . Pilt ’ ai , T . A .S open square bracketeptickaja 105 closing, An estimate square for bracket the absolute S period constant Knapowski comma On the mean values of certain functions in prime numberin the theory Goldbach period – Sˇ nirelman Acta problem . Issled . po teorii ˇcisel, Saratov No . 4 ( 1 972 ) , 35 – Math period Acad5 1 period Sci period Hungar period 1 0 open parenthesis 1 959 closing parenthesis comma 375\noindent endash 390[ period 105 ] S . Knapowski , On the mean values of certain functions in prime number theory . Acta Math . Acad . Sci . Hungar . 1 0 ( 1 959 ) , 375 −− 390 . open square bracket 106 closing square bracket H period(Russian von) Koch. comma Sur la distribution des nombres premiers[ 106 period ] H . Acta von Math Koch period , Sur 24 open la distribution parenthesis 1 90 1 des closing nombres parenthesis premiers comma 1 . 59 Acta endash Math 182 period . 24 ( 1 90 1 ) , 1 59 −− 182 . [ 105 ] S . Knapowski , On the mean values of certain functions in prime number theory . Acta [open 107 square ] L . bracket Kronecker 107 closing , Vorlesungen square bracket\ L” period{u} ber Kronecker Zahlentheorie comma Vorlesungen , I . p udieresisber . 68 , Teubner Zahlen- , Leipzig , 1 901 . Math . Acad . Sci . Hungar . 1 0 ( 1 959 ) , 375 – 390 . [ 106 ] H . von Koch , Sur la distribution theorie[ 108 comma ] P .I period Kuhn p , period Zur Vigo 68 comma Brun Teubner ’ s chen comma Siebmethode Leipzig comma I 1 . 901 Norske period Vid . Selsk . 14 ( 1 941 ) , 145 −− 148 . des nombres premiers . Acta Math . 24 ( 1 90 1 ) , 1 59 – 182 . [ 107 ] L . Kronecker , [open 109 square ] P . bracket Kuhn108 , Neue closing Absch square\”{ bracketa} tzungen P period auf Kuhn Grund comma der Zur Viggo Vigo Brunschen Brun quoteright Siebmethode s chen . 1 2 . Skand . Vorlesungen über Zahlentheorie , I . p . 68 , Teubner , Leipzig , 1 901 . [ 108 ] P . Kuhn , Zur SiebmethodeMat . Kongr I period . ( Norske 1 953 Vid ) period , 1 60 Selsk−− period1 68 14 . open parenthesis 1 941 closing parenthesis comma 145 Vigo Brun ’ s chen Siebmethode I . Norske Vid . Selsk . 14 ( 1 941 ) , 145 – 148 . [ 109 ] P . Kuhn endash[ 110 148 ]P.Kuhn period $ , \ddot{U} $ ber die Primteiler eines Polynoms . Proc . ICM Amsterdam 2 ( 1 954 ) , 35 −− 37 . , Neue Abschätzungenauf Grund der Viggo Brunschen Siebmethode . 1 2 . Skand . Mat . Kongr . [open 1 1 square 1 ] E bracket . Landau 109 closing , Gel square\”{o bracket} ste und P period ungel Kuhn\”{ commao} ste Neue Probleme Abschadieresistzungen aus der Theorie auf Grund der Primzahlverteilung und ( 1 953 ) , 1 60 – 1 68 . [ 1 10 ] P . Kuhn , U¨ ber die Primteiler eines Polynoms . Proc . ICM derder Viggo Riemannschen Brunschen Siebmethode Zetafunktion period 1 . 2 Jahresber period Skand . period\quad Deutsche Math . Ver . 2 1 \quad ( 1 9 1 2 ) , 208 −− 228 . Amsterdam 2 ( 1 954 ) , 35 – 37 . [ 1 1 1 ] E . Landau , Gelösteund ungelösteProbleme aus der [Mat Proc period . 5 Kongr th Internat period open . parenthesis Congress 1 of 953 Math closing . parenthesis , I , 93 comma−− 108 1 60 , endash Cambridge 1 68 period 1 9 13 ; Collected Works , 5 , Theorie der Primzahlverteilung und der Riemannschen Zetafunktion . Jahresber . Deutsche 240open−− square255 bracket , Thales 1 10 Verlag closing square . ] bracket P period Kuhn comma Udieresis ber die Primteiler eines Math . Ver . 2 1 ( 1 9 1 2 ) , 208 – 228 . [ Proc . 5 th Internat . Congress of Math . , I , Polynoms[ 1 1 2 period ] A Proc . Languasco period ICM , Amsterdam On the exceptional 2 open parenthesis set 1 of 954 Goldbach closing parenthesis ’ s problem comma 35in endash short intervals . Mh . Math . 93 – 108 , Cambridge 1 9 13 ; Collected Works , 5 , 240 – 255 , Thales Verlag . ] [ 1 1 2 ] A. 371 period 41 ( 2004 ) , 147 −− 1 69 . Languasco , On the exceptional set of Goldbach ’ s problem in short intervals . Mh . Math . 1 [open 1 13 square ] A bracket . Languasco 1 1 1 closing , Jsquare . Pintz bracket , E A period . Zaccagnini Landau comma , OnGelodieresisste the sum undof two ungelodieresisste primes and 41 ( 2004 ) , 147 – 1 69 . [ 1 13 ] A . Languasco , J . Pintz , A . Zaccagnini , On the sum of Probleme$ k $ powers aus der Theorie of two der . Primzahlverteilung und two primes and k powers of two . Bull . London Math . Soc . 39 ( 2007 ) , no . 5 , 771 – Bullder Riemannschen . London Math Zetafunktion . Soc . period 39 ( Jahresber 2007 ) period , no .. . 5Deutsche , \quad Math771 period−− 780 Ver period . 2 1 .. open 780 . [ 1 14 ] Hong Ze Li , Goldbach numbers in short intervals . Science in China 38 ( 1 995 ) , parenthesis[ 1 14 ] 1 Hong9 1 2 closing Ze Li parenthesis , Goldbach comma numbers 208 endash in 228 short period intervals . Science in China 38 ( 1 995 ) , 641 −− 652 . 641 – 652 . [ 1 1 5 ] Hong Ze Li , Primes in short intervals . Math . Proc . Cambridge Philos . [open 1 1 square 5 ] Hong bracket Ze Proc Li period , Primes 5 th Internat in short period intervals Congress of . Math Math period . Proc comma . Cambridge I comma 93 Philos endash . Soc . 1 2 2 ( 1 997 ) , Soc . 1 2 2 ( 1 997 ) , 108 comma Cambridge 1 9 13 semicolon Collected Works comma 5 comma 1 93 – 205 . \ centerline240 endash{ 2551 93comma−− Thales205 . Verlag} period closing square bracket [ 1 1 6 ] Hong Ze Li , The exceptional s et of Goldbach numbers . Quart . J . Math . Oxford Ser . open square bracket 1 1 2 closing square bracket A period Languasco comma On the exceptional set of ( 2 ) 5 0 , no . 200 ( 1 999 ) , 471 – 482 . [ 1 17 ] Hong Ze Li , The exceptional s et of Goldbach Goldbach\noindent quoteright[ 1 1 s 6 problem ] Hong in Zeshort Li intervals , The period exceptional Mh period s Math et period of Goldbach numbers . Quart . J . Math . Oxford Ser . ( 2 ) numbers , II . Acta Arith . 92 , no . 1 ( 2000 a ) , 501 41 , open no parenthesis. 200 ( 1 2004 999 closing ) , 471parenthesis−− 482 comma . 147 endash 1 69 period 71endash − eight8. [open 1 17 square ] Hong bracket Ze 1 Li 13 closing , The square exceptional bracket A s period et of Languasco Goldbach comma numbers J period , Pintz II .comma Acta A Arith period . 92 , no . 1 ( 2000 a ) , [ 1 18 ] Hong Ze Li , The number of powers of 2 in a representation of large even integers by sums Zaccagnini comma On the sum of two primes and k powers of two period of such powers and of two primes . Acta Arith . 92 ( 2000 b ) , 229 – 237 . [ 1 1 9 ] Hong Ze \ centerlineBull period{ London$ 71 Math endash period− Soce i g h period t 8 39 open . $ parenthesis} 2007 closing parenthesis comma no period Li , The number of powers of 2 in a representation of large even integers by sums of such powers 5 comma .. 771 endash 780 period and of two primes , II . Acta Arith . 96 ( 2001 ) , 369 – 379 . [ 1 20 ] Yu . V . Linnik , Prime \noindentopen square[ 1bracket 18 ] 1 Hong 14 closing Ze Li square , The bracket number Hong of Ze powersLi comma of Goldbach 2 in a numbers representation in short intervals of large even integers by sums numbers and powers of two . Trudy Mat . Inst . Steklov . 38 ( 1 951 ) , 1 52 – 1 69 ( Russian ) periodof such Science powers in China and 38 openof two parenthesis primes 1 995. Acta closing Arith parenthesis . 92 comma ( 2000 641 b endash ) , 229 652−− period237 . . [ 1 2 1 ] Yu . V . Linnik , Some conditional theorems concerning the binary Goldbach problem [open 1 1 square 9 ] Hong bracket Ze 1 Li1 5 closing , The square number bracket of powers Hong Ze of Li comma 2 in a Primes representation in short intervals of period large Math even integers by sums . Izv . Akad . Nauk . SSSR 1 6 ( 1 952 ) , 503 – 520 . [ 1 22 ] Yu . V . Linnik , Addition of periodof such Proc periodpowers Cambridge and of Philos two primes period Soc , period II . 1Acta 2 2 open Arith parenthesis . 96 ( 1 9972001 closing ) , parenthesis 369 −− 379 comma . prime numbers and powers of one and the same number . Mat . Sb . ( N . S . ) 32 ( 1 953 ) , 3 – [1 1 93 20 endash ] Yu 205 .period V . Linnik , Prime numbers and powers of two . Trudy Mat . Inst . Steklov . 38 ( 1 951 ) , 60 ( Russian ) . [ 1 23 ] J . Y . Liu , M . C . Liu , T . Z . Wang , The number of powers of 2 in 1open 52 −− square1 69 bracket ( Russian 1 1 6 closing ) .square bracket Hong Ze Li comma The exceptional s et of Goldbach numbers a representation of large even integers , I . Sci . China Ser . A 4 1 ( 1 998 a ) , 386 – 397 . [ period[ 1 2Quart 1 ] period Yu . J V period . Linnik Math period , Some Oxford conditional Ser period open theorems parenthesis concerning 2 closing parenthesis the binary Goldbach problem . Izv . 1 24 ] J . Y . Liu , M . C . Liu , T . Z . Wang , The number of powers of 2 in a representation of Akad5 0 comma . Nauk no .period SSSR 200 1 open 6 ( parenthesis 1 952 ) 1, 999 503 closing−− 520 parenthesis . comma 471 endash 482 period large even integers , II . Sci . China Ser . A 41 ( 1 998 b ) , 1 255 – 1 271 . [ 1 25 ] J . Y . Liu , [open 1 22 square ] Yu bracket . V .1 17 Linnik closing square , Addition bracketHong of prime Ze Li comma numbers The and exceptional powers s et of of Goldbach one and numbers the same number . Mat . M . C . Liu , T . Z . Wang , On the almost Goldbach problem of Linnik . J . Théor. Nombres commaSb. II (N.S. period Acta )32(1953) Arith period 92 comma ,3 no−− period60 ( 1 Russianopen parenthesis ) . 2000 a closing parenthesis comma Bordeaux 1 1 ( 1 999 ) , 133 – 147 . [ 1 26 ] Hong - Quan Liu , Jie Wu , Numbers with a large [71 1 endash-eight 23 ] J . Y 8 period . Liu ,M . C . Liu , T . Z . Wang , The number of powers of 2 in a representation of large prime factor . Acta Arith . 89 , no . 2 ( 1 999 ) , 1 63 – 187 . evenopen integers square bracket , I 1 .18 Sci closing . Chinasquare bracket Ser . Hong A4 Ze 1 (Li 1 comma 998 aThe ) number , 386 of−− powers397 .of 2 in a repre- [ 1 27 ] S . T . Lou , Q . Yao , A Chebychev ’ s type of prime number theorem in a short interval , sentation[ 1 24 of ] large J . even Y . integers Liu ,M by sums . C . Liu , T . Z . Wang , The number of powers of 2 in a representation of large II . Hardy – Ramanujan J . 1 5 ( 1 992 ) , 1 – 33 . [ 1 28 ] S . T . Lou , Q . Yao , The number evenof such integers powers and , ofII two . Sci primes . period China Acta Ser Arith . A41 period ( 921 998 open b parenthesis ) , 1 255 2000−− b closing1 271 parenthesis . of primes in a short interval . Hardy – Ramanujan J . 1 6 ( 1 993 ) , 2 1 – 43 . [ 1 29 ] H . Maier comma[ 1 25 229 ] endash J . Y 237 . period Liu ,M . C . Liu , T . Z . Wang , On the almost Goldbach problem of Linnik . J . Th\ ’{ e} or . , Small differences between prime numbers . Michigan Math . J . 35 ( 1 988 ) , 323 – 344 . Nombresopen square Bordeaux bracket 1 1 1 9 ( closing 1 999 square ) , 133 bracket−− Hong147 . Ze Li comma The number of powers of 2 in a [ 130 ] H . Maier , C . Pomerance , Unusually large gaps between consecutive primes . Trans . representation[ 1 26 ] Hong of large− evenQuan integers Liu , by Jie sums Wu , Numbers with a large prime factor . Acta Arith . 89 , no . 2 ( 1 999 ) , Amer . Math . Soc . 322 ( 1 990 ) , 201 – 237 . [ 131 ] E . Maillet , L ’ intermédiairedes math 1of 63 such−− powers187 . and of two primes comma II period Acta Arith period 96 open parenthesis 2001 closing . 1 2 ( 1 905 ) , p . 108 . [ 132 ] H . Mikawa , On the exceptional s et in Goldbach ’ s problem . parenthesis comma 369 endash 379 period Tsukuba J . Math . 1 6 ( 1 992 ) , \noindentopen square[ bracket 1 27 ]1 20 S closing . 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J . 35 ( 1 988 ) , 323 −− 344Akad . period Nauk period SSSR 1 6 open parenthesis 1 952 closing parenthesis comma 503 endash 520 period open square bracket 1 22 closing square bracket Yu period V period Linnik comma Addition of prime numbers and\noindent powers of[ one 130 and ] the H same . Maier number , C period . Pomerance Mat period , Unusually large gaps between consecutive primes . Trans . Amer . MathSb period . Soc open . parenthesis 322 ( 1 990N period ) , S 201 period−− closing237 parenthesis . 32 open parenthesis 1 953 closing parenthesis comma[ 131 3endash ] E . 60 Maillet open parenthesis , L ’ interm Russian\ closing’{ e} diaire parenthesis des period math . 1 2 ( 1 905 ) , p . 108 . [open 132 square ] H . bracket Mikawa 1 23 , closing On the square exceptional bracket J period s et Y period in Goldbach Liu comma ’ M s period problem C period . Tsukuba Liu comma J . Math . 1 6 ( 1 992 ) , T period Z period Wang comma The number of powers of 2 in a representation of large \ centerlineeven integers{5 comma 13 −− I period543 .Sci} period China Ser period A 4 1 open parenthesis 1 998 a closing parenthesis comma 386 endash 397 period open square bracket 1 24 closing square bracket J period Y period Liu comma M period C period Liu comma T period Z period Wang comma The number of powers of 2 in a representation of large even integers comma II period Sci period China Ser period A 41 open parenthesis 1 998 b closing parenthesis comma 1 255 endash 1 271 period open square bracket 1 25 closing square bracket J period Y period Liu comma M period C period Liu comma T period Z period Wang comma On the almost Goldbach problem of Linnik period J period Theacuteor period Nombres Bordeaux 1 1 open parenthesis 1 999 closing parenthesis comma 133 endash 147 period open square bracket 1 26 closing square bracket Hong hyphen Quan Liu comma Jie Wu comma Numbers with a large prime factor period Acta Arith period 89 comma no period 2 open parenthesis 1 999 closing parenthesis comma 1 63 endash 187 period open square bracket 1 27 closing square bracket S period T period Lou comma Q period Yao comma A Chebychev quoteright s type of prime number theorem in a short interval comma II period Hardy endash Ramanujan J period 1 5 open parenthesis 1 992 closing parenthesis comma 1 endash 33 period open square bracket 1 28 closing square bracket S period T period Lou comma Q period Yao comma The number of primes in a short interval period Hardy endash Ramanujan J period 1 6 open parenthesis 1 993 closing parenthesis comma 2 1 endash 43 period open square bracket 1 29 closing square bracket H period Maier comma Small differences between prime numbers period Michigan Math period J period 35 open parenthesis 1 988 closing parenthesis comma 323 endash 344 period open square bracket 130 closing square bracket H period Maier comma C period Pomerance comma Unusually large gaps between consecutive primes period Trans period Amer period Math period Soc period 322 open parenthesis 1 990 closing parenthesis comma 201 endash 237 period open square bracket 131 closing square bracket E period Maillet comma L quoteright intermeacutediaire des math period 1 2 open parenthesis 1 905 closing parenthesis comma p period 108 period open square bracket 132 closing square bracket H period Mikawa comma On the exceptional s et in Goldbach quoteright s problem period Tsukuba J period Math period 1 6 open parenthesis 1 992 closing parenthesis comma 5 13 endash 543 period 402 .. Jaacutenos Pintz \noindentopen square402 bracket\quad 133J closing\ ’{ a} squarenos Pintz bracket H period Mikawa comma On the intervals between consecutive numbers that are sums of two primes period \noindentTsukuba J[ period 133 ]Math H . period Mikawa 1 7 comma , On the No period intervals 2 open between parenthesis consecutive 1 993 closing parenthesis numbers thatcomma are sums of two primes . 443Tsukuba endash 453 J . period Math . 1 7 , No . 2 ( 1 993 ) , 443 −− 453 . [ 134 ] H . L . Montgomery , Zeros of $ L − $ functions . Invent . math . 8 ( 1 969 ) , 346 −− 354 . open square bracket402 134János closingPintz square bracket H period L period Montgomery comma Zeros of L hyphen functions[ 135 period ] H . Invent L . Montgomery period math period , R 8 . open C . parenthesis Vaughan 1 , 969 The closing exceptional parenthesis s comma et in 346 Goldbach endash 354 ’ s problem . 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[ 139 137 ] closingCheng square Dong Pan bracketOn the W representation period Narkiewicz of even numbers comma as The the sum Development of a prime and of Prime Number[ 139 Theory ] \quad periodCheng From Dong Euclid Pan to Hardy On the and representation of even numbers as the sum of a prime and a product of nota productmore ofthan not more 4 primes than 4 primes . Sci. Sci . . Sinica Sinica1 2 ( 1 1 2963 ( ) , 1 455 963 – 473 ) . , [ 140 455 ] A−− . Perelli473 . Littlewood period, J . Springer Pintz , On comma the exceptional 2000 period set for the 2k− twin primes problem . Compositio Math . 82 , [open 140 square ] A . bracket Perelli 138 closing , J . square Pintz bracket , On .. the Cheng exceptional Dong Pan On set the for representation the $ 2 of even k numbers− $ twin primes problemno . 3 ( 1 992 . Compositio ) , 355 – 372 . [ 141 ] A . Perelli , J . Pintz , On the exceptional s et for Goldbach as the sum of a prime’ s pro and blem a in short intervals . J . London Math . 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G period Poacutelya comma Heuristic reasoning in the theory of numbers period Amer period Math period Monthly 66 open parenthesis 1 959 closing parenthesis comma 375 endash 384 period open square bracket 1 56 closing square bracket H period Rademacher comma Beitradieresisge zur Viggo Brunschen Methode in der Zahlentheorie period Abh period Math period Sem period Hamburg 3 comma 1 2 endash 30 period open square bracket Collected Papers 1 open parenthesis 1 924 closing parenthesis comma 259 endash 277 comma MIT Press comma 1 974 period closing square bracket open square bracket 1 57 closing square bracket K period Ramachandra comma A note on numbers with a large prime factor period J period London Math period Soc period open parenthesis 2 closing parenthesis 1 open parenthesis 1 969 closing parenthesis comma 303 endash 306 period open square bracket 1 58 closing square bracket K period Ramachandra comma On the number of Goldbach numbers in small intervals period J period Indian Math period Soc period open parenthesis N period S period closing parenthesis 37 open parenthesis 1 973 closing parenthesis comma 1 57 endash 1 70 period open square bracket 1 59 closing square bracket .. 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Un . Math . single almost hyphen. Izv . prime Akad . number Nauk SSSR period1 2 Izv( 1 948 period ) , 57 Akad – 78 period( Russian Nauk ) . SSSR [ 1 65 ] 1S 2 z open . Gy parenthesis. Rev´ esz´ , 1 948 closingItal parenthesis . 15 ( 1comma 936 57 ) endash , 183 78−− open187 parenthesis . Russian closing parenthesis period [ 1 67 ] \quadEffectiveG . oscillation Ricci , theorems Su la for congettura a general class of di real Goldbach - valued remainder e la costante terms . Acta di Arith Schnirelmann . , II , III . Ann . Scuola open square bracket49 ( 1 1 988 65 ) closing , 481 – 505 square . [ bracket 1 66 ] SG z . period Ricci , GySu la period congettura Reacuteveacutesz di Goldbach e la comma costante Effectivedi oscillationNorm . theorems Sup . Pisa for a general , Cl . class Sci of . real ( hyphen 2 ) 6 valued ( 1 937 remainder ) , 71 −− 90 , 9 1 −− 1 1 6 . [ 1 68 ] \quadSchnirelmannG . Ricci. Boll , . La Un .differenza Math . Ital . di1 5 numeri( 1 936 ) , primi 183 – 187 consecutivi . [ 1 67 ] G . . Ricci Rendiconti, Sem . Mat . Univ . e Po − terms period ActaSu la Arith congettura period di 49Goldbach open parenthesise la costante di 1 988Schnirelmann closing parenthesis , II , III . Ann comma . Scuola 481 Norm endash . Sup 505 . period litecnicoopen square Torino bracket 1 1 66 1 closing , 149 square−− 200 bracket . Corr .. G period . ibidem Ricci comma1 2 ( Su 1 952la congettura ) , p .di 31 Goldbach 5 . e la [ 1 69 ] \quadPisaG , Cl . . Ricci Sci . ( 2 ,) 6 Sull( 1 937 ’ ) andamento , 71 – 90 , 9 1 – della 1 1 6 . differenza[ 1 68 ] G . Ricci di numeri, La differenza primi di consecutivi . Riv . Mat . Univ . costante di Schnirelmannnumeri primi period consecutivi Boll period. Rendiconti Un period Sem Math . Mat period . Univ . e Po - litecnico Torino 1 1 , 149 – 200 . ParmaItal period 5 ( 1 1 5 954 open ) parenthesis , 3 −− 54 1 936 . closing parenthesis comma 183 endash 187 period [ 1 70 ] H .Corr Riesel . ibidem ,1 R 2 .(1 C 952 . ) Vaughan , p . 31 5 . ,[ 1 Onsums 69 ] G . Ricci of primes, Sull ’ andamento . Arkiv della Mat differenza . 2 1di ( 1 983 ) , 45 −− 74 . open square bracketnumeri 1 primi 67 closing consecutivi square. Riv bracket . Mat .. . Univ G period . Parma Ricci5 comma( 1 954 ) Su, 3 – la 54 congettura . [ 1 70 ] H di . RieselGoldbach , e la costante[ 1 71 di ] Schnirelmann L . Ripert comma , L II ’ comma Interm III\ ’ period{ e} diaire Ann period des Scuola Math . 1 0 ( 1 903 ) , p . 66 . [ 1 72 ] N .R P . C. . Romanov Vaughan , On , On sums Goldbach of primes . Arkiv’ s problem Mat . 2 1 ( . 1 983 Tomsk ) , 45 – , 74 Izv . [ 1 . 71 Mat ] L . Ripert . Tek . I ( 1 935 ) , 34 −− 38 ( Russian ) . Norm period Sup, L ’ period Intermédiairedes Pisa comma Math Cl . 1 period 0 ( 1 903 Sci ) ,period p . 66 . open [ 1 72 parenthesis ] N . P . Romanov 2 closing, On parenthesis Goldbach ’ s 6 open parenthesis 1 937 closing parenthesis comma 71 endash 90 comma 9 1 endash 1 1 6 period \noindent [ 1problem 73 ] A. Tomsk . A , $Izv . . Mat\ .check Tek . I{ (S 1} 935$ ) , 34anin – 38 ( , Russian Determination ) . of constants in the method of Brun −− open square bracket[ 1 73 ] 1A.A 68 closing.Sˇ anin square, Determination bracket ..of constants G period in Riccithe method comma of Brun La – differenzaSˇ nirelman di numeri. Volzh primi consecutivi$ \check{ periodS} $ Rendiconti nirelman Sem . period Volzh Mat . Mat period . Univ period e Po hyphen Sb . 2(1964. Mat ) . , Sb 261 . 2 (−− 1 964265 ) , 26 ( 1 Russian– 265 ( Russian ) .) . litecnico Torino[ 1 74 1 comma] Y . Saouter 149 endash, Checking 200 the period odd Goldbach Corr period conjecture ibidem up to 1 210 open20. Math parenthesis . Comp . 167 952( 1 closing parenthesis comma p period 31 5 period \noindent [ 1998 74 ) ],863 Y .– 866 Saouter . , Checking the odd Goldbach conjecture up to $ 10 ˆ{ 2 open square bracket[ 1 751 ] A 69 . closingSchinzel square , W . Sbracket ierpinski´ .. G, Sur period certaines Ricci hypoth commae` Sulls es quoteright concernant les andamento nombres della differenza0 } . $ di numeri Mathprimi . Comp consecutivi . 67 ( period 1 998 Riv ) period , Mat period Univ period 863 −− 866 . premiers . Acta Arith . 4 ( 1 958 ) , 185 – 208 ; Erratum : ibidem 5 p . 259 . Erratum : ibidem 5 Parma 5 open parenthesis, p . 259 . [ 1 76954 ] J closing . - C . Schlageparenthesis - Puchta comma, The 3 endash equation 54ω period(n) = ω(n + 1). Mathematika open square bracket50 ( 2003 1 70 ) closing , no . 1 - square 2 , 99 – bracket 10 1 ( 2005 H period ) . [ 1 77 Riesel ] L . G comma . S chnirelman R period, COn period additive Vaughan properties comma On\noindent sums of primes[ 1 period75 ] A Arkiv . Schinzel Mat period ,W. 2 1 open S parenthesis ierpi \ ’{ 1n} 983ski closing , Sur parenthesis certaines comma hypoth 45 endash $ 74\grave{e} $ s es concernantof numbers les nombres. Izv . Donsk premiers . Politehn . . Inst . 14 ( 1 930 ) , 3 – 28 ( Russian ) . [ 1 78 ] L.G. period Schnirelman , U¨ ber additive Eigenscha f − t en von Zahlen . Math . Ann . 1 7 ( 1 933 ) , Actaopen Arith square bracket . 4 ( 1 1 71 958 closing ) , square 185 −− bracket208 L period; Erratum Ripert : comma ibidem L quoteright 5 p . 259 Intermeacutediaire . Erratum : des ibidem 5 , p . 259 . [ 1 76 ] J . − C . Schlage − Puchta , The649 equation – 690 . $ \omega ( n ) = \omega Math period 1 0 open[ 1 79 parenthesis ] A . Selberg 1 903, The closing general parenthesis sieve method comma and its place p period in prime 66 number period theory . Proc . Inter (open n square + bracket 1 ) 1 72 .$ closing Mathematika50(2003) square bracket N period P period ,no.1 Romanov− comma2 , On Goldbach quoteright 99 −− 10 1 (- 2005 nat . Congress ) . of Math . , Cambridge , Mass . 1 ( 1 950 ) , 286 – 292 . [ 180 ] A . Selberg , s problem period TomskCollected comma Papers Izv , Vol period . II . Mat Springer period , Berlin Tek , 1 period 991 . I open parenthesis 1 935 closing parenthesis comma[ 1 77 34 endash ] L . 38 G open . S parenthesis chnirelman Russian ,¨ On closing additive parenthesis properties period of numbers . Izv . Donsk . Politehn . Inst . 14 ( 1 930 ) , 3[ 181−− ] 28C ( . L Russian . Siegel , )U .ber die Klassenzahl quadratischer Körper . Acta Arith . 1 , 83 – 86 open square bracket. [ Gesam 1 73 - melteclosing Abhandlungen square bracket1 ( 1 A 936 period ) , 406 A – 409 period , Springer Scaron , Berlin anin –comma Heidelberg Determination , 1 966 . ] of constants[ 1 78 in ] the L method. G . Schnirelmanof Brun endash Scaron $ , nirelman\ddot{ periodU} $ Volzh ber period additive Mat period Eigenscha $ f−t $ en von Zahlen[ . 182 Math ] C . . SpiroAnn ., Thesis 1 7 (. Urbana 1 933 , 1 ) 98 , 1 . [ 183 ] J . J . Sylvester , On the partition of Sb period 2 openan even parenthesis number into 1 964 two closing primes parenthesis. Proc . London comma Math 26 . 1 Soc endash . 4 ( 1871265 open) , 4 – parenthesis 6 . [ Collected Russian closing parenthesis period \ centerline {649Math−− . Papers690 ., 2}, 709 – 71 1 , Cambridge , 1 908 . ] [ 184 ] W . Tartakowski , Sur quelques s open square bracketommes 1 du 74 type closing de Viggo square Brun bracket. C .Y R .period Acad . Saouter Sci . URSS comma , N . Checking S . 23 ( 1 the 939 odda ) , 1Goldbach 2 1 – 1 25 conjecture up to 10 to the power of 2 0 period Math period Comp period 67 open parenthesis 1 998 closing parenthesis \noindent [ 1. 79 [ 185 ] A ] W . .Tartakowski Selberg ,, TheLa méthode general du crible sieve approximatif method “ électif and ” its. C . place R . Acad in . Sci prime . number theory . Proc . Inter − comma URSS , N . S . 23 ( 1 939 b ) , 1 26 – 1 29 . [ 186 ] P . Turan´ , On the remainder term of the nat863 . endash Congress 866 period of Math . , Cambridge , Mass . 1 ( 1 950 ) , 286 −− 292 . [ 180 ] A . Selbergprime number , formula Collected I . Acta Papers Math . Hungar , Vol . 1 .( II 1 950 . ) Springer , 48 – 63 . [ , 187 Berlin ] S . Uchiyama , 1 991, . open square bracketOn the 1 difference 75 closing between square consecutive bracket primeA period numbers Schinzel. Acta comma Arith . W27 period( 1 975 S ) ierpinacuteski , comma Sur certaines hypoth egrave s es concernant les nombres premiers period \noindent [ 181 ] \quad C.L. Siegel $1 53 , – 1 57\ddot . {U} $ ber die Klassenzahl quadratischer K\”{o} rper . Acta Arith . 1 , 83 −− 86 . [ Gesam − Acta Arith period[ 188 4 ] openC parenthesis . J . de la 1Vall 958ee´ closing - Poussin parenthesis, Recherches comma analytiques 185 endash sur la 208 théoriedes semicolon nombres Erratum : ibidemmelte 5 p Abhandlungen period 259 period 1 Erratum ( 1 936 : ibidem ) , 406 5 comma−− 409 p period , Springer 259 period , Berlin −− Heidelberg , 1 966 . ] [ 182 ] \quadpremiersC . Spiro , I – III ,. Thesis Ann . Soc . . SciUrbana . Bruxelles , 120 98( 1896 1 . ) , 183 – 256 , 281 – 362 , 363 – 397 . open square bracket[ 189 ]1 76C closing . J . de square la Vall bracketee´ - Poussin J period, Sur hyphen la fonction C periodζ( Schlages) de Riemann hyphen et Puchta le nombre comma des The equation[ 183 omega ] J . open J . parenthesis Sylvester n closing , On theparenthesis partition = omega of open an even parenthesis number n plus into 1 closing two primesparenthesis . Proc . London Math . Soc . 4 ( 1871nombres ) , 4 premiers−− 6 inférieurs . [ Collecteda` une limite Math donnée . Papers. Mem Couronnésde , 2 , 709 l ’ Acad−− 71 . Roy 1 . , Sci Cambridge . , 1 908 . ] period MathematikaBruxelles 50 open , 5 parenthesis 9 ( 1899 ) . 2003 [ 1 90 closing ] R . C parenthesis . Vaughan , commaOn Goldbach no period ’ s problem 1 hyphen. Acta 2 Arith comma . 22 [99 184 endash ] W 10 . 1 Tartakowski open parenthesis , 2005 Sur closing quelques parenthesis s ommes period du type de Viggo Brun . C . R . Acad . Sci . URSS , N.S.23(1939a)( 1 972 ) , 2 1 – 48 ,121 . [ 1 9 1−− ] R .1 C 25. Vaughan . , On the estimation of Schnirelman ’ s constant . open square bracketJ . Reine 1 77 Angew closing . Math square . 290 bracket( 1 977L period ) , 93 – G 108 period . S chnirelman comma On additive properties of[ numbers 185 ] periodW . Tartakowski Izv period Donsk ,period La m\ Politehn’{ e} thode period du Inst crible period approximatif 14 ‘‘ \ ’{ e} lectif ’’ . C . R . Acad . Sci . URSS ,N . S.open 23(1939b) parenthesis 1 930 closing ,126 parenthesis−− 1 29 comma . 3 endash 28 open parenthesis Russian closing parenthesis period[ 186 ] P . Tur \ ’{ a}n , On the remainder term of the prime number formula I . Acta Math . Hungar . 1 (open 1 950 square ) , bracket 48 −− 1 7863 closing . square bracket L period G period Schnirelman comma Udieresis ber additive Eigenscha[ 187 ] f-t S en . von Uchiyama Zahlen period , On theMath difference period Ann period between 1 7 open consecutive parenthesis prime 1 933 closing numbers parenthesis . Acta Arith . 27 ( 1 975 ) , comma \ centerline649 endash{ 6901 53period−− 1 57 . } open square bracket 1 79 closing square bracket A period Selberg comma The general sieve method and its place\noindent in prime[ number 188 ] theory\quad periodC . Proc J . period de la Inter Vall hyphen\ ’{ e}e − Poussin , Recherches analytiques sur la th \ ’{ e} orie des nombres premiers , I nat−− periodIII . Congress Ann . of Soc Math . period Sci . comma Bruxelles Cambridge 20 comma( 1896 Mass ) , period 183 −− 1 open256 parenthesis , 281 −− 1 950362 closing , 363 −− 397 . parenthesis[ 189 ] comma\quad 286C . endash J . 292de periodla Vall \ ’{ e}e − Poussin , Sur la fonction $ \zeta ( sopen ) $ square de bracket Riemann 180 et closing le squarenombre bracket des Anombres period Selberg comma Collected Papers comma Vol period IIpremiers period Springer inf \ comma’{ e} r i Berlin e u r s comma $ \grave 1 991{ perioda} $ une limite donn \ ’{ e}e . Mem Couronn \ ’{ e} s de l ’ Acad . Roy . Sci . Bruxelles , 5open 9 ( square 1899 bracket ) . 181 closing square bracket .. C period L period Siegel comma Udieresis ber die Klassenzahl quadratischer[ 1 90 ] R Kodieresisrper . C . Vaughan period ,Acta On Arith Goldbach period 1 ’ comma s problem 83 endash . Acta 86 period Arith open . square 22 ( bracket 1 972 Gesam ) , 2 1 −− 48 . hyphen[ 1 9 1 ] R . C . Vaughan , On the estimation of Schnirelman ’ s constant . J . Reine Angew . Math . 290 (melte 1 977 Abhandlungen ) , 93 −− 1108 open . parenthesis 1 936 closing parenthesis comma 406 endash 409 comma Springer comma Berlin endash Heidelberg comma 1 966 period closing square bracket open square bracket 182 closing square bracket .. C period Spiro comma Thesis period Urbana comma 1 98 1 period open square bracket 183 closing square bracket J period J period Sylvester comma On the partition of an even number into two primes period Proc period London Math period Soc period 4 open parenthesis 1871 closing parenthesis comma 4 endash 6 period open square bracket Collected Math period Papers comma 2 comma 709 endash 71 1 comma Cambridge comma 1 908 period closing square bracket open square bracket 184 closing square bracket W period Tartakowski comma Sur quelques s ommes du type de Viggo Brun period C period R period Acad period Sci period URSS comma N period S period 23 open parenthesis 1 939 a closing parenthesis comma 1 2 1 endash 1 25 period open square bracket 185 closing square bracket W period Tartakowski comma La meacutethode du crible approximatif quotedblleft eacutelectif quotedblright period C period R period Acad period Sci period URSS comma N period S period 23 open parenthesis 1 939 b closing parenthesis comma 1 26 endash 1 29 period open square bracket 186 closing square bracket P period Turaacuten comma On the remainder term of the prime number formula I period Acta Math period Hungar period 1 open parenthesis 1 950 closing parenthesis comma 48 endash 63 period open square bracket 187 closing square bracket S period Uchiyama comma On the difference between consecutive prime numbers period Acta Arith period 27 open parenthesis 1 975 closing parenthesis comma 1 53 endash 1 57 period open square bracket 188 closing square bracket .. C period J period de la Valleacutee hyphen Poussin comma Recherches analytiques sur la theacuteorie des nombres premiers comma I endash III period Ann period Soc period Sci period Bruxelles 20 open parenthesis 1896 closing parenthesis comma 183 endash 256 comma 281 endash 362 comma 363 endash 397 period open square bracket 189 closing square bracket .. C period J period de la Valleacutee hyphen Poussin comma Sur la fonction zeta open parenthesis s closing parenthesis de Riemann et le nombre des nombres premiers infeacuterieurs agrave une limite donneacutee period Mem Couronneacutes de l quoteright Acad period Roy period Sci period Bruxelles comma 5 9 open parenthesis 1899 closing parenthesis period open square bracket 1 90 closing square bracket R period C period Vaughan comma On Goldbach quoteright s problem period Acta Arith period 22 open parenthesis 1 972 closing parenthesis comma 2 1 endash 48 period open square bracket 1 9 1 closing square bracket R period C period Vaughan comma On the estimation of Schnirelman quoteright s constant period J period Reine Angew period Math period 290 open parenthesis 1 977 closing parenthesis comma 93 endash 108 period 404 .. Jaacutenos Pintz \noindentopen square404 bracket\quad 1 92J \ closing’{ a} nos square Pintz bracket I period M period Vinogradov comma Representation of an odd number as a sum of three prime numbers period \noindentDoklady Akad[ 1 period 92 ] Nauk I . M period . Vinogradov SSSR 1 5 open , parenthesis Representation 1 937 closing of an parenthesis odd number comma as 29 1a endash sum of three prime numbers . 294Doklady open parenthesis Akad . Russian Nauk . closing SSSR parenthesis 1 5 ( 1 937period ) , 29 1 −− 294 ( Russian ) . [open 1 93 square ] Ibracket . M . Vinogradov1 93 closing square , A bracketcertain I period general M period property Vinogradov of the comma distribution A certain general of prime numbers . Mat . Sb . 7 ( 1 940404 ) ,János 365Pintz−− 372 ( Russian ) . property of the distribution[ 1 92 ] I . M of . prime Vinogradov numbers, Representation period Mat periodof an odd number as a sum of three prime numbers . [Sb 1 period94 ] A 7 open . I parenthesis. Vinogradov 1 940 , closing Application parenthesis of comma Riemann 365 endash ’ s $ 372\zeta open parenthesis( s Russian ) $ to the EratosthenianDoklady Akad sieve . Nauk . . SSSR Mat1 . 5 Sb( 1 937 . 41 ) , 29 1 – 294 ( Russian ) . [ 1 93 ] I . M . Vinogradov , closing parenthesisA period certain general property of the distribution of prime numbers . Mat . Sb . 7 ( 1 940 ) , 365 – 372 (open 1 957 square ) , bracket 49 −− 1 9480 closing ; Corr square . bracket: ibidem A period , 415 I period−− 41 Vinogradov 6 ( Russian comma )Application . of Riemann [ 1 95 ] A .( I Russian . Vinogradov ) . [ 1 94 ] A . , I .The Vinogradov density, Application hypothesis of Riemann for ’ s Dirichletζ(s) to the Eratosthenian $ L − $ s e r i e s . Izv . Akad . Nauk . SSSR quoteright s zeta opensieve parenthesis. Mat . Sb . s41 closing( 1 957 parenthesis ) , 49 – 80 ; to Corr the . : Eratosthenian ibidem , 415 – 41 sieve 6 ( Russian period ) Mat . [ 1 period 95 ] A.I. Sb period 4129 ( 1965 ) , 903 −− 934 ( Russian ) . Corr . : ibidem , 30 ( 1 966 ) , 71 9 −− 720 . [ 1 96 ] I . Vinogradov\quad M., The\quad densityVinogradov hypothesis for Dirichlet , \quadL−S ps e eries c i a l. Izv\quad . AkadVariants . Nauk . SSSR of2 the 9 Method of Trigonometric Sums . \quad Nauka , open parenthesis( 1 1 965 957 ) ,closing 903 – 934 parenthesis ( Russian ) comma . Corr . 49 : ibidem endash , 30 80( semicolon 1 966 ) , 71 Corr 9 – 720 period . [ 1 : 96 ibidem ] I.M comma 415 endashMoskva 41 6 , open 1 976 parenthesis ( Russian Russian ) closing. parenthesis period [ 1 97 ] Tianze. Wang Vinogradov , On, LinnikSpecial ’ s Variants almost of the Goldbach Method of Trigonometrictheorem . Sums Sci. . China Nauka , Ser . A 42 ( 1 999 ) , 1 1 55 −− open square bracketMoskva 1 95, 1 closing976 ( Russian square ) . bracket [ 1 97 ] ATianze period Wang I period, On VinogradovLinnik ’ s almost comma Goldbach The theorem density. hypothesis Sci for Dirichlet L hyphen s eries period Izv period Akad period Nauk period SSSR \ centerline {1. 1 China 72 Ser . } . A 42 ( 1 999 ) , 1 1 55 – 2 9 open parenthesis 1 965 closing parenthesis comma1 1 72903 . endash 934 open parenthesis Russian closing parenthesis period Corr period : ibidem comma 30 open parenthesis 1 966 closing parenthesis comma 71 9 \noindent [ 1[ 98 1 98 ] ] Yuan Wang Wang, On , On the therepresentation representation of a large even of integer a large as a sum even of a product integer of at as a sum of a product of at endash 720 period most 3 primes and a product of at most 4 primes . Acta Math . Sin . 6 ( 1 956 ) , 500 – 5 13 mostopen 3 square primes bracket and 1 a 96 product closing square of at bracket most I 4 period primes .. M . period Acta .. Math Vinogradov . Sin comma . 6 ( .. 1 Special 956 ) .. , 500 −− 5 13 Variants of the Method of Trigonometric Sums period ..( NaukaChinese comma). \ [(Moskva Chinese comma 1 976 ) open . parenthesis\ ] Russian closing parenthesis period open square bracket[ 1 99 1 ] 97Yuan closing Wang square, On sievebracket methods Tianze and Wang s ome comma of their Onapplications Linnik quoteright , I . Acta Math s almost . Sinica Goldbach theorem period Sci8 period( 1 958 China ) , 413 Ser – period 429 ( Chinese A 42 open ) . [ Englishparenthesis translation 1 999 : closing Sci . Sinica parenthesis8 ( 1 959 comma ) , 357 1 – 1 381 55 . endash \noindent1 1 72 period[ 1] 99 [ 200 ] ]YuanYuan Wang , Sheng , On - sieve gang Xie methods , Kunrin Yu and, Remarks s ome on of the theirdifference applications of consecutive , I . Acta Math . Sinica 8 (open 1 958 square ) , bracket 413primes−− 1. 98429 Sci closing . Sinica ( Chinese square14 ( 1 bracket 965 ) ) . , 786 [ Yuan English – 788 Wang . [ 201 comma translation ] E . OnWaring the representation, :Meditationes Sci . Sinica Algebraicae of a large 8 ( even 1 959 ) , 357 −− 381 . ] integer[ 200 as ] a sumYuan of. Wang a Cantabrigine product , Sheng of at. 1− 770gang . [ 3 rd Xie . ed ., 1782 Kunrin ; English Yu transla , Remarks - t ion : on American the Math difference . Soc . , of consecutive primesmost 3 primes . Sci andProvidence . Sinica a product , 1 99 14 of 1 . (at ] 1 most [ 202 965 ] 4N primes ). Watt , 786 period, Short−− intervals Acta788 Math . almost period all containing Sin period primes 6 open. Acta parenthesis Arith 1 ¨ 956[ closing201 ] parenthesis E . Waring. 72 comma( 1 995 , Meditationes) 500 , 131 endash – 1 67 . 5 [ 13 203 Algebraicae ] E . Westzynthius . 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Sheng hyphen gang Xie comma Kunrin Yu comma Remarks on the difference of consecutive \noindentprimes periodJ \ ’{ Scia} periodnos Pintz Sinica 14 open parenthesis 1 965 closing parenthesis comma 786 endash 788 period open square bracket 201 closing square bracket E period Waring comma Meditationes Algebraicae period Cantabrigine\noindent R period\ ’{ e} 1nyi 770 Mathematicalperiod open square Institute bracket 3 rd of period the Hungarian ed period 1782 Academy semicolon of English Sciences transla hyphenBudapest t ion : American Math period Soc period comma Providence comma 1 99 1 period closing square bracket \noindentopen squareRe bracket\ ’{ a} ltanoda 202 closing u square . 13 bracket−− 1 5 N period Watt comma Short intervals almost all containing primes period Acta Arith period 72 open parenthesis 1 995 closing parenthesis comma 131 endash 1 67 period \noindentopen squareH − bracket1053 203 , closingHungary square bracket E period Westzynthius comma Udieresis ber die Verteilung derE − Zahlenmail comma : p dieintz zu der $@$ n ersten renyi Primzahlen . hu teiler hyphen fremd sind period Comm period Phys period Math period Helsingfors open parenthesis 5 closing parenthesis 25 open parenthesis 1 93 1 closing parenthesis comma 1 endash 37 period open square bracket 204 closing square bracket M period Y period Zhang comma P period Ding comma An improvement to the Schnirelman constant period J period China Univ period Sci period Techn period 1 3 open parenthesis 1 983 closing parenthesis comma Math period Issue comma 3 1 endash 53 period Jaacutenos Pintz Reacutenyi Mathematical Institute of the Hungarian Academy of Sciences Budapest Reaacuteltanoda u period 13 endash 1 5 H hyphen 1053 comma Hungary E hyphen mail : p intz at renyi period hu