Hardy Littlewood Primes

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    SOME PR OBLE M S OF PA RT ITI0 NUM ERORUM ; III: ON THE

    EXPRESSION OF h NUM BER AS h SUM OF PRIMES.

    BY

    G . H . H A R D Y a n d J . E . L I T T L E W O O D .

    N e w C o l l e g e , T r i n i t y C o l l e g e ,

    O X F O R D C A M B R I D G E

    ~ . I n t r o d u c t i o n .

    z . I . I t w a s a s s er t e d b y G O L D B A C H , i n a l e t t e r t o E u L E R d a t e d 7 J u n e ,

    1742 , tha t e ve r y ev en n um b er 2m i s t h e s um o / t w o o d d pr i m e s a i~d t h i s pr o pos i -

    t i o n h a s g e n e r a l l y b e e n d e s c r i b e d a s G o l d b a c h s T h e o r e m . T h e r e is n o r e a s o n a b l e

    d o u b t t h a t t h e t h e o r e m i s c o r r e c t , a n d th a t t h e n u m b e r o f r e p r e s e n t a t i o n s i s

    l a r g e w h e n m i s l a r g e; b u t a l l a t t e m p t s t o o b t a i n a p r o o f h a v e b e e n c o m p l e t e l y

    u n s u c c e s s fu l . I n d e e d i t h a s n e v e r b e e n s h o w n t h a t e v e r y n u m b e r ( o r e v e r y

    l a r g e n u m b e r , a n y n u m b e r , t h a t i s t o s a y , f r o m a c e r t a i n p o i n t o n w a r d s ) i s t h e

    s u m o f x o p r im e s , o r o f i o o o o o o ; a n d t h e p r o b l e m w a s q u i t e r e c e n t l y cl a s s if i e d

    a s a m o n g t h o s e b e i m g e g e n w i i r t i g e n S t a n d e d e r W i s s e n s e h a f t u n a n g r e i f b a r . ~

    I n t h i s m e m o i r w e a t t a c k t h e p r o b l e m w i t h t h e a id o f o u r n e w t r a n s c e n -

    d e n t a l m e t h o d i n a d d i t iv e r Z a h l e n t h eo r i e . ~ W e d o n o t s o l v e i t: w e d o n o t

    i E . L A N D A U , G e l 6 s t e u n d u n g e lO s t e P r o b l e m e a u s d e r T h e o r i e d e r P r i m z a h l v e r t e i l u n g u n d

    d e r R i e m a n n s c h e n Z e t a f u n k t i o n ' , l~ oceedings of the fifth Infernational Congress of M athemat ic ians,

    C a m b r i d g e , i 9 t 2 , v ol . i , p p . 9 3 - - i o 8 ( p . . io s ) . T h i s a d d r e s s w a s r e p r i n t e d i n t h e Jah resbe r i ch t

    de r

    19eutscheu Math.-Vereinigung, v o l . 2 1 ( i 9 1 2 ) , p p . 2 o 8 - - 2 2 8 .

    W e g i v e h e r e a c o , n p t e t e l i s t o f m e m o i r s c o n ce r n e d w i t h t h e v a r i ou s a p p l i c a t i o n s o f

    t h i s m e t h o d .

    G . H . H A RD Y .

    I . A s y m p t o t i c f o r m u l a e i n c o m b i n a t o r y a na ly s i s ' , Com l tes rendus du quatri~me

    Congr~s des ma th ema t i c i ens Scandinaves h Stockholm, I 9 ,6 , p p . 4 5 - - - 5 3 .

    2 . O n t h e e x p r e s s i o n o f a n u m b e r a s th e s u m o f a n y n u m b e r o f s q u a r e s , a n d i n

    p a r t i c u l a r o f f i v e o r se v e n ' , Proceediugs of t h e N a t i on a l A cademy o f Sc iences , vol. 4 (19x8),

    p p . 1 8 9 - - 1 9 3 .

    Acta mathemat ica . 44. Imprim d le 15 fdvrier 1922. 1

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    G . H . H a r d y a n d J . E . L i t t le w o o d .

    e v e n p r o v e t h a t a n y n u m b e r i s t h e s u m o f x o o o o o o p r i m e s . I n o r d e r t o p r o v e

    a n y t h i n g , w e h a v e t o a s s u m e t h e t r u t h o f a n u n p r o v e d h y p o t h e s i s , a n d , ev e n

    o n t h i s h y p o t h e s i s , w e a r e u n a b l e t o p r o v e G o l d b a c h ' s T h e o r e m i t s e lf . W e s h o w ,

    h o w e v e r , t h a t t h e p r o b l e m i s n o t ' u n a n g r e i f b a r ' , a n d b r i n g i t i n t o c o n t a c t w i t h

    t h e r e c o g n i z e d m e t h o d s o f t h e A n a l y t i c T h e o r y o f N u m b e r s .

    3 . ' 8 o m e f a m o u s p r o b l e m s o f t h e T h e o ry . o f N u m b e r s , a n d i n p a r t i c u l a r W a r i n g' s

    P r o b l e m ' ( O x f o r d , C l a r e n d o n P r e s s , 1 92 o, p p . 1 - -3 4 ) .

    4 - ' O n t h e r e p r e s e n t a t i o n o f a n u m b e r a s t h e s u m o f a n y n u m b e r o f s q u a re s , a n d

    i n p a r t i c u l a r o f f i v e ' ,

    Transactions of the American Mathematical Society,

    vol. 2x (I92o), pp.

    255--z84.

    5 . ' N o t e o n R a m a n u j a n ' s t r i g o n o m e t r i c a l su m

    c~ (n) , .proceedings of the Cam bridge

    .philoso1~hical Society,

    vol . 2o (x92I) , pp. 263--z 7I .

    G. H . H xRDY a n d J. E. L1TTLEWOOD.

    Z . ' A n e w s o l u t i o n o f W a r i n g ' s P r o b l e m ' ,

    Quarterly Journa l of Irate and aFflied

    mathematics,

    vol . 48 (1919), pp.

    ZTZ--293.

    2 . ' N o t e o n M e s s r s . S h a h a n d W i l s o n ' s p a p e r e n t i t l e d : O n a n e m p i r i c a l f o r m u l a

    c o n n e c t e d w i t h G o l d b a c h ' s T h e o r e m ' , .proceedings of the Cambridge Philosophical Society,

    vol . 19 (1919) , pp. 245--z54.

    3 . ' S o m e p r o b l e m s o f ' P a r t i t i o n u m e r o r u m ' ; I : A n e w s o l u t io n o f W a r i n g ' s P r o -

    b l e m ' ,

    .u van der K. Ge.sdlschaft der Wissensehaften zu G6ttingen

    ( i9zo) , pp. 3 3--54.

    4 . ' S o m e p r o b l e m s o f ' P a r t i t i o n u m e r o r u m ' ; I I : P r o o f t h a t a n y la r g e n u m b e r is t h e

    s u m o f a t m o s t 2 x b i q u a d r a t e s ' , Mathematische Zeitschrift, voh 9 ( i 92 i ) , pp . 14 - - 27 .

    G. H. HARRY an d S. Is

    L ' U n e f o r m u l e a s y m p t o t i q u e p o u r l e h o m b r e d e s p a r t i t i o n s d e

    n , Com ptes rendus

    de l Acad~ mie des Sciences, 2

    Jan. I9x7.

    2 . ' A s y m p t o t i c fo r m u l a e i n c o m b i n a t o r y a n a l y s i s ' ,

    .Proceedings of the London Mathem .

    atical Society,

    ser . 2 , vol . 17 (xg18) , pp. 7 5~ II 5 .

    3 . ' O n t h e c o e f f i c ie n t s i n t h e e x p a n s i o n s o f c e r t a in m o d u l a r f u n c t i o n s ' ,

    Proceedings

    of the Royal Society of London

    (A), vol. 95 (1918), pp . x44--155.

    E .

    LANDAU

    I . ' Z u r H a r d y - L i t t l e w o o d ' s c h e n L 6 s u n g d e s ~ u P r o b l e m s ' ,

    Nachrichfen

    yon der K. Gesellschaft der Wissenschaften zu G6ttingen (192I) , pp. 88--92.

    L. J. MORDELL.

    I . ' O n t h e r e p r e s e n t a t i o n s o f n u m b e r s a s th e s u m o f a n o d d n u m b e r o f s q u a r e s ',

    Transactions o f the Cam bridge .philoso2hical Society,

    vol . z2 (1919), pp. 36t- - 37z .

    A OSTROWSKI

    L ' B e m e r k u n g e n z u r H a r d y - L i t t l e w o o d ' s c h e n L 6 s u n g d e s W a r i n g s c h e n P r o b l e m s ' ,

    Mathematische Zcitschrift~

    vol . 9 (19zI) , PP. 28--34.

    S RAMANUJAI~

    z ' O n c e r t a i n t r i g o n o m e t r i c a l s u m s a n d t h e i r a p p l i c a t i o n s i n t h e t h e o r y o f n u m -

    b e r s ' , Transactions of the Cambridge Philosophical Society, vo l . zz ( gx8) , pp. z59--276.

    N . M . SH A a n d B . M . WILSOn.

    L ' O n a n e m p i r i c a l f o r m u l a c o n n e c t e d w i t h G o l d b a c h ' s T h e o r e m ' ,

    .proceedings of

    the Cam bridge Philoserphical Society, vol. 19 (I919), pp. 238--244.

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    Parti tio numerorum. II I: On the expression of a number as a sum of primes. 3

    Our main result may be stated as follows: i / a c e r t a in h y p o t h e s i s (a natural

    generalisat ion of Riemann s hypothesi s concerning the zeros of his Zeta-function)

    i s t r u e , t h e n e v e r y l a r g e o d d n u m b e r n i s t h e s u m o / t h r e e o d d p r i m e s ; a n d th

    n u m b e r o / r e p r e s e n t a t i o n s i s g i v e n a s ym p t o t ic a l ly b y

    n ~

    w h e r e p r u n s t h r o u g h a l l o d d p r i m e d i v i s o r s o / n , a n d

    i . ~ 2) C ~ - ~ H i + , ~ 2 _ z ,

    t he prod uc t ex t en d i ng over a l l odd pr i m es v~ .

    H y p o t h e s i s R .

    x. z. We proceed to explain more closely the natur e of our hypothesis .

    Suppose that q is a posit ive integer , and that

    h = ~(q)

    is the numb er of numb ers less than q and pr ime to q. We denote by

    x n ) . = z k n ) k - I , 2 . . . . . h )

    one of the h Diriehlet s cha ract ers to modu lus 7 1: ZL is the prin cip al chara cter.

    By ~ we denote the complex num ber conju gate to : Z is a character .

    By L s , Z ) we denote the function def ined for a > i by

    L s ) = L c t + i t ) = L s , X ) = L s , g k ) = ~ . z n ) .

    ~ t n s

    n 1

    Unless the con trar y is stated the m odulus is q. We write

    / ~ s ) = L s ,

    ~).

    By

    ~-=fl ir

    Our notation, so far as the theory of L-functions is concerned, is that of Landau s

    Handbuch dcr Lehre yon der Verteilung der _Primzalden, vol. i, book 2, pp. 391 r seq., except that

    we use q for his k, k for his x, and ~ for a typical prime instead of 2. As regards the Farey

    dissection , we adhere to the notation of our papers 3 and 4.

    We do not profess to give a complete summary of the relevan t parts of the theory of

    the L-functions; but our references to Laudau should be sufficient to enable a reader to find

    for himself everything that is wanted.

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    4 G. H, Hard y and J . E . Li t t lewood,

    w e d e n o t e a t y p i c a l z e r o o f L s ) , t h o s e f o r w h i c h 7 ~ - o , f l < o b e i n g e x c lu d e d .

    W e c a l l t h e s e t h e non- t r iv ia l z e r o s . W e w r i t e N T ) f o r t h e n u m b e r o f Q s o f

    L s ) f o r w h i c h o < 7 < T .

    T h e n a t u r a l e x t e n s i o n o f R i e m a n n s h y p o t h e s i s i s

    H Y P O T H E S I S R * . E v e r y Q h a s i ts re a l p a r t l e ss th a n o r e q u al to ~ .~

    2

    W e s h a l l n o t h a v e t o u se t h e fu l l f o r c e o f t h i s h y p o t h e s i s . W h a t w e s h a l l

    i n f a c t a s s u m e i s

    H Y P O T H E S I S R . T h e re i s a n u m b e r 0 < 3 s uc h t h at

    4

    ~ o

    ]or euery ~ o] every L s) .

    T h e a s s u m p t i o n o f t h is h y p o t h e s i s is f u n d a m e n t a l i n a ll o u r w o r k ;

    all the

    resu l t s o[ the memoir , so jar as they are nove l , depend upon i t s ;

    a n d w e s h a ll n o t

    r e p e a t i t in s t a t i n g t h e c o n d i t i o n s o f o u r t h e o r e m s .

    W e s u p p o s e t h a t O h a s i t s s m a l l e s t p o s s i b le v a l u e , I n a n y e a s e O > I .

    = 2

    F o r , i f q i s a c o m p l e x z e r o o f L s ) , ~ i s o n e o f / ~ ( s) . H e n c e i - - ~ i s o n e o f

    L ( i ~ s ) , a n d s o , b y t h e f u n c t i o n a l e q u a t i o n s, o n e o f

    L s ) .

    Fur ther no ta t ion and te rmino logy .

    I . 3- W e u s e t h e fo l lo w i n g n o t a t i o n t h r o u g h o u t t h e m e m o i r .

    A i s a p o s i t i v e a b s o l u t e c o n s t a n t w h e r e v e r i t o c cu r s , b u t n o t t h e s a m e

    c o n s t a n t a t d i f f e r e n t o c c u r r e n c e s . B is a p o s i t i v e c o n s t a n t d e p e n d i n g o n t h e

    s i n g le p a r a m e t e r r . O s r e f e r t o t h e l i m i t p r o c e s s n - ~ r t h e c o n s t a n t s w h i c h

    t h e y i n v o l v e b e i n g o f t h e t y p e B , a n d o s a r e u n i f o r m i n a ll p a r a m e t e r s

    except r .

    i s a p r i m e , p ( w h i c h w i l l o n l y o c c u r i n c o n n e c t i o n w i t h n ) i s a n o d d

    p r i m e d i v i s o r o f n . p i s a n i n t e g e r . I f q = - ~ , p - -- -- o; o t h e r w i s e

    o < p < q , ( p , q ) = ~ ,

    ( r e, n ) i s t h e g r e a t e s t c o m m o n f a c t o r o f m a n d n . B y m [ n w e m e a n t h a t n is

    d i v i s i b l e b y m l b y m ~ n t h e c o n t r a r y .

    J / ( n ) , tt (n ) h a v e t h e m e a n i n g s c u s t o m a r y i n t h e T h e o r y o f N u m b e r s , T h u s

    . d ( n ) i s l o g ~ i f n = ~ a n d z e r o o t h e r w i s e : ~ ( n ) is ( - - I ) k i f n is a p r o d u c t o f

    T h e h y p o t h e s i s m u s t b e s t a t e d i n t h i s w a y b e c a u s e

    (a) i t has not been pro ved tha t no

    L s)

    has rea l zero s be tw een ~ and I ,

    ( b) t h e L - f u n c t i o n s a s o c i a te d w i t h impriraitive uneigent l ich) charac te rs ha ve zeros on the l ine a = o ,

    t~a tura l ly many of the resu l t s s ta ted inc identa l ly do not depend upon the hypothes is .

    8 Landau, p . 489 . Al l re fe ren ces to Landau are to h is

    Handbuch,

    unless the cont ra ry i s s ta ted .

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    Parti t io num eroru m. III: On the expression of a number as a sum of primes. 5

    k d i f f e r e n t p r i m e f a c t o r s , a n d z e ro o t h e r w i s e . T h e f u n d a m e n t a l f u n c t i o n w i t h

    w h i c h w e a r e c o n c e r n e d i s

    ( 9 3 I ) / ( Z ) = 2 l o g f f X ~ r

    T o s i m p l i f y o u r f o r m u l a e w e w r i t e

    e x ) = e 2 ~ I ~ , e q x ) = e q ) ,

    A l s o

    ( i , 3z)

    I f X k i s p r i m i t i v e ,

    ~ 33)

    P

    5

    Vk

    = v (Zk) = 2 eq (p) Xk (P) = 2 eq (m) Zk (m ). '

    p m ~ l

    T h i s s u m h a s t h e a b s o l u t e v a l u e ~ ~ q .

    T h e F a r e y d i s s e c t i o n .

    x . 4 . W e d e n o t e b y F t h e c i r c le

    1

    ( I. 4 I ) I x l = e - / / = e

    W e d i v i d e F i n t o a r c s ~ , q w h i c h w e c a ll F a r e y a r c s , i n t h e f o l l o w i n g m a n n e r .

    W e f o r m t h e F a r e y ' s s e ri es o f o r d e r

    ( I. 4 2 ) N = [ V n ] ,

    t h e f i r s t a n d l a s t t e r m s b e i n g o a n d _ I.

    I I

    p ' p

    s e r ie s , a n d ~ a n d ~ t h e

    ] 'p ,q ( q > i ) t he i n t e r va l s

    W e s u p p o s e t h a t -p i s a t e r m o f t h e

    q

    a d j a c e n t t e r m s t o t h e le f t a n d r i g h t, a n d d e n o t e b y

    ~

    ( I ) ( I i , i ) . T h e s e i n t e r v a l s j u s t

    y ]'o ,1 a n d ]'1,1 t h e i n t e r v a l s o , ~ - ~ - ~ a n d r - - N +

    7 , k m ) - - o i t m , 2 ) > ~.

    L a n d a u , p . 4 9 7 .

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    6 G . H . Ha r d y a n d J . E . L i tt le wo o d .

    f il l u p t h e i n t e r v a l ( o , I ) , a n d t h e l e n g t h o f e a c h o f t h e p a r t s i n t o w h i c h j p, q i s

    d i v i d e d b y -pq i s l e s s t h a n q -N I a n d n o t l e s s t h a n . . . .q N I I f n o w t h e i n t e r v a l s 3 p,~

    e~ rc c o n s i d e r e d a s i n t e r v a l s o f v a r i a t i o n o f 0 , w h e r e 0 ~ - a r g x , a n d t h e tw o

    2~v

    e x t r e m e i n t e r v a l s j o i n e d i n t o o n e , w e o b t a i n t h e d e s i r e d d i s s e c t i o n o f F i n t o a r c s ~ p, ~.

    W h e n w e a r e s t u d y i n g t h e a r c ~ p , q , w e w r i t e

    pal

    ( L 4 3 ) x f f i e 9 X f f i e ~ ( r ) X ~ e q f ~ ) e - r ,

    ~ , 4 4 ) Y ~ ~7 iO .

    T h e w h o l e o f o u r w o r k t u r n s o n t h e b e h a v i o u r of / (x ) a s ] x ~ - - . i , , / ~ o , a n d

    w e s h a l l s u p p o s e t h r o u g h o u t t h a t o < ~ < I -- . W h e n x v a r i e s o n ~ p,g , X v a r i e s

    ~ Z

    o n a c o n g r u e n t a r c ~ p ,g , a n d

    0 -~ - - ( a rg - 2 p ,- r~

    v a r i e s ( in t h e i n v e r s e d i r e c t io n ) o v e r a n i n t e r v a l - -O ~ v , g ~ O < O p , ~ . P l a i n l y O p, ~

    2Y'g ~T

    a n d 0 ~ ,~ a r e le s s t h a n ~ a n d n o t l e s s t h a n ~ _g , s o t h a t

    q = M s x ( O p , 4 , O ' p ,q ~ < : N

    I n a l l c a s e s

    Y - ' = (~i ~ - i 0 ) - :

    h a s i t s p r i n c i p a l v a l u e

    e x p ( ~ S l o g (~ +

    i 0 ) ) ,

    w h e r e i n ( s i n c e , / i s p o s i t i v e )

    - - ~

    rc < ~ l o g

    7 + i 0 ) < _ I ~ : r.

    2 2

    B y N r ( n ) w e d e n o t e t h e n u m b e r o f r e p r e s e n t a t i o n s o f n b y a s u m o f r p r i m e s,

    a t t e n t i o n b e i n g p a i d t o o r d e r , a n d r e p e t i t io n s o f t h e s a m e p r i m e b e i n g al lo w e d ,

    s o t h a t

    The d i s t inc t ion b e tween m ajor and minor a rcs , fundam enta l in our work on Wa r ing 's

    Problem. does not ar ise here.

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    Part i t io

    n u m e r o r u m I I I :

    On the

    e x p r e s s i o n o f a n u m b e r a s a s u m o f p r im e s

    B y v~(n) w e d e n o t e t h e s u m

    r , . (n ) ~ ~ l o g ~ l o g ~ . . . l o g W ~ ,

    ~tO-t+ , f f 2, . r . . . + ~T r - - n

    I . 4 6 )

    s o t h a i

    ( i . 4 7 )

    * , ( n ) x " = ( I ( , ) ) ' .

    F i n a l l y S . i s t h e s ingu lar s e r ie s

    I . 4 8 )

    flo r

    = ~ l t ( q ) t e I _

    ,

    q ~ . l l~ p ( q ) ~ , n ) .

    2 P r e l i m i n a r y l e m m a s

    2 . I . L e m m a r . I 1 ~ ---- ~ Y ) > o t h e n

    2 . I I

    l ( x ) ~ l , ( x ) + h ( x ) ,

    where

    2 , 1 2 )

    f , ~ ) = 2 l ~ . ) . . _ X l og . ~ x n ~ ,+ x ~r ~+ .

    -) ,

    q , . ) > 1

    (2. ~3)

    2 + i ~

    h x) =2,~i

    2 - - a e

    Y - has i t s pr inc ipa l va lue ,

    2 . I 4 )

    h t

    ~ ,~ L k(s)

    z ( ~ ) = , ~ , ~ k

    ~ ,

    k - - 1

    C~ depends only on p, q and 7~k,

    ( 2 . I 5 )

    a n d

    C = - - - -

    ~ q )

    2 . 1 6 )

    I C k [ _ _ < _ ? -

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    G H H a r d y a n d J E L i t t l e w o o d

    W e h a v e

    h : ~ ) = 1 : ~ ) - 1 , x ) = ~ ~ n ) x *

    q , n ) - - 1

    l _ < _ i < q , (q , * } - 1 l - 0

    2 + i o o

    t ] ' y _ s F ( s ) ( l q + ] ) _ s d s ,

    ~ . e , p i ) ~ _ 4 z ~ + j )

    i l 2 - i ~

    w h e r e

    2 + i Q o

    --2~ilY-~F s)Z s)ds,

    2 i ~

    S i n c e ( q , ] ) = I , w e h a v e 1

    J / ( l q + ? )

    ~ :Cff

    h

    I , ~ . . . . L % t s~ ;'

    h ~ z k ~ 7 ~

    k ~ l

    a n d s o

    4 - L ' z , ( s ) ,

    Z s ) = z ~ ; k

    w h e r e

    C k - - h i ~ _ ~ e q ( p T ) Z k ( ] )

    j - 1

    S i n c e ] 3 , ( j ) = o i f ( q , j ) > I , t h e c o n d i t i o n ( q, i ) = I m a y b e o m i t t e d o r r e t a i n e d

    a t o u r d i s c r e t i o n .

    T h u s ~

    I

    l_

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    P a r t i t i o n u m e r o r u m .

    H I : O n t h e e x p r e s s i o n O f a n u m b e r a s a s u m o f p r i m e s . 9

    A g a i n , i f k > I w e h a v e

    j 1 m 1

    I f Z r, i s a p r i m i t i v e c h a r a c t e r ,

    I C k l = ? -

    I f ~ i s i m p r i m i t iv e , i t b e Io n gs t o Q = w h e r e d > I . T h e . 7 ,k m ) h a s t h e

    p e r i o d Q , a n d

    QI d - 1

    m - - 1 n ~ l l - - 0

    T h e i n n e r s a m i s z e ro . He n c e C a = o , a n d t h e p ro o f o f t h e l e m m a is c o m p l e t e d , n

    2 . 2 . L emm a z . W e h a v e

    [ / , (x ) l < A ( log (q + I ) )a ~ -~

    2 . 2 1 )

    W e h a v e

    I t ( x ) ~ - ~ . . 4 n ) x n - - ~ . ~ l o g w ( x ~ + x ~ a + - . - ) = / 1 , 1 ( x ] - - / , , 2 ( x ) .

    (q , n) > 1 Z J

    co

    l / l a ( X ) l < ~ lo ~ ~ I ~ U

    z ~ [ q r - - I

    co o

    < A l o g ( q + I ) l o g q ~ 1 . 1 2 < A ( lo g ( q + ~ ) ) ' ~ e - , ,

    r - - 1 r ~ l

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    10

    G. H. Hardy and J. E. Lit t lewood.

    A l s o

    a n d s o

    2 l og ~ < A V~ ,

    I l ,~ (z) I< ~

    log

    ,~1~1~ < A ( , - - I ~ 1 ) ~ V ~ l ~,1

    r_~2, ~* n

    1 1

    < A ( I - - I x I ) - ~ < A ~ ~

    F r o m t h e s e t w o r e s u l ts t h e l e m m a f o ll o w s.

    2. 3. Lemma 3. We have

    2 . 3 1 ) L ( 8 ) 8 - - 1 ~ - 2 8 [ - - 0 '

    o

    where

    F (z)

    ~ ( z ) = r - - ( z~

    the

    ~ 's , b ' s , b ' s

    and

    b ' s

    are constants depending upon q and

    Z, a

    is o or 1,

    2 . 3 2 )

    a n d

    2. 33)

    B , = I , ~ = o k > I ),

    o ~ b < A lo g (q + i ) .

    A l l t h e s e r e s u l t s a r e c l a s s i c a l e x c e p t t h e l a s t 3

    T h e p r e c i s e d e f i n i t i o n o r b i s r a t h e r c o m p l i c a t e d a n d d o e s n o t c o n c e r n u s.

    W e n e e d o n l y o b s e r v e t h a t b d o e s n o t e x c e e d th e n u m b e r o f d i f fe r e n t pr i m e s

    t h a t d i v i d e q ,~ a n d s o s a t i s f i e s ( 2 . 3 3 ) .

    2. 41 .

    Lemma 4 I [ o < ~ < ~,

    then

    h

    (2. 4 i i ) / ( x ) - - + ~C k G ~ + P ,

    k - - 1

    where

    2 . 4 r 2 ) O k = ~ F ( q ) Y - ~ ,

    t L a n d a u , p p . 5 0 9 , 5 to , 5 x 9 .

    L a n d a u , p . 5 11 f o o tn o t e ) .

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    Parti t io num ero rum , l II: On the expression of a number as a sum of primes. 11

    (2. 413)

    h I 1 1

    k=l

    2 . 414) 0 - - a r c t a n

    I ~ l

    W e hav e , f r om ( 2 . x3) an d ( 2. x4),

    (2. 4z5)

    s a y . B u t I

    2 iQo

    z

    / r - r ( , ) Z ( s l d 8

    h : O = 2 ~ ~

    2 ioo

    2 i ao

    = ~ Y - . t O ) L - - ~ a , = ~ e , / ~ k x ) .

    k - 1 k - I

    2--iQa

    2 + i ~

    X

    f ,

    L ( 8) ~ ~ r ( r

    y o

    ( 2 . 4 1 6 ) 2 i L ( 8 )

    _ y- F s )~d s=--- V+ R + +

    2 i~ P

    w h e r e

    1

    f r-.r(.)n ( )-

    - ~ (8 ) a S

    1

    4

    L

    . , ( s )

    R - - { Y 1 ( 8 ) - ~ 7 ) } o ,

    f

    ~ / (s )j 0 d e n o t i n g g e n e r a l l y t h e r e s i d u e o f / (s ) f o r s = o .

    ~ o w ~

    L ( s ) , z r ,~ , , l og ~ ~ ~,, l og w ~

    2 7 ~ - - - 2 ~ v 2 L ( ~ - ~ )

    w h e r e Q i s t h e d i v i s o r o f q t o w h i c h Z b e l o n g s , c i s th e n u m b e r o f p r i m e s w h i c h

    d i v i d e q b u t n o t Q , ~ r, , z ~ , . . , a r e t h e p r im e s i n q u e s t i o n , a n d , ~ i s a r o o t o f

    u n i t y . H e n c e , i f a i

    - - - , w e h a v e

    T h i s a p p l i c a t i o n o f C a u c h y s T h e o r e m m a y b e j u s ti f i ed o n t h e l i n e s o f t h e c l as s ic a l

    p r o o f o f t h e e x p l i c i t f o r m u l a e f o r ~ ( x ) a n d = (x ): s e e L a n d a u , p p . 3 3 3 -- 3 6 8. I n t h i s e a s e t h e

    p r o o f i s m u c h e a s i e r , s i n c e Y - - s F ( s ) t e n d s t o z e r o, w h e n I t [ -~ Q o , l i k e a n e x p o n e n t i a l e a I r |

    C o m p a r e p p . x 3 4 -- *3 5 o f o u r m e m o i r : C o n t r i b u t i o n s to t h e t h e o r y o f t h e R i e m a n n Z e t a - f u n c t i o n

    a n d t h e t h e o r y o f t h e d i s t r i b u t i o n o f p ri m e s , Ac l a Ma t h em a t i ca , e e l . 4 1 0 9 1 7 ) , p p . I x 9 - - I 9 6 .

    L a n d a u , p . 5 1 7 .

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    2 G. H. Hardy and J. E. Littlewood,

    (2. 417)

    L ( , ) [

    < A lo g g + A r lo g q + A lo g ( I t l + 2 ) + A

    < A ( log ( ~, + i ) ) a log ( i t i + 2 ) .

    I

    A g a i n, if s = - - - + i t ,

    Y = ~ + i O ,

    w e h a v e

    4

    1

    , Y , . o p , . a r o t a o ) .

    f r - , r ( s ) l < A l r l ~ ( I t t + 2 ) - ~ e x p - ~ - a r c t a n l t l

    1

    1 i t l

    < A I Y J ~ l o g ( l t l + 2) e -' ~i t~

    a n d s o

    (2. 418)

    1

    - - - i n

    4 7 _ _ ~

    I I I ' L 't s~ I Y l J t ~ e - ~

    1 0

    4

    1 1

    < A ( l o g ( q + 1 ) ) a [ Y I 4 d ~

    2 , 4 2 . W e n o w c o n s i d e r R . S i n c e

    w e h a v e

    + - - -o ( s - - - o) ,

    ---- A~ ( b+ b ) - - C b- - b) ( A~ + A3 log Y) + Ct ( a ) + C~( a ) log Y,

    w h e r e e a c h o f t h e C s h a s o n e o f t w o a b s o l u t e c o n s t a n t v a l u e s , a c c o r d i n g t o t h e

    v a l u e o f a . S i n c e

    1

    o < b < I , o < b < A l o g ( q + I ) , I lo g

    Y I < A l o g I - < A r ,

    - - 2 ,

    w e h a v e

    1

    (2. 42x) IRl albl

    + A

    l o g ( q + i ): ~ ~

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    2. 422)

    2. 423)

    Partitio numerorum lII : On the expression of a number as a sum of primes.

    F r om 2 . 415), 2 . 416), 2 , ~ I 8) , 2 . 42I ) and 2 . I 5 ) we de du ce

    h , k ~ ) = - - y + G ~ + P ~ ,

    [ P k [ < A ( l o g (q + x ) )a ( i b l + v - ~ + l

    Y ] 6 ~),

    1

    x) h Y

    k

    IPl I a n d k i s a p r i m i t i v e a n d t h e re / or e n o n - p r i n c i p a l a)

    aeb s

    a = a q , X ) = a ~ ,

    1

    w

    ] L x ) l = ~ q

    2]L o)

    ( a = x ) ,

    1

    N

    I L ( r ) l = 2 q 2 lL '( o) l ( a = o ) .

    - - o < 9 ~ ~ ) s

    L I ) I < A log q + I ) ) A

    T h i s l e m m a i s m e r e l y a c o l l e c ti o n o f r e s u l t s w h i c h w i ll b e u s e d i n t h e p r o o f

    o f L e m m a s 6 a n d 7 - T h e y a r e o f v e r y u n e q u a l d e p t h . T h e fo r m u l a 2 . 5 I ) i s

    c l a ss i ca l . ~ T h e t w o n e x t a r e i m m e d i a t e d e d u c t i o n s f r o m t h e f u n c t i o n a l e q u a t i o n

    f o r

    L s ) . s

    T h e i n e q u a l i t ie s 2 . 5 3) f o ll o w f r o m t h e f u n c t i o n a l e q u a t i o n a n d t h e

    i Lan dau p. 480.

    Land au p. 507.

    8 La nd au pp. 496 497.

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    14

    G . H . H a r d y a n d J . E . L i t t l e w o o d .

    a b s e n c e ( f o r p r i m i t i v e

    t O G R O N W A L L . 1

    2. 6i. Lemma 6.

    ~ ) o f f a c t o r s i - - e ~ : ~ f r o m L . F i n a l l y (2 . 5 4) i s d u e

    If M T) is the number o] zeros Qo[ L s) [or which

    o

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    Par t i t io numerorum . H I : On the express ion of a number as a sum of pr imes . 15

    T h e n u m b e r o f ~ v s i s le s s t h a n A l o g ( q + i ) , a n d e a c h E ~ h a s a s e t o f z e r o s,

    o n a = o , a t e q u a l d i s t a n c e s

    2~f 2~rg

    l o g ~ > l o g (q + ~ )

    T h e c o n t ri b u t io n o f t h e s e z e r o s t o M T ) i s t h e r e f o r e l e s s t h a n A ( l o g ( q + i )) ,

    a n d w e n e e d c o n s i d e r o n l y a p r i m i t i v e ( a n d t h e r e f or e , i f q > I , n o n - p r i n c i p a l ) L s ) .

    W e o b s e r v e :

    ( a) t h a t ~ i s t h e s a m e f o r L s) a n d L ( , ) ;

    ( b ) t h a t L s) a n d L ( s ) a r e c o n j u g a t e f o r re a l. s , s o t h a t t h e b c o r r e s p o n d i n g t o

    L ( s ) i s 6 , t h e c o n j u g a t e o f t h e b o f - L ( s ) ;

    ( e) t h a t t h e t y p i c a l e o f /~ (s ) m a y b e t a k e n t o b e e i t h e r ~ o r ( in v i r t u e o f t h e

    f u n c t i o n a l e q u a t i o n ) i - - e , s o t h a t

    S = Z I i _ _ 0

    i s r e a l

    B e a r i f lg t h e s e r e m a r k s i n m i n d , s u p p o s e f ir s t th a t. ~ = I .

    f r o m ( 2 . 5x) an d ( 2. 52I ) ,

    W e h a v e t h e n ,

    s i n c e

    T h u s

    = e b ) + S ,

    I I ~=I.

    I

    I 8i-2~

    I - -

    2 . 6x2)

    ] 2 9 ~ ( b ) + S I< A l o g ( q + ~ ).

    O n t h e o t h e r h a n d , i f a = o , w e h a v e , f r o m ( 2. 5 I) a n d (2 . 5 2 2) ,

    4 _ I L ( I ) n I ) I 1

    a n d ( 2 . 6 x 2 ) f o l l o w s a s b e f o r e .

    2 . 62 . A g a in , b y 2 . 3x )

    L ( 1 )

    ( 2. 6 2 1 ) L ( I )

    I

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    1 6 G . H . Ha rd y a n d J . . E . L i tt le wo o d.

    f o r e v e r y n o n - p r i n c i p a l c h a r a c t e r ( w h e t h e r p r i m i t i v e o r n o t ) . I n p a r t i c u l a r , w h e n

    ; r i s p r i m i t i v e , w e h a v e , b y ( z . 6 2 I ) , ( z . 5 4 ) , a n d ( 2 . 3 3 ) ,

    ~ I ~ L ( I ) , i (

    ) l < A l o g q + I ) ) a .

    2 .

    C o m b i n i n g ( 2 . 6 12 ) a n d ( 2. 6 22 ) w e s e e t h a t

    8 < A ( lo g (q + i ) ) a

    a . 623)

    a n d

    (2 . 624)

    1 9 ~(b ) l < A ( lo g (q + x ) ) a .

    2 . 6 3. I f n o w q > x , a n d ;r i s p r i m i t i v e ( so t h a t 1 ~ o ) , a n d s ~ z + i T , w e

    h a v e , b y (2 . 3 I ) , ( z . 3 3 ) , a n d (2 . 6 2 4 ) ,

    2 - - / ~ I I

    < A + A l og ( q + l ) + A ( lo g ( q + 1 )) a + A lo g ( I T l + e )

    < A ( lo g ( q + i ) ) a l o g (I T I + 2 ) ,

    e - - f l < A l o g q + i ) ) a l o g l T [ + 2 ) .

    (2 - - f l)~ + ( T - 7 ) ~

    IT 71~I

    E v e r y t e r m o n t h e l e f t h a n d s i d e i s g r e a t e r t h a n A , a n d t h e n u m b e r o f t e r m s

    i s n o t l e ss t h a n M ( T ) . H e n c e w e o b t a i n t h e r e s u l t o f t h e l e m m a . W e h a v e

    e x c l u d e d t h e c a s e q ~ 1 , w h e n t h e r e s u l t i s o f c o u r s e c l a s s i c a l ?

    2. 7 r . Lem ma 7 . We have

    (2. 711) [ b i < A q ( lo g ( q +

    I ) ) A .

    S u p p o s e f i r s t t h a t x i s n o n - p r i n c i p a l . T h e n , b y ( 2 . 6 21 ) a n d ( 2. 5 4) ,

    L a n d a u , p . 3 3 7 .

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    P a r t i t i o n u m e r o r u m . I I I : O n t h e e x p r e s s i o n o f a n u m b e r a s a s u m o f p r i m e s . 1 7

    W e w r i t e

    2.7i ) 2 = 2 , 2 ;

    w h e r e ~ i i s e x t e n d e d o v e r t h e z e r os f o r w h i c h

    1 - - e < ~ e ) < e

    a n d i ~ e o v e r

    t h o s e f o r w h i c h 9 ~ ( q ) = o . N o w ~ 1 - - -- - 8' , w h e r e S ' i s t h e 8 c o r r e s p o n d i n g t o a

    p r i m i t i v e L ( s ) f o r m o d u l u s Q , w h e r e Q [ q . H e n c e , b y (2 . 6 2 3) ,

    ( 2 . 7 1 4 ) [ ~ t [ < A ( l o g ( Q + x ) ) a < a ( l o g ( q + 1 ) ) ~ .

    A g a i n , t h e q 's o f ~ e a r e t h e z e r o s ( o t h e r t h a n s = o ) o f

    [ I / ,

    p

    t h e ~ ' s b e i n g d i v i s o rs o f q a n d r~ a n m - t h r o o t o f u n i t y , w h e r e m ~ e p Q ) < q l ;

    s o t h a t t h e n u m b e r o f ~ , ' s i s l e s s t h a n A l o g q a n d

    ~, ~ e2 ~ i r ,

    w h e r e e i t h e r ~o~ = o o r

    ny

    q~ i s o f t h e f o r m

    q _ < _ l o , _ _ < - ~

    L e t u s d e n o t e b y r a z e r o ( o t h e r t h a n s - - -- o ) o f i - *~w T-~ , b y q ' , a # ,' f o r w h i c h

    i q , i _< _ i, a n d b y q , a q , f o r w h i c h I q , l > I . T h e n

    2 ~ i m + o , )

    q - ~ l o g ~ , '

    w h e r e m i s a n i n t e g e r . H e n c e t h e n u m b e r o f z e r o s d ~ i s l e ss t h a n A l o g ~Y~ o r

    t h a n A l o g ( q + i ) ; a n d t h e a b s o l h t e v a l u e o f t h e c o r r e s p o n d i n g t e r m i n o u r s u m

    i s l e s s t h a n

    A < A l o g ~

    (2 . 716 ) ]q ] io j~ ] < A q l o g q + I ) ;

    I Fo r (Landau , p . 482 ) .%-- - -X(v~) , wh ere X i s a cha ra c te r to modu lus Q .

    Acta mathematiea. 44. Imprim~ e 15 f6v rler 1922.

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    18 O. H. Hard y and J. E. Littlewood.

    s o t h a t

    (2.

    727

    Also

    2 . 7 ~ 8 )

    ] ~ < ~ i _ ~ < :t

    < A ( log ~ , ) ~-~ < A ( log (q + ~) )~ .

    F r om ( ~. 715) ,. ( 2. 7z7) and ( 2. 718) we d ed uc e

    (2. 719)

    I ~ 1 _ 3 a n d 0 > 1 , ~ - . r < r - - i - - r - - l + O . , a n d f r o m ( 3 . I 4 2 ) ,

    - - 2 2 4 - -

    (3 . I54) , an d (3- 155) we ob ta in

    (3. 156)

    v r ( n ) - - ( r _ i ) e q ( - - n p ) +

    n - i t q ) l

    ( l o g n ) )

    - - ( r _ _ i i q < ~ N / ~ - ~

    c e ( - - n ) +

    3 . 1 6, I n o r d e r t o c o m p l e t e t h e p r o o f of T h e o r e m A , w e h a v e m e r e l y t o

    s h o w t h a t t h e f i n i t e s e r i e s in ( 3. 1 56 ) m a y b e r e p l a c e d b y t h e i n f i n i t e se r i e s S ~ . N o w

    r -1 I I ' (q)~" c

    B n r - 1 ~ q x - ~ ( l o g

    q ) B < Bn - i ~

    ( log n ) B ,

    q ~ ( ~ ] q ( - - n ) < q > N

    a n d

    X - r < r - - l + ( O - - 3 - ] .

    H e n c e t h i s e r r o r m a y b e a b s o r b e d i n t h e s e c o n d t e r m

    2

    / 4

    o f ( 3 . 1 5 6 ) , a n d t h e p r o o f o f t h e t h e o r e m i s c o m p l e t e d ,

    S u m m a t i o n o / t h e s i n g u l a r s e r i e s .

    3 . 2 1 . L e m m a i t . I ]

    (3 - 2 1 i ) c q ( n ) - ~ e q ( n p ) ,

    wh e r e n i s a p o s i t i v e i n t e g e r a n d t h e s u m m a t i o n e x t e n d s o v e r a l l p o s i t i v e v a l u e s o / p

    l e s s t h a n a n d p r i m e t o q , p = o b e i n g i n c l u d e d wh e n q - ~ 1 , b u t n o t o t h e r wi s e , t h e n

    (3- 212)

    (3. 213)

    i [ ( q , q ' ) = I ; a n d

    (3. 214)

    c q - -n )= c q n );

    e q r n ) = c q n ) C q , n )

    w h e r e ~ i s a c o m m o n d i v i s o r o ] q a n d n .

    T h e t e r m s i n p a n d

    q - - p

    a r e c o n j u g a t e .

    a n d cq --n) r e c o n j u g a t e w e o b t a i n ( 3. 2 1 2 ) .a

    He nc e r i s rea l . As cq(~ )

    i T he arg um ent fails if q---- i or q---- 2; bu t G(n)= G(- -n) = i , c~(n)= c~( - n) -~ - - i .

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    P a r t i t i o n u m e r o ru m . I I I : O n t h e e x p r e s s io n o f a n u m b e r a s a s u m o f p r i m e s . 2 7

    A g a i n

    w h e r e

    ( ( ) ~ 1 2 n P J r i i

    c q ( n ) e q , ( n ) - - - 2 e x p 2 n ~ v i

    p,p, p, pr

    P = p q p q .

    W h e n p a s s u m e s a s e t o f 9 ( q ) v a l u e s , p o si O i ve , p r i m e t o q , a n d i n c o n g r u e n t t o

    m o d u l u s r a n d p ' a s i m i l a r s o t o f v a h t e s f o r m o d u l u s q ' , th e n P a s s u m e s a s e t

    o f r r -----9 ( qq ') v a l u e s , p l a i n l y a l l p o s i t iv e , p r i m e t o q q ' a n d i n c o n g r u e n t t o

    m o d u l u s q q ' . H e n c e w e o b t a i n ( 3- 2 1 3) .

    F i n a l l y , i t is p h i n t h a t

    dlq h- -O

    w h i c h i s z e r o u n l e s s q I n a n d t h e n e q u a l to q . H e n c e , if w e w r i t e

    w e h a v e

    a n d t h e r e f o r e

    ~ ( q ) = q ( q I ) , , ~ ) = o ( q n ) ,

    ~ c a n ) = ~ q ) ,

    dlq

    die

    b y t h e w e l l - k n o w n i n v e r s i o n f o r m u l a o f M S b i u s . t

    3 . 22 . L e m r a a z z . S u p p o s e t h a t r > 2 a n d

    T h i s i s ( 3 . 2 1 4 ) 3

    ~ - l ~ P ( q ) c ~ ( . .. n ) .

    T h e n

    3 . 2 2 o ) S ~ ~ o

    t Landau, p.

    577.

    The formula (3- 214) is proved by RXMXt~UaAN On certain trigonometrical sums and their

    applications in the theory of numbers , Trans. Camb. Phi l . Soc. , eel. zz (~918), pp. z 5 9 - - z 7 6 (p. 26o)).

    It had alr ead y be en g iven for n ---- i by LANDAU Handbuch (19o9), p. 572: Landau refer s to it as

    a k nown result), and i n the general case by JExs~g ( E~ nyt Udtr yk for den talteoretiske Funk-

    tion 2 I , ( n ) = M ( n ) , D e n 3 . S k a n d i n a vi s k r ~ l a le m a t i ke r - K o n g re s , K ~ t i a n i a 1 91 3, Kri sti ani a (~915),

    P. 145). Rama nujan mak es a large numb er of very beautif ul applications of the sums in ques-

    tion, and they may well be associated with his name.

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    2 8 G . H . Ha r d y a n d J . E . L i t tl e wo o d .

    i ] n a n d r a r e o ] o p p o s i t e p a r i t y . Bu t i ] n a n d r a r e o ] b i k e p a r i t y t h e n

    (~.

    223

    2~r

    , ~ - ~ ) ~ - - - - ~ ) ~ ~

    w h e r e p i s a n o d d p r i m e d i v i s o r o ] n a n d

    (3 . 224)

    L e t

    (3- 225)

    T h e n

    , , x - - ~ ) ~ t

    ~

    (q}V ,

    ~ )

    c q - - n ) =

    Aq.

    ~ e ( q q ) = ~ e ( q ) ~ L ( q ) , 9 ( q q ) = 9 ( q ) ~ P ( q ) , c ~ , ( - - n ) - - c q ( - - n ) e q , ( - - n )

    i f ( q, q ' ) = I ; a n d t h e r e f o r e ( o n t h e s a m e h y p o t h e s i s )

    Aqq = A~ A~ .

    3- 226)

    H e n c e t

    w h e r e

    (3. 227)

    S ~ . = A ~ A . , A , . . . . I A 2 . . . . l l z g

    g o

    =

    I + A . + A . , + A . . + . . . . I + A . ,

    s i n ce A ~ , A g , , . . . v a n i s h i n v i r t u e o f t h e f a c t o r p ( q ) .

    3 . 2 3- I f ~ n , w e h a v e

    ~ e ( ~ ) = - - ~ , ~ p ( ~ ) = ~ - - ~ , c ~ ( n ) = ~ e ( ~ ) = - - ~ ,

    ( 3. 2 3 1) A ~ =

    I f o n t h e o t h e r h a n d ~ i n , w e h a v e

    (3. 232}

    - - I r

    - - I ) ~

    I

    S i n c e ] c q ( n ) l _ < _ ~ 3 , w h e r e O [ n , w e h a v e c q n ) - - -O 1 ) w h e n n i s f ix e d a n d q - - , ~ . Also

    b y L o m ma i o, ? ( q ) > A q logq) -A . Her~ ce the se r i e s an d p roduc t s conce rned a re abso lu te ly

    convergen t .

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    Partifio numerorum.

    Henc ~

    3 . 2 3 3 )

    II I: On the expression of a number as a sum of primes. 29

    , =

    If n is even and r is odd the f irst factor vanishes in vir tue of the factor

    for which w-----2; if n is odd and r even the second factor vanishes similarly.

    Thus Sr = o when ever n a nd r are of opposite pari ty.

    I f n an d r are of l ike par i ty the factor correspo nding to w = 2 is in any

    case z; and

    ~ = 2 H ~ ( ~ -~ ) '/ = , ( p - ~ ) ~ - ( - ~ ) ~

    '

    as stated in the lemma.

    Prool o/ the / inal /orraulae.

    3. 3. Th eo rem B.

    S u p p o s e t h a t r > 3 . T h e n , i / n a n d r a r e o / u n l i k e p a r i ty ,

    3 . 3 I ) ~ , ~ n ) =

    a n~ - l ) .

    B u t i ~ n a n d r a r e o / l i k e p a r i t y t h e n

    2 o ~ ~ - + . ~ ) r p - ~ ) i ,

    ( 3 . 3 2 )

    r~ (n ) c ~ ( r - - I ) t n ~ - l f l

    l

    ( P I) r I ) r - - ( - - I ) r ]

    where p i s an odd pr ime d iv i sor o / n and

    3 = f i

    ( w - - i ) ~ /

    Er

    This follows immediately from Theorem A and Lemm a i9..1

    3 . 4 . L e m ma i3 . I / r ~ 3 and n and r are o] l i ke par i ty , then

    u~(n) > B n ~ - 1 ,

    / o r n > = n o r ) .

    i R e s u l t s e q u i v a l e n t t o t h e s e a r e s t a t e d i n e q u a t i o n s 5 . I I ) - - 5 . 2 2) o f o u r n o t e 2, b u t

    i n c o r r e c t l y , a f a c t o r

    lo g n ) r

    b e i n g o m i t t e d i n e a c h , o w i n g t o a m o m e n t a r y c o n f u s io n b e t w e e n , r n ) a n d N r n ) . T h e v r n )

    o f 2 i s t h e N r n ) o f t h i s m e m o i r .

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    3 0 G . H . H a r d y a n d J . E . L i t tl e w o o d .

    T h i s l e m m a i s r e q u i r e d f o r t h e p r o o f o f T h e o r e m C . I f r i s e v e n

    I

    ~

    ( ~ - ~ ) - ~

    I > ~

    I f r i s o d d

    ~ff- 8

    I n e i t h e r c a s e t h e c o n c l u s i o n f o l l o w s f r o m ( 3. 32 ).

    3 . 5 . T h e o r e m C . I ] r > 3 a n d n a n d r a r e o I l i k e p a r i t y , t h e n

    q , , n )

    N , ( n ) c ~

    ( l o g n ) ~ "3 . 5 I )

    W e o b s e r v e f i r s t th a t

    ~ i + ~'2 + ' - + % . : n

    a n d

    ( 3 . 5 I I )

    z ~ ~ I o , a , a r . . . > o ) ,

    N,. n) ----o

    ~ - 3

    P . STXCKEL, Die Dars t e l lung de r go raden Z ah len a l s Sum men yon zwe i P r im zah len , 8

    Augus t x916 ; Die L f ickenzah len r - t e r S tu fe und d ie Dars t e l lung de r ge raden Zah len a l s Sum-

    men und D i f fe renzen unge rade r P r im zah len , I . Te i l 27 Deze mbe r I917 , I I . T e i l I9 Janu a r x9x8,

    III . Tel l 19 Jol i 1918.

    T h r o u g h o u t 4 . 2 A i s the same cons tan t .

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    Part i t io numerorum. III : On the express ion of a number as a sum of pr imes.

    W r i t e

    (4. 24)

    ~ 2 n ) = A n l I ~ - I )

    - -2- n even) , ~2 n)=o (n odd).

    T h e n , b y ( 4 - 2 1) a n d T h e o r e m C , n o w v a l i d i n v i r t u e o f (4 . 2 ) ,

    ( 4. 2 5 ) v ~ ( n ) - 2 ] l o g w l o g ~ c,~ ~ ( n ) ,

    i t b e i n g u n d e r s t o o d t h a t , w h e n n is o d d , t h i s fo r m u l a m e a n s

    ~ , ~ n ) = o n ) .

    F u r t h e r l e t

    / s ) = - - ~ . . . . ~

    ~

    t h e s e se r ie s b e in g a b s o lu t e l y c o n v e r g e n t i f ~ ( s ) > 2 , ~ ( u ) > ][. T h e n

    (4. 26)

    s a y .

    T h e n

    H e n c e

    (4 . z7)

    / = )= A Z n - = I I ~ - ] [

    A ~ 2 - ~ = p - " " p ' -~ ' =

    a > 0

    ( ~ - - I ) ( r I ) . . .

    ' ( p - - 2 ) ( r

    2 - " A ] [ ( w ~ ] [ W - ~ ) - - 2 - " A ~ (u )

    S u p p o s o n o w t h a t u - - * i , a n d l e t

    ~0 - u W ~a

    *0 =3 ][

    ,Cff--u

    [ ~ - ] [ ) ] [

    A A ~ A

    3 5

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    36

    G. H. Hardy and J. E. Littlewood.

    On the other hand, when x~i,

    and so

    4 . 2 s )

    I

    ~(~) + ~(z) +--- + ~( n) ~ -n ~.

    2

    It is an elem entary ded uction ~ that

    ~'~ 9 8 - - 2

    when s~ 2 ; and hence* tha t (under the hyp othes es (4. 21) and (4. 22))

    4 . 2 9 )

    l s ) ~ I

    8 ~ 2

    Comparing (4. z7) and (4. 29), we obtain the res ult of the theorem.

    4. 3. The fact that both Sylvester s and Br un s formulae contain an

    erroneous con stan t factor, an d th at this factor is in each case a simple functi on

    of the number e e, is not so remarkable as it may seem.

    In the first place we observe t hat any formula in the theory of primes,

    deduced /re in coasiderat ions o/ probabil i ty ,

    is likely to be erroneous in just this

    way. Consider, for example, the problem

    wha t is the chance that a large num ber

    n should be pr ime?

    We know that the ans wer is that the chance is approxim-

    i

    ate ly log n

    Now the chance th at n should not be divisible by any prime less than a

    / i z e d number x is asymptotically equivalent to

    W e h e r e u s e T h e o r e m 8 o f o u r p a p e r ' T a u b e r i a n t h e o r e m s c o n c e r n i n g p o w e r s e r i e s a n d

    D i r i c h l e t ' s s e r i e s w h o s e c o e f f i c i e n t s a r e p o s i t i v e ' ,

    Prec. London Math. See.

    s e r . 2 , v o l . I 3 , p p .

    i ; 4 - - I 9 2 . T h i s i s t h e q u i c k e s t p r o o f , b u t b y n o m e a n s t h e m o s t e l e m e n t a r y . T h e f o r m u l a

    ( 4. 2 8) i s e q u i v a l e n t t o t h e f o r m u l a

    n a

    2 ( l o g n ) ~

    1

    u s e d b y L a n d a u i n h i s n o t e q u o t e d o n p . 3 3.

    2 F o r g e n e r a l t h e o r e m s i n c l u d i n g t h o s e u s e d h e r e a s v e r y sp e c i a l c a s es , s e e K .

    KNow

    D i v e r g e n z c h a r a c t e r o g e w i s s o r D i r i c h l e t ' s c h e r R e i h e n ' ,

    Acta Mathematica

    v o l . 3 4 , t 9 o 9 , p p . I 6 5 - -

    z o ~ ( e . g . S a t z I I I , p . I 7 6 ) .

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    Par t i t io numero rum.

    a n d i t w o u l d

    e q u i v a l e n t t o

    B u t ~

    l l I : On the express ion o f a num ber a s a sum o f p r imes . 37

    b e n a t u r a l t o i n f er ~ t h a t t h e c h a n c e r e q u i r e d i s a s y m p t o t i c a l l y

    va> Vn log n

    a n d o u r i n f e r e n c e i s i n c o r r e c t , t o t h e e x t e n t o f a f a c t o r 2 e - C .

    I t i s t r u e t h a t B r u n ' s a r g u m e n t i s n o t s t a t e d in t e r m s of p r o b a b i l i t i e s a,

    b u t i t i n v o l v e s a h e u r i s t i c p a s s a g e to t h e l i m i t o f e x a c t l y t h e s a m e c h a r a c t e r

    a s t h a t i n t h e a r g u m e n t w e h a v e j u s t q u o t e d . B r u n f i n d s f i r st ( b y a n i n g e n i o u s

    u s e o f t h e ' s ie v e o f E r a t o s t h e n e s ' ) a n a s y m p t o t i c f o r m u l a fo r t h e n u m b e r o f

    r e p r e s e n t a t i o n s o f n a s t h e s u m o f t w o n u m b e r s ,

    neither div isible by any / ixed

    number o / pr imes

    T h i s f o r m u l a i s c o r r e c t a n d t h e p r o o f v a l i d . S o i s t h e f i r s t

    s t a g e i n th e a r g u m e n t a b o v e ; i~ r e s t s o n a n e n u m e r a t i o n o f c a s e s, a n d a l l r e fe -

    r e n c e t o ' p r o b a b i l i t y ' ~ i s e a s i l y e l i m i n a t e d . I t i s i n t h e p a s s a g e t o t h e l i m i t

    t h a t e r r o r i s i n t r o d u c e d , a n d t il e n a t u r e o f t il e e r r o r i s t h e s a m e in o n e c a s e

    a s i n t h e o t h e r .

    4 . 4 . Stu Ar t a n d W I LS O N h a v e t e s t e d C o n j e c t u r e A e x t e n s i v e l y b y c o m p a r i s o n

    w i t h t h e e m p i r i c a l d a t a c o l l e c t e d b y C A ~T O R, A U B R Y , H A V S SN E R , a n d R 1 P E a T .

    W e r e p r i n t t h e i r t a b l e o f r e s u l ts ; b u t so m e p r e l i m i n a r y r e m a r k s a r e r e q u i r e d .

    I n t h e f i r s t p l a c e i t i s e s s e n ti a ] , i n a n u m e r i c a l t e s t , t o w o r k w i t h a f o r m u l a

    N ~ ( r t ) , s u c h a s ( 4. x ~ ), a n d ' n o t w i t h o n e f o r v ~ ( n ) , s u c h a s ( 4. 2 5 ). I n o u r

    a n a l y s i s , o n t h e o t h e r h a n d , i t i s r ~ ( n) w h i c h p r e s e n t s i t s e lf f i r s t, a n d t h e f o r m u l a

    f o r N 2 ( n ) i s s e c o n d a r y . I n o r d e r t o d e r i v e t h e a s y m p t o t i c f o r m u l a f o r N ~ ( n ) ,

    w e W r i t e

    ~ q ( n ) = ~ l o g ~ l o g z ~ ' c ~ ( l o g n )~ N 2 ( n ) .

    ~- F ~ff = n

    T h e f a c t o r ( l og n ) ~ i s c e r t a i n l y i n e r r o r t o a n o r d e r l o g n , a n d i t is m o r e n a t u r a l 5

    t o r e p l a c e v 2 ( n ) b y

    ( ( l o g n ) ~ - - 2 l o g n + - - - ) N 2 ( n ) .

    On e mi g h t we l l r e pl a c e ~ < l / n b y ~ < n , i n wh i c h c a s e we s ho u l d o b t a i n a p r o b a b i li t y

    ha l f a s l a rge. Th i s r em ark i s in i t s e l f enough to show the unsa t i s f ac to ry cha rac te r o f the a rgumen t .

    2 Lan dau, p . z i8 .

    a W h e t h e r S y l v e s t e r' s a r g u me n t wa s o r wa s n o t we h a v e n o d i r e c t me a n s of j u d g in g .

    Probability i s no t a no t ion oE pure m a them at ic s , bu t o f ph i lo so phy o r phys ic s .

    6 Com pare S hah and Wi l son , l . c ., p . 238 . The sam e conclus ion may be a r r ived a t in

    other ways.

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    P a r t i t i o n u m e r o r u m I I I: O n t h e e x p r e s s i o n o f a n u m b e r a s a s u m o f p r i m e s 3 9

    Table

    n q , ( n ) p ( n )

    Q,(n) : p(n)

    3 0 = 2 . 3 . 5

    32- -2 s

    3 4 = 2 . 1 7

    36=2~ .3 ~

    210----2 .3 .5 .7

    2 1 4 = 2 . 1 0 7

    2 1 6 = 2 a . 3 8

    256=2 s

    2,048 = 2

    2,25 o= 2.32 .5 a

    2,304 -- 2 s 9 3~

    2,306 = 2. 1153

    2 ,310= 2"3 '5 "7 ' 1 I

    3,888

    = 2 4 3 g

    3,898 = 2. I949

    3,990 = 2. 3 - 5. 7 9 19

    4,096 = 21~

    4,996 = 27 9 1249

    4,998 = 2 -3 9 7 ~. 17

    5,000 = z 7 "54

    8 , 1 9 o = 2 , f f . 5 . 7 . I 3

    8,192 = 2Is

    8 , I 9 4 ~ - ~ 2 . I 7 . 2 4 I

    lO,OO8 = 2-'. 37. 139

    IO, OIO = 2 . 5" 7 9 I I . 13

    10,014 .-- 2 .3 9 1669

    30,03 0 ----- 2 . 3. 5 9 7 9 I i . 13

    36,96o- - 2z.3 9 5 9 7. II

    39,27o---- 2. 3. 5- 7 9 11. 17

    41,58o = 2 ~.3 ~. 5. 7. It

    i

    6+ 4 =1o

    4+ 7 = i i

    7 + 6 = 1 3

    8 + 8 - - 1 6

    42 + o =4 2

    17 + o = 17

    28 + o =2 8

    i6 + 3 = I9

    5 ~ + 17= 67

    1 7 4 + 2 6 = 2 o o

    I34 + 8 --.--I4 z

    67 + 20 =87

    228 + 1 6=2 44

    186 + 24 =21 o

    99 + 6 =1 o 5

    328+ 20-----348

    IO4 + 5 =1 o9

    124 + I 6= I4O

    228 + 20= 308

    i5o + z6 =

    578 + 2 6=

    15o + 3 2=

    192 + io =

    388 + 3 ~

    384 + 3 6=

    408+8 =

    1,8oo + 54 =

    1,956 + 38=

    2,152 + 36=

    2 , 1 4 o + 4 4 =

    50,026 =2. 25o I3 702 8 =

    5o,1 44=2 ~.15 67 607 + 32=

    1 7 o , i 6 6 = 2 . 3 . 7 9 . 3 5 9

    1 7 o , 1 7 o = 2 . 5 . 7 . 1 1 . 1 3 . 1 7

    1 7 o , 1 7 2 = 2 ~ . 3 - . 2 9 . 1 6 3

    3,734 + 46=

    3,784 + 8 =

    3,732 + 48=

    8

    9

    I7

    49

    i6

    32

    63

    I79

    136

    69

    244

    I97

    99

    342

    lO2

    I I 9

    305

    I76 157

    q - - - - -

    604 597

    182 I71

    202 .__219

    418 1 396

    42o 384

    416 396

    1854 I I795

    1994 I937

    2188

    2213

    2184 2125

    71o 692

    706 694

    378o

    376z

    379 z 3841

    378o 3866

    O. 45

    I. 38

    1 . 4 4

    0. 94

    o. 85

    i.

    07

    o. 88

    I. 10

    I . 06

    I . . . .

    I ,

    04

    I .26

    1 .00

    I . 0 6

    1 . 0 6

    I . 0 2

    I . O 6

    I. 18

    I . O I

    I 1 2

    I . 0 I

    I .

    06

    o. 92

    t . o 6

    i . 09

    x . 0 5

    I .. o 3

    i . 03

    o. 99

    i. 03

    1 . 0 3

    I . 0 2

    1 .00

    O. 99

    O. 98

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    40

    G. H. Hardy and J. E. Littlewood.

    5 O t h e r p r o b l e m s

    5. I. This last section is fra nkly conjectural, and is not to be judged by

    the same standards as w167--3.

    The problems to which we have applied our method may be divided roughly

    into three classes. The typical problem of the first class is Waring s Problem.

    Our solution of this problem is not yet as conclusive as we should desire, and

    we have not exhausted the possibilities of our method, even when allowance is

    made for still unpublished work; we cannot at present prove, for example, that

    every large numbe r is the sum of 7 cubes or x6 biqua drate s. But our proofs,

    so far as they go, are complete.

    The ty pical problem of the second class is th at considered in w167--3. The

    arguments by which we prove our results are rigorous, but the results depend

    upon the unproved hypothesis R.

    There is a th ird class of problems, of which Goldbaeh s Prob lem is typical.

    Here we are unable (with or without Hypothesis R) to offer anyth ing approaching

    to a rigorous proof. Wha t our meth od yields is a [ormula, and one which seems

    to stan d the test of comparison with the facts. In this concluding section we

    propose to stat e a number of fur ther formulae of the same kind. Our apology

    for doing so must be (I) tha t no similar formulae have been suggested before,

    and that the process by which t hey are deduced has at any rate a certain

    algebraical interest, and (2) tha t it seems to us very desirable that (in defa ult

    of proof) the formulae should be checked, and th at we hope th at some of the

    many mathematicians interested in the eomputative side of the theo ry of numbers

    may find them worthy of their attention.

    Conjugate problems: the problem o/ prime-pairs .

    5- 2. The problems to which our method is applicable group themselves in

    pairs in an interesting manner which will be most easily understood by an example.

    In Goldbach s Problem we have to st udy the integral

    where

    I dx

    1

    4 i9

    l(x) = log ~ xe , x ~- Re i~' = e ,

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    Parti t io numero rum . III : On the expression of a number as a sum of primes. 41

    or

    (5. 21)

    2~

    0

    T h e f o r m a l t r a n s f o r m a t i o n s o f t h i s i n t e g r a l t o w h i c h w e a r e le d m a y b e s t a t e d

    s h o r t l y a s f o l l o w s . W e d i v i d e u p t h e r a n g e o f i n t e g r a t i o n i n t o a l a r g e n u m -

    b e r o f p ie c e s b y m e a n s o f t h e F a r e y a r e s

    ~p ,q, ~p

    v a r y i n g o v e r t h e i n t e r v a l

    ( 2 P : ~ - - O ~ , q , 2 P T ~ + O p ,ql

    w h e n x v a r i es o v e r ~ , q . W e th e n r e p l a c e

    ] ( x ) b y

    t h e

    ~ ~

    q

    a p p r o p r i a t e a p p r o x i m a t i o n

    9( q) log (-e-q-(xP-)) 9 ( q ) I i(~v 2 P ,( )

    q t

    (p 2 p ~ b y u , a n d th e i n t e g r a l

    q

    (5. 22)

    b y

    e q { _ n p ) ) p q e~- ~iu

    . . . . . ~ d u

    5- 23)

    oo

    [ ~ e l - - i w

    n eq ( - - n P ) J i i - - - - i w ) d w = 2 ~ n e q ( - - p ) .

    - -o o

    W e a r e t h u s l e d t o t h e s i n g u l a r s e r i e s S z .

    N o w s u p p o s e t h a t , i n s t e a d o f t h e i n t e g r a l (5 . 2 1), w e c o n s i d e r t h e i n t e g r a l

    (5- 24)

    2~

    0

    w h e r e n o w k i s a

    / i x e d

    p o s i t i v e i n te g e r . I n s t e a d o f (5 . 2 2 ), w e h a v e n o w

    Op q

    e q ( k p ) f e k i U

    oo

    d u c~ eq (kp ) f -i -d--u

    ~ ~

    u~

    A c t a m a t h e m a t / c a .

    44. Impvim~ le 16 f~ vr ie r 1922. ~

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    42 G. H. Hardy and J. E. Litttewood.

    We are thus led to suppose that

    (5. 25) J ( R ) ~ 2 n ~ ~ - -~ - ) e q ( k p )

    1

    when R= e , , n--.oo.

    The series here (which we call for th e mom ent S 2) is the singu lar series S~

    with --k in the place of n. On the other hand

    2 ~

    J( R) =~ lo g~ . ~_.~ og ~R ~ e -- . e ~ i ~ d q J = ~

    ]

    o

    where

    a ~ ~ l o g ~

    log (~

    + k

    i ] b o th ~ a n d ~ + k a r e pr/me, and a~ ~ o otherwise. Hence we obtai n

    Here R ~e n but the result is easily extended to the case of continuous ap-

    proach to the limit z, and we deduce ~

    (5- 2 6) 2 a ~ c ~ n S ' 2 .

    ~ n

    And from this it would be an easy deduction that the number of prime pairs

    differing by k, and less than a large number n, is asymptotic ally equivalent to

    n

    ( q o - g ; f . 9

    We are thus led to the following

    Conjecture B.

    T h e r e a r e i n ] i n i t e l y m a n y p r i m e p a i r s

    /o r e v e r y e v e n k . I ] P k (n ) iS th e n u m b e r o / ? a i r s l e s s th a n n , t h e n

    n ~ I

    wh e r e C~ i s th e c o n s ta n t o ] w 4 a n d . p i s a n o d d p r im e d iv i s o r e l k .

    i W e a p p e a l a g a i n h e r e t o t h e T a u b e r i a n t h e o r e m r e f e r r e d t o a t t h e e n d o f 4 - z ( f, n . i ).

    T h i s t i m e , o f c o u r s e , t h e r e i s n o q u e s t i o n o f a n a l t e r n a t i v e a r g u m e n t .

    ~ N o te t h a t S 2 = o i f k i s o d d , a s i t s h o u l d b e .

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    Par t i t io numerorum . I I I : On the express ion of a num ber as a sum of pr imes . 43

    I t w il l b e o b s e r v e d t h a t t h e a n a l y s i s c o n n e c t e d w i t h C o n j e c t u r e s A a n d B ,

    w h i c h d e a l r e s p e c t i v e l y w i th t h e e q u a t i o n s

    n ~ + ~ r , ~ r = ~ o ~ + / c ,

    i s s u b s t a n t i a l l y t h e s a m e . I t i s p a i r s o f p r o b l e m s c o n n e c t e d in t h i s m a n n e r t h a t

    w e c a l l conjugate p r o b l e m s .

    Numerical verilications.

    5 . 3 1. F o r k ~ 2 , 4 , 6 w e o b t a i n

    2 C 2 n

    (5 . 311)

    P z n ) c ~ - ~ - g ~ ,

    2 C ~ n

    ( 5 , 3 1 2 ) P ( )

    4 C~n

    5 313) P0 n)

    Th us there should be approxim ately equal numbers o/ prim e-pairs di //er ing by 2 and

    by 4, but about twice as many di/[ering by 6 . T h e a c t u a l n u m b e r s of p a i r s,

    b e l o w t h e l i m i t s

    z o o , 5 0 0 , I O O O , 2 0 0 0 , 3 0 0 0 , 4 0 0 0 , 5 o 0 0

    a r e

    I l O 3

    9 2 4 _ _ 9 5 _ 6 I 8 1 1 2 5

    9 6 , ~ 7 ~ 63 86

    2 I

    . . . . I . . . . . . . . . . . . . .

    ,25 x68 I 2ox I 241

    T h e c o r r e s p o n d e n c e i s a s a c c u r a t e a s c o u l d b e d e s ir e d .

    5 . 3 2. T h e f i r s t f o r m u l a ( 5 . 3 1 1) h a s b e e n v e r i f i e d m u c h m o r e s y s t e m a t i c -

    a l l y . A l i t t l e c a u t i o n h a s t o b e e x e r c i s e d i n u n d e r t a k i n g s u c h a v e r i f i c a t i o n .

    T h e f o r m u l a ( 5. 2 6) i s e q u i v a l e n t , w h e n k = 2 , to

    (5 . 321) ~ l l m ) l l m + 2 ) c~ 2C~n;

    rn n

    a n d , w h e n w e p a s s f r o m t h i s f o r m u l a t o o n e f o r t h e n u m b e r o f p r i m e - p a i rs , th e

    f o r m u l a w h i c h a r i s e s m o s t n a t u r a l l y i s n o t ( 5. 3 i l ) b u t 1

    1 This formu la fo l lows f rom (5. 321) in exac t ly the same way tha t

    ~ ( x) o o L i x

    follows from

    A(m) c,~ x.

    m

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    44

    G . H . H a r d y a n d J . E . L i t t l e w o o d .

    " d z .

    ( 5 . 3 2 2 ) P 2 ( n ) c ,, o 2 0 2 (l--og~ ) q '

    indeed it is not unreasonable to expect this appro xima tion to be a really good

    one, and much bette r than the formu lae of 4. 4. The formula 5. 322) is nat-

    urally equiv alent to 5. 3Ix). But

    11

    = + +

    and the second f actor on the right hand side is for such values of n as we

    have to consider) far from negligible. It is for this reason tha t Brun, whe n he

    attempt ed to deduce a value of the constant in 5. 3z I) fr om the statistical

    results, wa s led to a value seriously in error.

    We the refore take the formu la 5. 322) as our basis for comparison, choosing

    the lower limit to be ~. For our statistics as to the actual number of prime-

    pairs we are inde bted to a) a c oun t up to too,coo made by Gr.AIsHm~ in T878 z

    and b) a muc h more extensive count made for us recen tly by Mrs. G. A.

    STR~ATF~;ILD. The results obtained by Mrs. Streatfeild are as follows.

    I OO0 0 0

    2 0 0 0 0 0

    300000

    400000

    5 0 0 0 0 0

    600300

    700o00

    800000

    900o00

    IOOOOOO

    9 1

    z C ~ d x

    1224 1246.3

    2159 2179.5

    2992 3035.4

    38o i 3846 . I

    4562 4625.6

    5328 I 538~ .5

    6058 t 6118.7

    6763 684o.2

    7469 7548.6

    8164 8z45.6

    R a t i o

    i

    o i 8

    1 ,009

    i .o15

    I .012

    o14

    I . 010

    I 010

    I .01I

    I . OlI

    I . 010

    i T h e . s e r i e s i s o f c o u r s e d i v e r g e n t , ~ i n d m u s t b e .r e g a r d e d a s c l o s e d a f t e r a f i n i t e n u m b e r

    o f t e rm s , w i t h a n e r r o r t e r m o f l o w e r o rd e r t h a n t h e l a s t t e r , u r e t a i n e d .

    J . W . L . G s Am ~ v m, ' A n e n u m e r a t i o n o f p r i m e - p a i r s ' , Messenger of Mathematics, vol . 8

    ( 13 78 ), p p . 2 8 - -3 3 . G l a i s h e r c o u n t s 0 , 3 ) a s a p a i r , s o t h a t h i s f i g u r e e x c e e d s o u r s b y I .

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    Par t i t io numero rnm. I I I : On the express ion o f a num ber a s a sum o f p r imes . 45

    5 - 3 3 . S i m i l a r r e a s o n i n g l e a d s u s t o t h e f o l l o w i n g m o r e g e n e r a l r e s u l ts .

    C o n j e c t u r e ( ] . I [ a , b a r e f ix e d p o s i t iv e i n t e g er s a n d ( a , b ) = i , a n d N ( n ) i s

    t h e n u m b e r o / r e p r e s e n t a t i o n s o / n i n t h e / o r m

    t h e n

    n ~ a ~ + b y e ,

    u n l e s s ( n , a ) = i , ( n , b ) - - i , a n d o n e a n d o n l y o n e e l n , a , b i s e v e n ) B u t i/

    t h e s e c o n d i t i o n s a r e s a t i s / i e d t h e n

    2 0 z n ( p - - I )

    N n ) a b l o g - l l

    wh e r e C7. i s t h e c o n s t a n t o / w 4 , a n d t h e p r o d u c t e x t e n d s o v e r a l l o d d p r i m e s p wh i c h

    d i v i d e n , a , o r b .

    0 o n j e c t u r e

    D . I ] ( a , b ) = i a n d P ( n ) i s t h e n u m b e r o / p a i r s o / s o l u t io n s o /

    a v~ - - b v~ ~ ]c

    s u c h t h a t ~ < n , t h e n

    u n l e s s ( k , a ) == I , ( k , b ) - - I , a n d j u s t o n e o / k , a , b i s e v en . B u t t / t h e s e c o n d i t io n s

    a r e s a t i s / i e d t h e n

    p ( n ) c 2C .~ n

    ( ~ )

    9 l I '

    wh e r e p i s a n o d d p r i me / a c t o r o / k , a , o r b .

    I t s h o u l d b e cl e a r t h a t t h e t h e o r e m s p r o v e d i n w167- 3 m u s t b e c a p a b l e o f

    a s i m i l a r g e n e r a l i s a ti o n . T h u s w e m i g h t i n v e s t i g a t e th e n u m b e r o f r e p r e s e n t a -

    t i o n s o f n i n t h e f o r m

    n - - a ~ + b v o ~ + c ~ ;

    a n d h e r e p r o o f w o u l d b e p o ss i b le , t h o u g h o n l y w i t h t h e a s s u m p t i o n o f h y p o -

    t h e s i s R . W e h a v e n o t p e r f o r m e d t h e a c t u a l c a l c u la t i o n s .

    1 Th i s i s t r iv i a l . I f n and a have a common fac to r , i t d iv ides bw , and mu s t the re fo re

    be ~r wh ich i s thus r e s t r i c t ed to a f in it e num ber o f va lues. I f n , a , b are a l l odd, w or w

    mus t be z .

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    Par t i t io numerorum. I I I : On the express ion o f a num ber a s a sum o f p r imes . 47

    a n d w e a r e l e d t o th e f o r m u l a

    (5 . 4 ix ) J ( R ) c ~ I V2 z n S ,

    4

    w h e r e S i s t h e s i n g u l a r s e r i e s

    (5 . 412) S ~ ~ ~ (q ) S p, ( - -p ) .

    R e p e a t i n g t h e a r g u m e n t s o f w 5 . 2 , w e c o n c l u d e t h a t the number P (n ) o / p r imes

    o/ the ]orm me + I and less than n is g iven asymp tot ical ly by

    ~

    (5 . 413) P( n) c,O~og n S .

    5. 4 2 .

    W r i t i n g

    T h e s i n g u l a r s e r ie s (5- 4 12 ) m a y b e s u m m e d b y t h e m e t h o d o f w 3 - 2 .

    8 = 2 A q = I + A ~ + A ~ + . , - , .

    t h e r e i s n o d i f f i c u lt y i n p r o v i n g t h a t A q q , = A q A q , i f ( q , q ' ) = x . H e n c e w e

    w r i t e 1

    w h e r e

    S = H Z ~ ,

    ; g ~ r= I + A ~ + A ~ r ~+ . . . . I + A . .

    I f i ' d = 2 , A ~ = o , Z . = I . I f ' ~ > 2 , ~

    a n d

    1 1

    1 Eve n this is a forma l process , for (5. 412) is not ab solutely conv ergen t .

    2 See D[RIOHLm.T-D~DEKINDVorlesungen i~ber Zahlenlheorie , ed. 4 I 8 9 4 0 , PP- 293 et seq.

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    4 8 G . H . H a r d y a n d J . E . L i t tl e w o o d .

    T h u s f i n a l l y w e a r e l ed t o

    C o n j e c t u r e E . There are in/initeIy many primes o] the ]orm m -t-x.

    number P(n) o/ such primes less than n is given asymptotically by

    ~

    P(n) c,z C ~g n'

    where

    = f l I

    ~Y=3

    The

    5 . 4 3 . G e n e r a l i s i n g t h e a n a l y s i s o f w 167. 4 I , 5 . 4 2 , w e a r r i v e a t

    C o n j e c t u r e F . S~ppose thai a, b, c are integers and a is positive; that (a, b, c) = i ;

    that a + b and c are not both even; and that D =b ~ - 4ac is not a square. Then

    there are in/ ini tely many pr imes o/ the ]orm amJ + bm + c. The number P( n) o/

    such primes less than n is given asymptotically by

    p( n) c ~C Vn (~-~-~I)

    where p i s a common odd prime div isor o/ a and b, e is i i] a + b is odd and 2

    i / a + b is even, and

    ( 5. 4 3 2 x ) C ~ H I - - f f . _ I

    I t i s i n s t r u c t i v e h e r e t o o b s e r v e t h e g e n es i s o f t h e e x c e p t i o n a l c a s e s. I f

    ( a , b , c ) = d > I , t h e r e c a n o b v i o u s l y b e a t m o s t o n e p r i m e o f t h e f o r m r e q u i r e d .

    I n t h i s c a s e Z ~ v a n i s h e s f o r e v e r y m fo r w h i c h ~ L d . I f a + b a n d c a r e b o t h

    e v e n , am ~+bm+ c i s a l w a y s e v e n : i n t h i s c a s e Z2 v a n i s h e s . I f D =k ' , t h e n

    a n d

    4a(am ~ +bin

    + c ) = ( 2 a m + b ) ~ - k ~ ,

    4a ~= (2am + b)'--

    k ~

    i n v o l v e s 2am + b:t: k j4a, w h i c h c a n b e s a t i s f ie d b y a t m o s t a f in i t e n u m b e r o f

    v a l u e s o f m . I n t h i s c a s e n o f a c t o r Z ~ v a n i s h e s , b u t t h e p r o d u c t (5 . 4 3 21 )

    d i v e r g e s t o z e r o .

    5 . 4 4. T h e c o n j u g a t e p r o b l e m i s t h a t o f t h e e x p r e s s i o n o f a n u m b e r n

    i n t h e f o r m

    ( 5 . 4 4 ) n = am ~ + bm + ~.

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    P a r t i t i o n u m e r o r u m , l l I : O n t h e e x p r e s s io n o f a n u m b e r a s a s u m o f p r i m e s . 4 9

    H e r e w e a r e l e d ~o

    C o n j e c t u r e G . S u p l : o s e t h a t a a n d b a r e i n te ~ e r s, a n d a > o , a n d l e t N ( n )

    b e t h e n u m b e r o / r e p r e s e n t a t i o n s o / n i n t h e ] o r m a m ~ + b m + ~ r . T h e n i ~ n , a , b

    h a v e a c o m m o n / a c l or , o r i / n a n d a + b a r e bo t h e v e n , or i / b ~ + 4 a n i s a s q u a r e , t h e n

    ( 5 - 4 4 z )

    I n a l l o t h e r c a s e s

    ( 5 - 4 4 3 )

    N ( n ) r

    N .

    f f f . ~ 3 , ~ r l 1 a

    w h e r e ~ i s a c o m m o n o d d p r i m e d i v i s o r o ] a a n d b , a n d ~ i s x i / a + b i s o d d a n d

    2 i / a + b i s e v e n .

    T h e f o l l o w i n g a r e p a r t i c u l a r l y i n t e r e s t i n g s p e c ia l c a s es o f t h i s p r o p o s i t i o n .

    C o n j e c t u r e H . E v e r y l a rg e n u m b e r n i s e i t h e r a s q u a r e o r t h e s u m o [ a p r i m e

    a n d a sq u a re . T h e n u m b e r N ( n ) e l r e p r e se n t a ti o n s i s g i v e n a s y m p t o t i c a l l y b y

    ( 5 . 4 4 4 ) N ( n ) ~ - - I . . . . . . . .

    l o g n ~ - -

    ~ r - - 3

    T h e r e d o e s n o t s e e m t o b e a n y t h i n g i n m a t h e m a t i c a l l i t e r a tu r e c o r r e s p o n d i n g

    t o t h i s c o n j e c t u r e : p r o b a b l y b e c a u s e , t h e i d e a t h a t

    e v e r y

    n u m b e r i s a sq u a r e ,

    o r t h e s u m o f a p r i m e a n d a s q u a r e , i s r e f u t e d ( e v e n i f I i s c o u n t e d a s a p r i m e )

    b y s u c h i m m e d i a t e e x a m p l e s a s 3 4 a n d 5 8 . B u t t h e p r o b l e m o f t h e r e p r e s e n t a -

    t i o n o f a n

    o d d

    n u m b e r i n t h e f o r m t ~+ 2 m ~ h a s r e c e i v e d s o m e a t t e n t i o n ; a n d

    i t h a s b e e n v e r i f i ed t h a t t h e o n l y o d d n u m b e r s l e ss th a n 9 o o o , a n d n o t o f t h e

    f o r m d e s i r e d , a r e 5 777 a n d 5 993 ?

    C o n j e c t u r e I .

    E v e r y l ar g e o d d n u m b e r n i s t he s u m o / a p r i m e a n d th e d o u bl e

    o / a s~ u a re : T h e n u m b e r N ( n ) o / r e p r es e n t at i o n s i s g i v e n a s y m p t o t i c a l l y b y

    ( 5 . 4 4 5 ) N (n ) c , z lo-g -n I ~ - - i

    v a ~ 3

    1 By S~ Ea ~ ; a n~ l h i s pup i l s in x856 . Se e D ic k son ' s

    Historg

    ( referred to on p. 32) p. 424.

    T h e t a b l e s c o n s t r u c t e d b y S t e r n w e r e p r e s e r v e d i n t h e l i b r a r y o f H u r w i t z , a n d h a v e b e e n v e r y

    k i n d l y p l a c e d a t o u r d i s p o s a l b y M r . G . P 6 1 y a . T h e s e m a n u s c ri l ~t s a l so c o n t a i n a t a b l e o f

    d e c o m p o s i t i o n s o f p r i m e s q - - 4 m + 3 i n t o s u m s q - ----p+ 2 p ~, w h e r e p a d d p r a r e p r i m e s o f t h e

    f o r m 4 m + i , e x t e n d i n g a s f a r a s q ---- 2 o 98 3 , T h e c o n j e c t u r e t h a t s u c h a d e c o m p o s i t i o n i s a l w a y s

    p o s s i b l e {I b e in g c o u n t e d a s a p r i m e ) w a s m a d e b y L a g r a n g o i n

    I775

    (see Di cks on , t. c., p. 4z4).

    Aeta mathen~allea 44 Imprlm~

    le 17 f~v rle r 1922 7

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    50

    G. II . Hardy and J. E. Lit t lewood.

    5 . 4 5 W e m a y e q u a l l y w o r k o u t t h e n u m b e r o f r e p r e s e n t a t i o n s o f n a s

    t h e s u m o f a p r i m e a n d a n y n u m b e r o f s q u a r e s . T h u s , f o r e x a m p l e , w e f i n d

    C o n j e c t u r e J . T he numbers o / r epresen ta t ions o / n in the [orms

    ~ ~ m ~ m ~ m ~ ,

    = ~ + m ~ + m ~ , n - ~ ~ + ~+ m,~

    are g iven asympto t ica l ly by the /ormulae

    w h e r e

    5 - 4 5 I I )

    a n d

    5. 452)

    w here

    5 , 452~)

    C

    ~

    + , ~ ~ . ~ ,

    V~= 3

    Here p i s an odd pr ime d iv i sor o / n , and represen ta t ions which d i / / e r on ly in the

    s ign or order O/ the num bers m~, m 2, . . . are counted a s dis t inct .

    T h e l a s t p a i r o f f o r m u l a e s h o u l d b e c a p a b l e o f r ig o r o u s p r o o f .

    Problems wi th cubes .

    5 . 5 - T h e c o r r e s p o n d i n g p r o b l e m s w i t h c u b e s h a v e , s o f a r a s w e a r e a w a r e ,

    n e v e r b e e n f o r m u l a t e d . T h e p r o b l e m w h i c h s u g g e s t s i ts e l f f i r s t i s t h a t o f t h e

    e x i s t e n c e o f a n i n f i n i t y o f p r i m e s o f t h e f o r m m S + 2 o r , m o r e g e n e r a l l y , m S + k,

    w h e r e k i s a n y n u m b e r o t h e r t h a n a p o s i t i v e o r n e g a t i v e ) c u b e ,

    H e r o a g a i n o u r m e t h o d m a y b e u s ed , b u t t h e a l g e b ra i c a l t r a n s f o r m a t i o n s ,

    d e p e n d i n g , a s o b v i o u s l y t h e y m u s t , o n t h e t h e o r y o f c u b ic r e s i d u a c it y , a r e

    n a t u r a l l y a l i t tl e m o r e c o m p l e x . A s t h e r e i s i n a n y c a s e n o q u e s t i o n o f p r o o f ,

    w e c o n t e n t o u r s e l v e s w i t h s t a t i n g a f e w o f t h e r e s u l t s w h i c h a r e s u g g e s t e d .

    C o n j e c t u r e K .

    I ] ~ is an y ] ixed num ber other than a posi t ive or negative)

    cube , then there are in] in i t e ly m an y pr im es o] the ]orm mS+ k . T he nu mber P n)

    o / such pr imes l e s s than n i s g iven asympto t ica l ly by

    5. 51) P n ) c , ~ l ~ n ~ I

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    Parti t io num eroru m. III: On the expression of a number as a sum of primes. 51

    where

    ~ I ( r ood . 3 ) , '

    and - - k )~ r i s equa l t o i or t o - - I accord i ng as - - k i s or i s no t a cub ic r e s idue o / ~ r .

    C o n j e c t u r e L . E v e r y l a rg e n u m b e r n i s e it h e r a c u b e o r t h e s u m o / a p r i m e

    a n d a p o s i ti v e ) cu b e . T h e n u m b e r N n ) o / r e p r e s e n t a t i o n s i s g i v e n a s y m p t o t i c a l ly b y

    N n ) l o g n ~ ~ - - I

    ~ n ) ~ ,

    t h e r a n g e o ] v a l u e s o / ~ b e i n g d e fi n e d a s i n K .

    C o n j e c t u r e M . I [ k i s any / i xed number o t her t han zero , t here are i n / i n i t e l y

    m a n y p r i m e s o ] t h e [o r m l a + m S + k , w h e r e l a n d m a r e b o th p o s i t iv e . T h e n u m b e r

    P n ) o / s u c h p r i m e s l e ss t h a n n , e v e r y p r i m e b e i n g c o u n te d m u l t i p l y a c c o r d in g to

    i t s n u m b e r o / r e p r e s e n t a t i o n s , i s g i v e n a s y m p t o t i c a l l y b y

    I i ~ ) ) 2 r ~ _ I 1 - 2 A o ,) ,

    w h e re ~ a n d v x a re o d d p r i m e s o / t h e [ o r m 3 r + I , p t k , ~ r ~ -k , a n d

    i / - - k i s a cub i c r e s i due o / ~ r ,

    A - - 2

    A w = g ' ( ~ - - I )

    I --A 9 - -B - -2

    Av ~

    i n t he con t rary case. T he po s i t i ve s i gn i s to be chosen i [

    ~o ~ a + bQ be i ng t ha t com pl ex pr i m e / ac t or o / ~ / or whi c h a ~ - - I , b ~ o ( m o d . 3 );

    t he nega t i ve i n t he con t rary even t. A nd A an d B are de] i ned by

    A = 2 a - - b , 3 B = b , 4 ~ = A 2 + 2 7 B : .

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    52 G .H . Hardy and J . E . Li tt lewood.

    I n p a r t i c u l a r , w h e n k = r , t h e n u m b e r o [ p r i m e s l a + m S + I i s g i v e n a s y m p -

    to t i c a l l y b y

    P n ) ~ . . . . . . . . . . . . . .. . . A - - 2

    I ,

    1 . (~ ) I o g n l I ~ . ( I - W ' ( ~ - I

    p r i m e s s u s c e p t i b l e o / m u l t i p l e r e p r e s e n t a t i o n b e i n g c o u n t e d m v l t i p l y .

    C o n j e c t u r e N . T h e r e a r e i n f i n i t e l y m a n y p r i m e s o / t h e [ o r m k s + l 3 + r i d ,

    w h e r e k , l , m a r e a l l p o si t iv e . T h e n u m b e r P ( n ) e l s u c h p r i m e s t es s t h a n n , p r i m e s

    s u s c e p ti b l e o f m u l t i p l e r e p r e s e n t a l i o n b e i n g c o u n t e d m u l t i p l y , i s g i v en a s y m p t o t i c a l l y b y

    P ( n ) ( I 4 z n

    w h e r e ~ i s a p r i m e o / t he / o r m 3 m + I , a n d A h a s t h e m e a n i n g e x p l a i n e d u n d e r M .

    T r i p l e t s a n d o t h e r s e q u e n c e s o / w i m e s .

    5. 6 1. I t is p l a i n t h a t t h e n u m b e r s ~ , ~ + 2 , ~ + 4 c a n n o t a ll b e p r i m e ,

    f o r a t l e a s t o n e o f t h e t h r e e i s d i v i s i b l e b y 3 . B u t i t i s p o s s i b l e t h a t

    ~ , ~ - 2 , ~ + 6 o r ~ , m ~ 4 , ~ + 6 s h o ul d a ll b e p r im e . I t is n a t u r a l t o e n q u i re

    w h e t h e r o u r m e t h o d i s a p p l i c a b l e i n p r i n c i p l e t o t h e i n v e s t i g a t i o n o f t h e

    d i s t r i b u t i o n o f t r i p l e t s a n d l o n g e r s e q u e n c e s .

    T h e g e n e r a l c a s e r a i se s v e r y i n t e r e s t i n g q u e s t i o n s a s to t h e d e n s i t y o f t h e

    d i s t r i b u t i o n o f p r i m e s , a n d i t w i ll b e c o n v e n i e n t , t o b e g i n b y d i s c u s s i n g t h e m .

    W e w r i t e

    5 - 6 ~ ) . o fx ) . .. . i i m ~ ,,: n ~ x ) - - M n ) ) ,

    s o t h a t ~ x ) = r is t h e g r e a t e s t n u m b e r o f p r i m e s t h a t o c c u r s i n d e f i n it e l y

    o f t e n i n a s e q u e n c e n + I , n 2 .. .. , n + [ x ] o f [ x ] c o n s e c u t i v e i n t e g e rs . T h e

    e x i s te n c e o f a n i n f i n i t y o f p r im e s s h o w s t h a t Q ( x ) > I f o r x > i , a n d n o t h i n g

    m o r e t h a n ~ hi s i s k n o w n ; b u t o f c o u r s e C o n j e c t u r e B i n v o l v e s q ( x ) > : 2 f o r x__>~3.

    I t i s pl a in " t h a t t h e d e t e r m i n a t i o n o f a l o w e r b o u n d f o r q (x ) i s a p r o b l e m o f

    e x c e p t i o n a l d e p t h .

    T h e p r o b l e m o f a n u p p e r b o u n d h a s g r e a t e r p o ss i b il i ti e s. W e p r o c e e d t o

    p r o v e , b y a s i m p l e e x t e n s i o n o f a n a r g u m e n t d u e t o L e g e n d r e l ,

    t See Landau, p. 67.

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    Par t i t io numerorum . I I I : On the express ion o f a num ber a s a sum o f p r imes . 53

    T h e o r e m G . I / ~ > o t h e n

    U X

    e ( x ) < ( I + ~ ) e f o g - ] o g x

    ( X > X o = X A e ) ) ,

    where C i s E u l e r s cons t an t . Mo re genera l l y , i [ N ( x , n ) i s t he num ber o[ t he i n t egers

    n + i , n + 2 , . . . . . n + I x ]

    t h a t a r e n o t d i v i s ib l e b y a n y p r i m e l e s s t h a n o r e q u a l to

    l o g

    x , t hen

    ........ - -C X

    o ~ ( x ) = ~ l i m N ( x , n ) < ( r + e ) e l o g - ] o g x ( x > x . , ( e ) ) .

    I t i s w e l l - k n o w n t h a t t h e n u m b e r o f t h e i n t e g e r s 1 , 2 . . . . . [ y ], n o t d i v i si b l e

    b y a n y o n e o f t h e p r i m e s p , , P2 . . . . . p ~ , i s

    z

    w h e r e t h e i - t h s u m m a t i o n i s t a k e n o v e r a l l c o m b i n a t i o n s o f t h e v p r i m e s i a t

    a t i m e . S i n c e t h e n u m b e r o f t e r m s i n t h e t o t a l s u m m a t i o n i s 2 , t h i s i s

    - - - )

    - + 0 ( 2 ~ ) = y I - - I I - - ~ I

    W e n o w t a k e p t , P 2 , - . - , P ~ to b e t h e f i rs t v p r i m e s , w r i t e n + x a n d n

    s u c c e s s i v e l y f o r y , s u b t r a c t , a n d t a k e t h e u p p e r l i m i t o f t h e d i f f e r e n c e a s n - - ~ .

    W e o b t a i n

    B u t

    = _ :/ I - c ~ l o g y

    a s y ~ 0 o.1 I f w e t a k e y = l o g x , a n d p ~ t o b e t h e g r e a t e s t p r i m e n o t le s s t h a n y ,

    w e h a v e

    r < p~,

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    54 G. H. Hard y and J . E . L i t t l ewood.

    A n e x a m i n a t i o n o f t h e p r i m e s l e s s t h a n 2 0 0 s u g g e s t s f o r c i b l y t h a t

    e x)s x> 2) .

    B u t a l t h o u g h t h e m e t h o d s w e a r e a b o u t t o e x p l ai n l e a d t o s t r ik i n g c o n j e c -

    t u r a l l o w e r b o u n d s , t h e y t h r o w n o l i g h t o n t h e p r o b l e m o f a n u p p e r b o u n d .

    W e h a v e n o t s u c c e e d e d i n p r o v i n g , e v e n w i th o u r a d d i t i o n a l h y p o t h e s i s , m o r e

    t h a n t h e ~ > el em e n ta ry ~ T h e o r e m G . W e p a s s o n t h e r e f o r e t o o u r m a i n t o p i c .

    5 . 6 2. W e c o n s i d e r n o w t h e p r o b l e m o f t h e o c c u r r e n c e o f g r o u p s o f p r i m e s

    o f t h e f o r m

    n , n + a t , n b a . ~, . . . , n + a m ,

    w h e r e a ~ , a ~ , . . . , a m a r e d i s t i n c t p o s i t i v e i n t e g e r s . W e w r i t e fo r b r e v i t y

    I r a ( x ) = 2 A ( a ~) ~ . 1 ( ~ a , ) . . J l ( ~ r + am ) x ~ .

    T h e n , i f ( h , k ) = i , w e h a v e

    ( 5 . 6 z I ) r a ' l m ( r ' e k ( h ) ) = ~ t ( ~ ) _ / 1 ( ~ + a ~ ) . . . _/ l( ~r + a , n ) r 2 ~ + , ' ~ e u ( ~ r h )

    2s~

    -- 2; , r 2 ~ 4 ( ~ ) . . . ~ . l (~ r + a m _ l ) r ~ e ~ i ~ ~ 2 d l ( ' a ~ , r ~ e - ~ i ~ e m i~ ~

    O

    2. r

    i t . _ ,

    0

    2_p~r +

    I f cp = 0 , r - - I , 0 ~ o , a n d 0 i s s u f f i c i e n t l y s m a l l i n c o m p a r i s o n w i t h

    q

    I - - r , t h e n

    ] ( r e _ i ~ ) c x ~ z ( q )

    I - - r e - i O '

    w h e r e

    l ( q )

    z ( ~ ) , p ( q )

    L e t u s a s s u m e f o r t h e m o m e n t t h a t

    ] , ,, - 1 ( r d * ) c ~ g i n - 1 ~ I - - r d o

    i f qJ - ~ -P O , r ~ z a n d 0 is s u f f i c i e n t l y s m a l l . T h e n o u r m e t h o d l e a d s u s t o w r i t e

    , +

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    Partitio numerorum. III : Oil the expression of a number as a sum of primes. 55

    2 ~

    I 2kar

    i d O

    P q ~p , q

    C . Z l _ i . ~ z ( q ) g , , , _ k I q l '

    P , q

    on replacing the integral

    suggests that

    by one extended from --zr to ~.

    5 . 6 2 2 ) ] ,~ r ) c ,~ g m o )

    X - - r

    Thus (5. 621)

    where gm is determined by the recurrence formula

    and

    (S. 624)

    P, q

    From this recurrence formula we obtain without difficulty

    (5-6 25) .q,~ (o) = Sm ---- ~ l l % ( q r ) g ( Q ) e a , p , . ,

    P h q t , 9 9 P r o , q m r ~ l

    where

    qr

    runs through all positive integral values,

    P r

    through all positive values

    legs than and prime to q,~ and Q is the number such that

    P p ~ + p 2 + . . . + p _~ , ( P , Q ) = 1 .

    = - q l q 2 q

    If we sum with respect to the p s, we obtain a res ult which we shall write in

    the form

    (5. 6251) S ,,~ = ~ A ( q ~ , q 2 . . . . q ,, ,) .

    q t ~ q 2 , 9 q t } t

    We shall see pre sently tha t the multiple series (5. 6251) is absolutely con-

    vergent.

    For greater precision of statement we now introduce a detinite hypothesis.

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    5 6 G , I I . H a r d y a n d J . E . L i t t le w o o d .

    ypothesis

    X . I [ m > o , a n d r ~ I , l h e n

    m

    ( 5 . 6 2 6 ) [ , , ( r ) o o . . . . ,

    I r

    w h e r e S m i s . q i v e n b y ( 5 - 6 2 5 ) a n d ( 5 . 6 2 5 1 ) .

    O u r d e d u c t i o n s f r o m t h i s h y p o t h e s i s w i ll b e m a d e r i g o r o u sl y , a n d w e s h a ll

    d e s c r i b e t h e r e s u l t s a s T h e o r e m s X I , X 2 . . . .

    5 . 5 3 . F r o m ( 5 . 5 2 6 ) i t f o l l o w s , b y t h e a r g u m e n t o f 4 - 2 , t h a t

    x

    ( 5 - 6 3 1 ) P ( x ; o , a ~ , a 2 . . . . . a , n ) c ,~ S m ( l o g

    x ) m

    a s x ~ , w h e r e t he l e / t -h a n d s i d e d en o t e s t he n u m b e r o / g r o u p s o / m + I p r i m e s

    n , n + a~ . . . . . n + a , , o ] w h i c h a l l l h e m e m b e r s a r e l e s s t h a n x :

    W e p r o c e e d t o e v a l u a t e S in . I n t h e f i rs t p la c e w e o b s e r v e t h a t A ( g , q 2 . . . . q , ,, )

    i s z e r o if a n y q h a s a s q u a r e f a c t o r . N e x t w e h a v e

    ( 5 . 6 3 2 ) A ( q , q , , q 2 q