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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