The First Fify Million Prime Numbers Essay

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    The First 50 MillionPrime Numbers*Don Zagier

    To my parents

    I w o u l d l i k e t o te l l y o u t o d a y a b o u t as u b j e c t w h i c h , a l t h o u g h I h a v e n o t w o r k e di n it m y s e l f , h a s a l w a y s e x t r a o r d i n a r i l yc a p t i v a t e d m e , a nd w h i c h h a s f a s c i n a t e dm a t h e m a t i c i a n s f r o m t h e e a r l i e s t t i m e su n t i l t h e p r e s e n t - n a m e l y , t h e q u e s t i o no f t h e d i s t r i b u t i o n o f p r i m e n u m b e r s .

    Y o u c e r t a i n l y a l l k n o w w h a t a p r i m enumb er is: it is a natu ral numbe r bigg ert h a n 1 w h i c h i s d i v i s i b l e b y n o o t h e rnatu ral nu mber ex cept for I. That atl e a s t i s t h e n u m b e r t h e o r i s t ' s d e f i n i -t i o n; o t h e r m a t h e m a t i c i a n s s o m e t i m e s h a v eo t h e r d e f i n i t i o n s . F o r t h e f u n c t i o n - t h e -orist, for instance, a prime number isa n i n t e g r a l r o o t o f t h e a n a l y t i c f u n c -t i o n

    ::The following a rticle is a revised ver sion ofthe author's inaugural lecture ~Antrittsvor-lesung) held on May 5, 1975 at Bonn University.Additional remarks and references to the lit-erature may be found at the end.Translated from the German by R. Perlis..Theoriginal German version will also be publishedin Beihefte zu Elemente der Mathematik No. 15,Birkh~user Verlag, Basel.

    7sin ~F(s)

    I - s ;sin --s

    for the alge brai st it is"the char acte rist ic of a finite field"

    or

    or"a point in Spec ~"

    " a n o n - a r c h i m e d e a n v a l u a t i o n " ;t h e c o m b i n a t o r i s t d e f i n e s t h e p r i m e n u m -bers in duct ivel y by the recu rsio n (I)

    1Pn+1 = [I - log2 ( ~ +n (-1)r

    r~1 ISil

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    8T o s u p p o r t t h e f i r s t o f t h e s e c l a i m s ,

    l e t m e b e g i n b y s h o w i n g y o u a l i s t o f t h ep r i m e a nd c o m p o s i t e n u m b e r s u p t o 1 0 0( w h e re a p a r t f r o m 2 I h a v e l i s t e d o n l yt h e o dd n u m b e r s ) .p r i m e c o m p o s i t e2 43 9 633 47 15 655 53 21 697 59 25 75

    11 61 27 7713 67 33 8117 71 35 8519 73 39 8723 79 45 9129 83 49 9331 89 51 9537 97 55 9941 57a n d l i s t s o f t h e p r i m e s a m o n g t h e 1 O On u m b e r s i m m e d i a t e l y p r e c e d i n g a n d f o l -l o w i n g 1 0 m i l l i o n :T h e p ri m e n u m b e r s b e t w e e n 9 , 9 9 9 , 9 0 0 a n d1 0 , 0 0 0 , 0 0 09 , 9 9 9 , 9 0 19 , 9 9 9 , 9 0 79 , 9 9 9 , 9 2 99 , 9 9 9 , 9 3 19 , 9 9 9 , 9 3 79 , 9 9 9 , 9 4 39 , 9 9 9 , 9 7 19 , 9 9 9 , 9 7 39 , 9 9 9 , 9 9 1T h e p ri m e n u m b e r s b e t w e e n 1 0 , 0 O O , O O O a n d1 0 , 0 O O , 1 0 010,0OO,019I 0 , O O O , 0 7 9

    I h o p e y o u w i l l a g r e e t h a t t h e r e i sn o a p p a r e n t r e a s o n w h y o n e n u m b e r i sp r i m e a n d a n o t h e r n o t . T o t h e c o n t r a r y ,u p o n l o o k i n g a t t h e s e n u m b e r s o n e h a st h e f e e l i n g o f b e i n g i n th e p r e s e n c e o fo n e o f t h e i n e x p l i c a b l e s e c r e t s o f c r e a -t io n . T h a t e v e n m a t h e m a t i c i a n s h a v e n o tp e n e t r a t e d t h i s s e c r e t i s p e r h a p s m o s tc o n v i n c i n g l y s h o w n b y t he a r d o u r w i t hw h i c h t h e y s e a r c h f o r b i g g e r a n d b i g g e rp r i m e s - w i t h n u m b e r s w h i c h g r o w r e g u -l a r l y , l i k e s q u a r e s o r p o w e r s o f t w o , n o -b o d y w o u l d e v e r b o t h e r w r i t i n g d o w n e x -a m p l e s l a r g e r t h a n t h e p r e v i o u s l y k n o w no n e s , b u t f o r p r i m e n u m b e r s p e o p l e h a v eg o n e t o a g r e a t d e a l o f t r o u b l e t o d oj u s t t ha t . F o r e x a m p l e , i n 18 7 6 L u c a sp r o v e d t h a t t h e n u m b e r 2 12 7 - I i s p r i m e ,a n d f o r 75 y ea r s t h i s r e m a i n e d u n s u r p a s -s e d - w h i c h i s p e r h a p s n o t s u r p r i s i n gw h e n o n e s e e s t h i s n u m b e r :

    2127 - 1 =1 7 0 1 4 1 1 8 3 4 6 0 4 6 9 2 3 1 7 3 1 6 8 7 3 0 3 7 1 5 8 8 4 1 0 5 7 2 7 .N o t u n t i l 1 9 51 , w i t h t h e a p p e a r a n c e o fe l e c t r o n i c c o m p u t e r s , w e r e l a r g e r p r i m en u m b e r s d i s c o v e r e d . I n th e a c c o m p a n y i n gt a b l e , y o u c a n s e e t h e d a t a o n t h e s u c -c e s s i v e t i t l e - h o l d e r s (3). A t t h e m o m e n t ,t h e l u c k y f e l l o w i s t he 6 0 0 2 - d i g i t n u m -b e r 2 1 99 3 7 - I ( w h i c h I w o u l d n o t c a r e t ow r i t e d o w n ) ; i f y o u d o n ' t b e l i e v e m e , y o uc a n l o o k i t u p i n t h e G u i n n e s s b o o k o fr e c o r d s .

    T h e l a r g e s t k n o w n p r i m e n u m b e r

    ~ ~ ,~-,~ 0 0~' ~ ~ o o~ O ~ , ~

    2 1 2 7 - I 3 9 1 8 7 6 L u c a s~ ( 2 1 4 8 + 4 4 1 9 51 F e r r i e r)1 1 4 ( 2 1 2 7 - I) + I 4 1 ~ M i l l e r . +1 8 0 ( 2 1 2 7 - I ) 2 + I 7 9 ~ 1 9 5 1 W h e e l e r +E D S A C I2521 - I 157

    2607 - I 183 I Le hm er +2 12 79 - I 3 8 6 1 9 5 2 R o b i n s o n22203 - I 664 + SW AC22281 - I 687

    R i e s e l +23217 - 1 969 195 7 B E S K24253 - I 1281 ~ Hur wit z +24423 - I 1332 # 1961 Se lf ri dg e+ I B M 7 0 9 029689 - I 291 729941 - I 2993 ~ 1963 Gi ll ie s +I L I A C 2211213 - I 3376

    T u c k e r m a n219937 - 1 600 2 1971 + I B M 3 6 0

    M o r e i n t e r e s t i n g , h o w e v e r , i s t h eq u e s t i o n a b o u t t h e l a w s g o v e r n i n g p r i m en u m b e r s . I h a v e a l r e a d y s h o w n y o u a l i s to f t h e p r i m e n u m b e r s u p t o 1 O O. H e r e i st h e s a m e i n f o r m a t i o n p r e s e n t e d g r a p h i c a l -l y. T h e f u n c t i o n d e s i g n a t e d ~ ( x) ( a b o u tw h i c h I w i l l b e c o n t i n u a l l y s p e a k i n g f r o mn o w o n) i s t h e n u m b e r o f p r i m e n u m b e r sn o t e x c e e d i n g x ; t h u s ~( x) b e g i n s w i t h Oa n d j u m p s b y I a t e v e r y p r i m e n u m b e r 2 ,3, 5 e t c . A l r e a d y i n t h i s p i c t u r e w e c a ns e e th a t , d e s p i t e s m a l l o s c i l l a t i o n s ,~ (x ) b y a n d l a r g e g r o w s q u i t e r e g u l a r l y .

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    10s e e t h a t, a l t h o u g h t h e f u n c t i o n x / l o g xq u a l i t a t i v e l y m i r r o r s t h e b e h a v i o u r o f~ ( x) , i t c e r t a i n l y d o e s n o t a g r e e w i t h~ (x ) s u f f i c i e n t l y w e l l t o e x p l a i n t h es m o o t h n e s s o f t h e l a t t er :

    60005000

    &O00

    30002000

    1000

    l o g x

    I i I I I Xv1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0

    T h e r e f o r e i t i s n a t u r a l t o a s k f or b e t t e ra p p r o x i m a t i o n s . I f w e t ak e a n o t h e r l o o ka t o u r t a b l e o f t h e r a t i o s o f x t o ~ ( x ) ,w e f i n d t h a t t h i s r a t i o i s a l m o s t e x a c t -l y l og x - I. W i t h a m o r e c a r e f u l c a l c u -l a t i o n a nd wi t h m o r e d e t ai l e d d a t a o n~ ( x ) , L e g e n d r e (5) f o u n d i n 1 8 0 8 t h a t ap a r t i c u l a r l y g o o d a p p r o x i m a t i o n i s o b -t a i n e d i f i n p l a c e o f I w e s u b t r a c t1 . 0 8 3 6 6 f r o m l o g x , i . e .

    x~(x) % log x - 1.083 66A n o t h e r g o o d a p p r o x i m a t i o n t o ~ ( x ),w h i c h w a s f i r s t g i v e n b y G a u s s , i s o b -

    t a i n e d b y ta k i n g a s s t a r t i n g p o i n t t h ee m p i r i c a l f a c t t h a t t he f r e q u e n c y o fp r i m e n u m b e r s n e a r a v e r y l a r g e n u m b e r xi s a l m o s t e x a c t l y I / l o g x . F r o m t h i s ,t h e n u m b e r o f p r i m e n u m b e r s u p t o xs h o u l d b e a p p r o x i m a t e l y g i v e n b y t h el o g a r i ~ m i c s um

    _ I I + + ILS (X ) l og 2 + log---~ "'" log-----~o r , w h a t i s e s s e n t i a l l y t h e s a m e ( 6) , b yt h e l o g a r i ~ m i c i n te g ra l

    x ILi( x) = 2/ lo--~ -~t.

    I f w e n o w c o m p a r e t h e g r a p h o f L i ( x) w i t ht h a t of ~ ( x ) , t h e n w e s e e t h a t w i t h i n t h ea c c u r a c y o f o u r p i c t u r e t h e t w o c o i n c i d ei e x a c t l y .T h e r e i s n o p o i n t i n s h o w i n g y ou t h e p i c-t u r e of L e g e n d r e ' s a p p r o x i m a t i o n a s w e l l ,f o r i n t h e r a n g e o f t h e g r a p h i t i s a ne v e n b e t t e r a p p r o x i m a t i o n t o ~( x ).

    600050004000

    30002000

    1000~

    t~

    501000 .I X

    1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0

    T h e r e i s o n e m o r e a p p r o x i m a t i o n w h i c hI w o u l d l i k e t o m e n t i o n . R i e m a n n ' s r e -s e a r c h o n p r i m e n u m b e r s s u g g e s t s t h a tt h e p r o b a b i l i t y f o r a l a r g e n u m b e r x tob e p r i m e s h o u l d b e e v e n c l o s e r t o 1 / l o g xi f o n e c o u n t e d n o t o n l y t h e p r i m e n u m b e r sb u t a l s o t h e powers o f p r im e s , c o u n t i n gt h e s q u a r e o f a p r i m e a s h a l f a p r im e ,t h e c u b e o f a p r i m e a s a t h i r d , e t c . T h i sl e a d s t o t he a p p r o x i m a t i o n

    ( x ) + 8 9 ~ ( / ~ ) + ~ ~ ( 3 ~ s + . . . = L i ( x )o r , e q u i v a l e n t l y ,

    ( x) = L i ( x ) - 8 9 ~ i ( ~ ) - 8 9 ~ i ( 3 / ~ ) _ . . .(7)

    T h e f u n c t i o n o n t h e r i g h t s i d e o f t h i sf o r m u l a i s d e n o t e d b y R ( x ) , i n h o n o u r o fR i e m a n n. I t r e p r e s e n t s a n a m a z i n g l y g o o da p p r o x i m a t i o n t o ~ ( x ), a s t h e f o l l o w i n gv a l u e s s h ow :

    X ~ (X) R(X )I 0 0 , 0 O O , O 0 02 0 0 , 0 0 0 , 0 0 03 0 0 , 0 0 0 , 0 0 04 0 0 , 0 0 0 , 0 0 05 0 0 , 0 0 0 , 0 0 06 0 0 , 0 0 0 , 0 0 07 0 0 , 0 0 0 , 0 0 08 0 0 , 0 0 0 , 0 0 09 0 0 , 0 0 0 , 0 0 0

    1 , 0 0 0 , O O 0 , 0 0 0

    5 , 7 6 1 , 4 5 51 1 , 0 7 8 , 9 3 71 6 , 2 5 2 , 3 2 52 1 , 3 3 6 , 3 2 62 6 , 3 5 5 , 8 6 73 1 , 3 2 4 , 7 0 33 6 , 2 5 2 , 9 3 14 1 , 1 4 6 , 1 7 94 6 , 0 0 9 , 2 1 55 0 , 8 4 7 , 5 3 4

    5 , 7 6 1 , 5 5 21 1 , O 7 9 , 0 9 01 6 , 2 5 2 , 3 5 52 1 , 3 3 6 , 1 8 52 6 , 3 5 5 , 5 1 73 1 , 3 2 4 , 6 2 23 6 , 2 5 2 , 7 1 94 1 , 1 4 6 , 2 4 84 6 , 0 0 9 , 9 4 95 0 , 8 4 7 , 4 5 5

    F o r t h o s e i n t h e a u d i e n c e w h o k n o w a l i t -t l e f u n c t i o n t h e o r y , p e r h a p s I m i g h t a d dt h a t R (x ) i s a n e n t i r e f u n c t i o n o f l o g x ,g i v e n b y t h e r a p i d l y c o n v e r g i n g p o w e r s e -r i e s

    R(x) = I + ~ I (log x) nn=1 n~ (n+1) n! '

    w h e r e ~ ( n +1 ) i s t h e R i e m a n n z e t a f u n c -ti on (8).

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    A t t h i s p o i n t I s h o u l d e m p h a s i z e t h a tG a u s s ' s a n d L e g e n d r e ' s a p p r o x i m a t i o n s t o~ (x ) w e r e o b t a i n e d o n l y e m p i r i c a l l y , a n dt h a t e v e n R i e m an n , a l t h o u g h h e w a s l e dt o h i s f u n c t i o n R ( x) b y t h e o r e t i c a l c o n -s i d e r a t i o n s , n e v e r p r o v e d t h e p r i m e n u m -b e r t h e o r em . T h a t w a s f i r s t a c c o m p l i s h e di n 18 9 6 b y H a d a m a r d a n d ( i n d e p e n d e n t l y )d e l a V a l l ~ e P o u s s i n ; t h e i r p r o o f s w e r eb a s e d o n R i e m a n n ' s w o r k .

    W h i l e s t i l l o n t h e t h e m e o f t he p r e -d i c t a b i l i t y o f t h e p r i m e n u m be r s , I w o u l dl i k e t o g i v e a f ew m o r e n u m e r i c a l e x a m -p l e s . A s a l r e a d y m e n t i o n e d , t h e p r o b a b i l -i t y f o r a n u m b e r o f t h e o r d e r o f m a g n i -t u d e x t o b e p r i m e i s r o u g h l y e q u a l t oI / l o g x ; t h a t i s , t h e n u m b e r o f p r i m e si n a n i n t e r v a l o f l e n g t h a a b o u t x s h o u l db e a p p r o x i m a t e l y a / l o g x , a t l e a s t ift h e i n t e r v a l i s l o n g e n o u g h t o m a k e s t a -t i s t i c s m e a n i n g f u l , b u t s m a l l in c o m p a r -i s o n t o x . F o r e x a m p l e , w e w o u l d e x p e c tt o f i n d a r o u n d 8 1 4 2 p r i m e s i n t h e i n t e r -v a l b e t w e e n 1 0 0 m i l l i o n a n d 1 0 0 m i l l i o np l u s 1 5 0 , 0 0 0 b e c a u s e

    1 5 0 , O O O 1 5 0 , O O Ol o g ( 1 O O , O 0 0 , O O 0 ) = 1 8 . 4 2 7 . % 8 1 4 2

    C o r r e s p o n d i n g l y , t h e p r o b a b i l i t y t ha tt w o r a n d o m n u m b e r s n e a r x a r e b o t h p r i m ei s a p p r o x i m a t e l y I / ( l o g x ) 2 . T hu s i f o n ea s k s h o w m a n y p r i m e t w i n s ( i .e . p a i r s o fp r i m e s d i f f e r i n g b y 2, l i k e 1 1 a n d 1 3 o r5 9 a n d 6 1) t h e r e a r e i n t h e i n t e r v a l f r o mx t o x + a t h e n w e m i g h t e x p e c t a p p r o x i -m a t e l y a / ( l o g x ) 2. A c t u a l l y , w e s h o u l de x p e c t a b i t m o r e , s i n c e t h e f a c t t h a t ni s a l r e a d y p r i m e s l i g h t l y c h a n g e s t h ec h a n c e t h a t n + 2 is p r i m e ( f or e x a m p l en + 2 i s t h e n c e r t a i n l y o d d ). A n e a s y h e u -r i s t i c a r g u m e n t (9) g i v e s C . a / ( l o g x ) 2a s t h e e x p e c t e d n u m b e r o f t w i n p r i me s i nt h e i n t e r v a l [ x, x + a ] w h e r e C i s a . co n-s t a n t w i t h v a l u e a b o u t 1 . 3 ( m o r e e x a c t l y :C = 1 . 3 2 0 3 2 3 6 3 1 6 . . . ) . T h u s b e t w e e n 1 O Om i l l i o n a n d 1 O O m i l l i o n p l u s 1 5 0 t h o u s a n dt h e r e s h o u l d b e a b o u t

    150~000 5 84( 1 . 3 2 ' ' ' ) ( 1 8 . 4 2 7 ) 2 =

    p a i r s o f p r i m e t w i n s. H e r e a r e da t a c o m -p u t e d b y J o n e s , L a l a n d B l u n d o n ( io ) g i v - -i n g t he e x a c t n u m b e r o f p r i m e s a n d p r i m et w i n s i n t h i s i n t e r v a l , a s w e l l a s i ns e v e r a l e q u a l l y l o n g i n t e r v a l s a r o u n dl a r g e r p o w e r s o f 1 0:

    1 1

    I n t e r v a l P r i m e P r i m en u m b e r s t w i n s

    I IO ~ ~

    1 O O , O O 0 , O O O - 8 1 4 2 8 1 5 4 58 4 6 011 O O , 1 5 0 , O O O1 , O O O , O 0 0 , O O O - 7 2 3 8 7 2 4 2 46 1 4 6 61 , O O O , 1 5 0 , O O O

    1 0 , O O O , O 0 0 , O O O - 6 5 1 4 6 5 11 3 7 4 3 8 91 0 , O O O , 1 5 0 , O O O1 0 0 , O O O , O 0 0 , O O O - 5 9 2 2 5 9 74 3 0 9 2 7 61 0 0 , 0 0 0 , 1 5 0 , 0 0 0

    1 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 - 5 4 29 5 4 33 2 59 27 61 , 0 0 0 , 0 0 0 , 1 5 0 , 0 0 01 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 - 5 0 11 5 0 6 5 2 21 2 0 81 0 , 0 0 0 , 0 0 0 , 1 5 0 , 0 0 0

    1 0 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 - 4 6 5 3 4 6 4 3 1 9 1 1 861 0 0 , 0 0 0 , 0 0 0 , 1 5 0 , 0 0 01 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 - 4343 4251 166 1611 , 0 0 0 , 0 0 0 , 0 0 0 , 1 5 0 , 0 0 0

    A s y o u c an s e e , t h e a g r e e m e n t w i t h t h et h e o r y i s e x t r e m e l y g o o d . T h i s i s e s p e -c i a l l y s u r p r i s i n g i n t h e c a s e o f th ep r i m e p a i r s , s i n c e i t h a s n o t y e t e v e nb e e n p r o v e d t h a t t h e r e a r e i n f i n i t e l ym a n y s u c h p a i r s , l e t a l o n e t h a t t h e y a r ed i s t r i b u t e d a c c o r d i n g t o t h e c o n j e c t u r e dl a w .

    I w a n t t o g i v e o n e l a s t i l l u s t r a t i o no f t h e p r e d i c t a b i l i t y o f p r i m e s , n a m e l yt h e p r o b l e m o f t h e g a p s b e t w e e n p r i m es .I f o n e l o o k s a t t a b l e s o f p r i m e s , o n es o m e t i m e s f i n d s u n u s u a l l y l a r g e i n t er -v a l s , e . g . b e t w e e n 1 1 3 a n d 1 2 7, w h i c hd o n ' t c o n t a i n a n y p r i m e s a t al l . L e t g ( x )b e t he l e n g t h o f t h e l a r g e s t p r i m e - f r e ei n t e r v a l o r " g a p " u p t o x . F o r e x a m p l e ,t h e l a r g e s t g a p b e l o w 2 0 0 is t h e i n t e r -v a l f r o m 1 1 3 t o 1 2 7 j u s t m e n t i o n e d , s og ( 2 O O ) = 14 . N a t u r a l l y , t h e n u m b e r g ( x)g r o w s v e r y e r r a t i c a l l y , b u t a h e u r i s t i ca r g u m e n t s u g g e s t s t h e a s y m p t o t i c f o r m u -la (II)

    g(x) ~ (log x) 2

    I n t h e f o l l o w i n g p i c t u r e , y o u c a n s e eh o w w e l l e v e n t h e w i l d l y i r r e g u l a r f u n c -t i o n g ( x) h o l d s t o t he e x p e c t e d b e h a v -i o u r .

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    127 0 06 0 05 0 0& O 03 0 02 0 01 0 0

    01 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 2

    U p to n o w , I h a v e s u b s t a n t i a t e d m yc l a i m a b o u t t h e o r d e r l i n e s s o f t h e pr i m e sm u c h m o r e t h o r o u g h l y t h a n m y c l a i m a b o u tt h e i r d i s o r d e r l i n e s s . A l s o , I h a v e n o ty e t f u l f i l l e d t h e p r o m i s e o f m y t i t l e t os h o w yo u t h e f i r s t 5 0 m i l l i o n p r i m e s , b u th a v e o n l y s h o w n y o u a f e w t h o u s a n d . S oh e r e i s a g r a p h o f ~( x) c o m p a r e d w i t h t h ea p p r o x i m a t i o n s o f L e g e n d r e , G a u s s , a n dR i e m a n n u p t o 1 0 m i l l i o n (12) . S i n c e t h e s ef o u r f u n c t i o n s l i e s o c l o s e t o g e t h e r t h a tt h e i r g r a p h s a r e i n d i s t i n g u i s h a b l e t o t h en a k e d e y e - a s w e a l r e a d y s a w i n t h e p i c -t u r e u p t o 5 0 , 0 0 0 - I h a v e p l o t t e d o n l yt h e d i f f e r e n c e s b e t w e e n t h em :

    T h i s p i c t u r e , I t h i n k , s h o w s w h a t t h ep e r s o n w h o d e c i d e s t o s t u d y n u m b e r t h e -o r y h a s l e t h i m s e l f i n f o r . A s y o u c a ns e e, f o r s m a l l x ( u p t o a p p r o x i m a t e l y Im i l l i o n) L e g e n d r e ' s a p p r o x i m a t i o nx / ( l o g x - 1 . O 8 36 6 ) i s c o n s i d e r a b l y b e t -t e r t h a n G a u s s ' s L i ( x ) , b u t a f t e r 5 m i l -l i o n Li ( x ) i s b e t t e r , a n d i t c a n b e s h o w nt h a t Li ( x ) s t a y s b e t t e r a s x g r o w s .

    B u t u p t o 10 m i l l i o n t h e r e a r e o n l ys o m e 6 0 0 t h o u s a n d p r i m e n u m b e r s ; t o s h o wy o u th e p r o m i s e d 5 0 m i l l i o n p r i m e s, Ih a v e t o g o n o t t o 1 0 m i l l i o n b u t a l l t h ew a y o u t t o a b i l l i o n ( A m e r i c a n s t y l e :1 0 ) . I n t h i s r a n g e , t h e g r a p h o fR(x) - ~(x) lo oks li ke this(13) :

    600~ Rx)-X(x) /~ AA. A , . , , , A , f L .;o

    - 3 0 0 | - - O ~ x ~ < 1 ,0 0 0 ,0 0 0 ,0 0 0T h e o s c i l l a t i o n o f t h e f u n c t i o n K ( x ) - ~ ( x )b e c o m e l a r g e r a n d l a r g e r , ; b u t e v e n f ort h e s e a l m o s t i n c o n c e i v a b l y l a r g e v a l u e so f x t h e y n e v e r g o b e y o n d a f e w h u n d r e d .

    I n c o n n e c t i o n w i t h t h e s e d a t a I c a nm e n t i o n y e t a n o t h e r f a c t a b o u t t h e n u m -b e r o f p r i m e n u m b e r s ~ ( x ) . I n t h e p i c -t u r e u p t o 1 0 m i l l i o n , G a u s s ' s a p p r o x i m a -t i o n w a s a l w a y s b i g g e r t h a n ~ ( x ) . T h a tr e m a i n s s o u n t i l a b i l l i o n , a s y o u c a ns e e i n t h e p i c t u r e o n t h e f o l l o w i n g p a g e( in w h i c h t h e a b o v e d a t a a r e p l o t t e d l o g -a r i t h m i c a l l y ) .

    X(X)+3OO-

    z(x}+200-i~ ( x ) + l ~ 1 7 6

    3 ~ ' ( x ]~(x)-1ool

    LegendreV ~ R i e m a n n /

    i , f V V m" R ,e m a n n , M - - R le m a n n " V ~ / ~ ~ ' " - ~ Nx ( m r m l il o n s

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    13

    5 0 0 L t ( x ) - R ( x )" ~ / / // t

    2 0 0

    200C

    100C

    10C I I i I I 1 J I1 0 2 0 3 0 5 0 7 0 1 0 0 2 0 0 5 0 0 1 0 0 0x ( in m i l l i o n s )

    S u r e l y t h i s g r a p h g i v e s u s t he i m p r e s -s i o n t h a t w i t h i n c r e a s i n g x t h e d i f f e r -e n c e L i ( x) - ~ ( x) g r o w s s t e a d i l y t o i n -f i n i t y , t h a t is , t h a t t h e l o g a r i t h m i c i n -t e g r a l L i (x ) c o n s i s t e n t l y o v e r e s t i m a t e st h e n u m b e r o f p r i m e s u p t o x ( t hi s w o u l da g r e e w i t h t h e o b s e r v a t i o n t h a t R( x) i sa b e t t e r a p p r o x i m a t i o n t h a n L i ( x) , s i n c eR ( x) i s a l w a y s s m a l l e r t h a n L i ( x ) ) . B u tt h i s i s f a l s e : i t c a n b e s h o w n t h a t t h e r ea r e p o i n t s w h e r e t h e o s c i l l a t i o n s o fR ( x ) - ~( x ) a r e s o b i g t h a t ~ (x ) a c t u a l -l y b e c o m e s l a r g e r t h a n L i ( x ) . U p t o n o wn o s u c h n u m b e r s h a v e b e e n f o u n d a n d p e r -h a p s n o n e e v e r w i l l b e f o u n d , b u t L i t t l e -w o o d p r o v e d t h a t t h e y e x i s t a n d S k ew e s (1 4 )p r o v e d t h a t t h e r e i s o n e t h a t i s s m a l l e rt h a n

    1 0 1 0 1 0 3 4(a n u m b e r o f w h i c h H a r d y o n c e s a i d t h a ti t w a s s u r e l y t h e b i g g e s t t h a t h ad e v e rs e r v e d a n y d e f i n i t e p u r p o s e i n m a t h e m a t -i c s ) . I n a n y c a s e , t h i s e x a m p l e s h o w sh o w u n w i s e i t c a n b e t o b a se c o n c l u s i o n sa b o u t p r i m e s s o l e l y o n n u m e r i c a l d a t a .

    I n t h e l a s t p a r t o f m y l e c t u r e I w o u l dl i k e t o ta l k a b o u t s o m e t h e o r e t i c a l r e -s u l t s a b o u t ~ ( x) s o t h a t y o u d o n ' t g oa w a y w i t h t h e f e e l i n g o f h a v i n g s e e n o n l ye x p e r i m e n t a l m a th . A n o n - i n i t i a t e w o u l dc e r t a i n l y t h i n k t h a t t h e p r o p e r t y o fb e i n g p r i m e i s m u c h t o o r a n d o m f o r u s t ob e a b l e t o p r o v e a n y t h i n g a b o u t i t . T h i sw a s r e f u t e d a l r e a d y 2 , 2 0 0 y e a r s a g o b yE u c l i d , w h o p r o v e d t h e e x i s t e n c e o f i n -f i n i t e l y m a n y p r i m e s . H i s a r g u m e n t c a nb e f o r m u l a t e d i n o n e s e n t e n c e : I f t h e r ew e r e o n l y f i n i t e l y m a n y p r i m e s, t h e n b ym u l t i p l y i n g t h e m t o g e t h e r a n d a d d i n g I,o n e w o u l d g e t a n u m b e r w h i c h i s n o t d i -v i s i b l e b y a n y p r i m e a t al l , a n d t h a t i si m p o s s i b l e . I n t h e 1 8 t h c e n t u r y , E u l e rp r o v e d m o r e , n a m e l y t h a t t h e s u m o f t h er e c i p r o c a l s o f t he p r i m e n u m b e r s d i -

    v e r g e s , i .e . e v e n t u a l l y e x c e e d s a n y p r e -v i o u s l y g i v e n n u m b er . H i s pr o o f , w h i c hi s a l s o v e r y s i m p l e , u s e d t h e f u n c t i o n

    ~ ( s ) = I + i _ + I _ _ + . . . ,2 s 3 s

    w h o s e i m p o r t a n c e f o r t h e s t u d y o f ~ (x )w a s f u l l y r e c o g n i z e d o n l y l at e r , w i t ht h e w o r k o f R i e m a n n . I t i s a m u s i n g t o r e -m a r k t h a t , a l t h o u g h t h e s u m o f t h e r e c i p -r o c a l s o f a l l p r i m e s i s d i v e r g e n t , t h i ss u m o v e r a l l t h e k n o w n p r i m e s ( l e t ' s s a yt h e f i r s t 5 0 m i l l i o n ) i s s m a l l e r t h a nf o u r ( 1 5 ) .

    T h e f i r s t m a j o r r e s u l t i n t he d i r e c -t i o n of t h e p r i m e n u m b e r t h e o r e m w a sp r o v e d b y C h e b y s h e v i n 1 8 5 0 ( 16 ) . H es h o w e d t h a t f or s u f f i c i e n t l y l a r g e x

    xx < ~(x) < 1.11 log x. 8 9 l o g xi . e . , t h e p r i m e n u m b e r t h e o r e m i s c o r -r e c t w i t h a r e l a t i v e e r r o r o f a t m o s t1 1% . H i s p r o o f u s e s b i n o m i a l c o e f f i c i e n t sa n d i s s o p r e t t y t h a t I c a n n o t r e s i s t a tl e a s t s k e t c h i n g a s i m p l i f i e d v e r s i o n o fi t ( w it h s o m e w h a t w o r s e c o n s t a n t s ) .

    I n t h e o n e d i r e c t i o n , w e w i l l p r o v ex(x) < 1. 7 ~ .

    T h i s i n e q u a l i t y i s v a l i d f o r x < 1 2O O .A s s u m e i n d u c t i v e l y t h a t i t h a s b e e np r o v e d f o r x < n a n d c o n s i d e r t h e m i d d l eb i n o m i a l c o e f f i c i e n t

    S i n c e n2 2 n = ( I + 1 ) 2 n = ( o 2 n ) +( ~ n ) + ' '' e ( ~ n )+ ' ' ' + ( 2n 2 n )t h i s c o e f f i c i e n t i s a t m o s t 2 2 n . O n t h eo t h e r h a n d

    2n (2n) ! (2n) x(2n-1) x... x2xl( n ) = ~ = (nx(n -1)X. x2xl) "

    E v e r y p r i m e p s m a l l e r t h a n 2 n a p p e a r s i nt h e n u m e r a t o r , b u t c e r t a i n l y n o p b i g g e rt h a n n c a n a p p e a r i n t h e d e n o m i n a t o r .

    i s d i v i s i b l e b y e v e r y p r i m e b e t w e e n n a n d2n:

    2 nP ( n ) 9n < p < 2 nB u t t h e p r o d u c t h a s ~ (2 n ) - ~ ( n) f a c t o r s ,e a c h b i g g e r t h a n n , s o w e g e t

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    14(2n) -~ (n) 2n 22 nn < ~ p < ( n ) 1 2 0 0 ) .

    H e n c e t h e t h e o r e m i s v a l i d a l s o f o r 2 n .S i n c e

    n~ ( 2 n + I ) < ~ ( 2 n ) + I < 3 . 0 9 ~ o g n + I2 n + I ( n > 1 2 0 0 )< 1.7 log (2n+i")

    i t is a l s o v a l i d f o r 2 n + I , c o m p l e t i n g t h ei n d u c t i o n .F o r t h e b o u n d i n t h e ot h e r d i r e c t i o n ,w e n e e d a s i m p l e l e m m a w h i c h c a n b ep r o v e d e a s i l y u s i n g t h e w e l l - k n o w n f o r -m u l a f o r t he p o w e r o f p w h i c h d i v i d e s n !(17) :L e m m a : L e t p b e a p r i m e . I f p V p i s t h e l a r -

    g e s t p o w e r o f p d i v i d i n g ( ~ ) ,t h e np V P < n .

    C o r o l l a r y : E v e r g b i n e m i a l c o e f f i c i e n t " ( ~ )s a t i s f i e s

    (~) = ~ p V p _ < n~(n)p ~ n

    I f w e a d d t h e i n e q u a l i t y o f t h e c o r o l l a r yf o r a l l b i n o m i a l c o e f f i c i e n t s (~)w i t h g i v e n n , t h e n w e f i n d

    n2n = (I+I ) n = Z (~) S ( n+ l) .n ~(n)k = O

    a n d t a k i n g l o g a r i t h m s g i v e s~ ( n ) > n l o g 2 l o g ( n + 1 )

    - ' l o g n - l o g n> 2 n (n > 20 0) .3 l o g n

    I n c l o s i n g , I w o u l d l i k e t o s a y a f e ww o r d s a b o u t R i e m a n n ' s w o r k . A l t h o u g h R i e -m a n n n e v e r p r o v e d t h e p r i m e n u m b e r t h e o -

    r a m, h e d i d s o m e t h i n g w h i c h i s i n m a n yw a y s m u c h m o r e a s t o n i s h i n g - h e d i s c o v -e r e d a n e x a c t f o r m u l a f o r ~ ( x ). T h i s f o r -m u l a h a s t h e f o r m

    + 8 9 + 8 9 += L i ( x ) - Z L i ( x )P

    w h e r e t h e s u m r u n s o v e r t h e r o o t s o f t h ez e t a f u n c t i o n ~ (s )( 18 ) . T h e s e r o o t s( a p a r t f r o m t h e s o - c a l l e d " t r i v i a l " r o o t s0 = - 2 , - 4 , - 6 , . . . , w h i c h y i e l d a n e g l i -g i b l e c o n t r i b u t i o n t o t h e f or m u l a) a r ec o m p l e x n u m b e r s w h o s e r e a l p a r t s l i e b e -t w e e n 0 a n d 1 . T h e f i r s t t e n o f t h e m a r ea s f o l l o w s ( 1 9 ) :

    IP l = ~ + 1 4 . 1 3 4 7 2 5 i ,IP 2 = ~ + 2 1 . 0 2 2 0 4 0 i ,IP 3 = ~ + 2 5 . 0 1 0 8 5 6 i ,IP 4 = ~ + 3 0 . 4 2 4 8 7 8 i ,IP 5 = ~ + 3 2 . 9 3 5 0 5 7 i ,

    Pl ~ 8 9 - 1 4 . 1 3 4 7 2 5 i ,IP 2 = ~ - 2 1 . 0 2 2 0 4 0 i ,1P 3 = ~ - 2 5 . 0 1 0 8 5 6 i ,

    54 = 8 9 - 3 0 . 4 2 4 8 7 8 i ,55 = 8 9 - 3 2 . 9 3 5 0 5 7 i .

    I t i s e a s y t o s h o w t h a t w i t h e a c h r o o ti t s c o m p l e x c o n j u g a t e a l s o a p p e a r s . B u tt h a t t h e r e a l p a r t o f e v e r y r o o t i s e x -a c t l y I / 2 i s s t i l l u n p r o v e d : t h i s i s t h ef a m o u s R i e m a n n h y p o t h e s i s , w h i c h w o u l dh a v e f a r - r e a d h i n g c o n s e q u e n c e s f o r n u m -b e r t h eo r y( 2 0) . I t h a s b e e n v e r i f i e d f o r7 m i l l i o n r o o t s .W i t h t h e h e l p o f t h e R i e m a n n f u n c t i o nR (X ) i n t r o d u c e d a b o v e w e c an w r i t e R i e -m a n n ' s f o r m u l a i n t h e f o r m

    ~ ( x ) = R ( x ) - Z R ( x p )P

    T h e k t h a p p r o x i m a t i o n t o ~ (x) w h i c h t h i sf o r m u l a y i e l d s i s t h e f u n c t i o n

    R k ( X ) = R( x ) + T I ( X ) + T 2 ( x ) + . . . + T k ( X ) ,

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    15w h e r e T n ( X ) = - R ( x P n ) - R ( x p n ) i s t h ec o n t r i b u t i o n o f th e n t h p a i r o f r o o t s o ft h e z e t a f u n c t i o n . F o r e a c h n t h e f u n c -t i o n Tn ( x ) i s a s m o o t h , o s c i l l a t i n g f u n c -t i o n o f x. T h e f i r s t f e w l o o k l i k e t h i s(21):

    0.30.2 io ~

    -0.1-0 .2- 0 . 3

    ~ 0N= T1 ( x )

    0 .2

    0 .1i/ ~ N= T2 ( X )

    ~.

    -0"I- 0 . 2

    N T3 ( x )

    0 .20.1

    0-0.1-0.2~ / ~~o N Tz. (x )

    ~ A~ A A A

    I t f o l l o w s t h a t R k ( x ) i s a l s o a s m o o t hf u n c t i o n f o r e a c h k . A s k g r o w s , t h e s ef u n c t i o n s a p p r o a c h ~ ( x ) . H e r e, f o r e x a m -p l e , a r e t h e g r a p h s o f t h e 1 0 t h a n d 2 9 t ha p p r o x i m a t i o n s ,

    15--

    XI I ..-50 100

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    16

    R 2 9 ( x )

    I I50 t00

    X t x ). . . . .. . . . . .. . . R 1 0 ( x )R 2 9 ( x )

    I I X50 100

    a n d i f w e c o m p a r e t h e s e c u r v e s w i t h t h eg r a p h o f ~ ( x ) u p t o 1 0 0 ( p. 9) w e g e tt h e f o l l o w i n g p i c t u r e :

    I h o p e t h a t w i t h t h i s a n d t h e o t h e rp i c t u r e s I h a v e s h o wn , I h a v e c o m m u n i c a t -e d a c e r t a i n i m p r e s s i o n o f t h e i m m e n s e

    i b e a u t y o f t h e p r i m e n u m b e r s a n d o f t h ei e n d l es s s u r p r i s e s w h i c h t h e y h a v e i ni s t o r e f o r u s .

    Remarks

    (I) J. M . G a n d h i , F o r m u l a e f o r t h e n t hp r i m e , P r oc . W a s h i n g t o n S t a t e U ni v .C o n f. o n N u m b e r T h e o r y , W a s h i n g t o nS t a t e U n i v . , P u l l m a n , W a s h . , 1 9 719 6 - 1 0 6

    (2) J . P . J o n e s , D i o p h a n t i n e r e p r e s e n t a -t i o n o f t h e s e t o f p r i m e n u m b e r s ,N o t i c e s o f t h e A M S 2 2 ( 1 9 7 5 ) A -3 2 6 .

    (3) T h e r e i s a g o o d r e a s o n w h y s o m a n yo f t h e n u m b e r s i n t h i s l i s t h a v et h e f o r m M k = 2 k - 1: A t h e o r e m o fL u c a s s t a t e s t h a t M k ( k >2 ) i s p r i m ei f a n d o n l y i f M k d i v i d e s L k - 1 ,w h e r e t h e n u m b e r s L n a r e d e f i n e d i n -d u c t i v e l y b y L I = 4 a n d L n + I = L ~ - 2( s o L 2 = 1 4 , L 3 = 1 9 4 , L 4 = 3 7 6 3 4 ,. .. ) a n d h e n c e i t i s m u c h e a s i e r t ot e s t w h e t h e r M k i s p r i m e t h a n i t i st o t e s t a n o t h e r n u m b e r o f t he s a m eo r d e r o f m a g n i t u d e .

    T h e p r i m e n u m b e r s o f t h e f o r m2 K - I ( f o r w h i c h k i t s e l f m u s t n e c -

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    11/13

    e s s a r i l y b e p r i me ) a r e c a l l e d M e r -s e n n e p r i m es ( a f te r t h e F r e n c h m a t h -e m a t l c i a n M e r s e n n e w h o i n 1 64 4 g a v e9 79a l i s t o f s u c h p r i m e s u p t o 1 0 ,c o r r e c t u p t o 1 0 18 ) a n d p l a y a r o l ei n c o n n e c t i o n w i t h a c o m p l e t e l y d i f -f e r e n t p r o b l e m o f n u m b e r t h e o r y .E u c l i d d i s c o v e r e d t h a t w h e n 2 P - Ii s p r i m e t h e n t h e n u m b e r 2 P - I ( 2 P -I )i s " p e r f e c t " , i . e . i t e q u a l s t h es u m o f i t s p r o p e r d i v i s o r s ( e. g.6 = I + 2 + 3 , 2 8 = I + 2 + 4 + 7 +% 4 , 4 9 6 = I + 2 + 4 8 + 1 6 + 3 1 +6 2 + 1 2 4 + 24 8 ) a n d E u l e r s h o w e dt h a t e v e r y e v e n p e r f e c t n u m b e r h a st h i s f o r m. I t i s u n k n o w n w h e t h e rt h e r e a r e a n y o d d p e r f e c t n u m b e r sa t a ll ; i f t h e y e x i s t , t h e y m u s t b ea t l e a s t 1 0 Q0 . T h e r e a r e e x a c t l y2 4 v a l u e s o f p < 2 0 , 0 0 0 f o r w h i c h2P - 1 is pri me.

    ( 4 ) C . F . G a u s s , W e r k e I I ( 1 8 9 2 ) 4 4 4 -4 4 7 . F o r a d i s c u s s i o n o f t he h i s t o -r y o f th e v a r i o u s a p p r o x i m a t i o n s t o~ ( x) , i n w h i c h a n E n g l i s h t r a n s l a -t i o n o f t h i s l e t t e r a l s o a p p e a r s ,s e e L . J . G o l d s t e i n : A h i s t o r y o ft h e p r i m e n u m b e r t h e o r e m , A m e r .M a t h . M o n t h l y , 8 0 ( 1 9 7 3 ) 5 9 9 - 6 1 5 .

    (5) A. M . L e g e n d r e , E s s a i s u r l a t h e o r i ed e N o m b r e s , 2 n d e d i t i o n , P a r i s ,1808, p. 394.( 6 ) M o r e p r e c i s e l y

    Ls(x) - 1.5 < Li(x) < Ls( x),i .e . t h e d i f f e r e n c e b e t w e e n L i( x )a n d L s( x ) i s b o u n d e d . W e s h o u l d a l s om e n t i o n t h a t t h e l o g a r i t h m i c i n t e -g r a l i s o f t e n d e f i n e d a s t h e C a u c h yp r i n c i p a l v a l u e

    X d t = l i m / i- ~ d tLi(x) = o log t

    )s oTG~qI - 5b u t t h i s d e f i n i t i o n d i f f e r s f r o mt h a t g i v e n i n t he t e x t o n l y b y ac o n s t a n t .

    (7) T h e c o e f f i c i e n t s a r e f o r m e d a s f o l -l o w s : t h e c o e f f i c i e n t o f L i ( n / x ) i s+ 1 / n i f n i s t h e p r o d u c t o f a n e v e nn u m b e r o f d i s t i n c t p r i m e s , - I / n i fn i s t h e p r o d u c t o f a n o d d n u m b e ro f d i s t i n c t p r i m e s , a n d 0 if n c o n -t a i n s m u l t i p l e p r i m e f a c t o r s .

    17(8) R a m a n u j a n h a s g i v e n t h e f o l l o w i n ga l t e r n a t i v e f o r m s f o r t h i s f u n c t i o n :

    ( l O g X) "t d tR(x) = 7 t r(t +i) ~(t +l)o( ~( s) = t h e R i e m a n n z e t a f u n c t i o na n d F ( s) = t h e g a m m a f u n c t i o n ) a n d

    R ( e 2 ~ X ) ' 2 ( 2 x 4 3 6 5 1= ~ ~ + 3 - -~ 4 x + ~ x + . .( 2 5 2 5 )_ ~2 12 x + 40 x 3 + --~--x +. ..

    ( B = k t h B e r n o u l l i n u m b e r ; t h e s y m -b o l ~ m e a n s t h a t t he d i f f e r e n c e o ft h e t w o s i d e s t e n d s t o O a s x g r o w st o i n f i n i t y ) . S e e G . H . H a r d y , R a m a -n u j a n: T w e l v e L e c t u r e s o n S u b j e c t sS u g g e s t e d b y Hi s L i f e a n d W or k , C a m -b r i d g e U n i v e r s i t y P r e s s , 1 9 40 , C h a p -ter 2.

    (9) N a m e l y : T h e p r o b a b i l i t y t h a t f o r ar a n d o m l y c h o s e n p a i r ( m, n) o f n u m -b e r s b o t h m a n d n a r e ~ 0 ( m o d p )i s o b v i o u s l y [ ( p - 1) / p ] 2 w h i l e f o ra r a n d o m l y c h o s e n n u m b e r n , t h e.p r o b a b i l i t y t h a t n a n d n + 2 a r e b o t hO (mod p) is I /2 for p = 2 an d( p - 2 ) / p f o r p ~ 2. T h u s t h e p r o b a -b i l i t y f o r n a n d n + 2 m o d u l o p t o b ep r i m e t w i n s d i f f e r s b y a f a c t o rp - 2 2p

    for p r 2 and by 2 for p = 2 fromt h e c o r r e s p o n d i n g p r o b a b i l i t y f o rt w o i n d e p e n d e n t n u m b e r s m a n d n . A l -t o g e t h e r , w e h a v e t h e r e f o r e i m p r o v e do u r c h a n c e s b y a f ac t o r

    C = 2p2 _ 2p

    p>2 p2 _ 2p + Ip p r i m e= 1 . 3 2 0 3 2 . F o r a s o m e w h a t m o r e

    c a r e f u l p r e s e n t a t i o n o f t h i s a r g u -m e n t , s e e H a r d y a n d W r i g h t , A n I n -t r o d u c t i o n t o th e T h e o r y o f N u m b e r s ,C l a r e n d o n P r e s s , O x f or d , 1 9 60 ,w 22 .2 0 (p. 371 - 373)( 1 0 ) M . F . J o n e s , M . L a l , a n d W . J . B l u n d o n ,S t a t i s t i c s o n c e r t a i n l a r g e p r i m e s ,Math . Co mp. 21 (1967) 103 - 107.( 1 1) D . S h a n k s , O n m a x i m a l g a p s b e t w e e ns u c c e s s i v e p r i m e s , M a t h . C o m p . 1 8( 1 96 4 ) 6 4 6 - 6 5 1 . T h e g r a p h o f g ( x )w a s m a d e f r o m t h e t a b l e s f o u n d i n

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    18t h e f o l l o w i n g p a p e r s: L . J. L a n d e ra n d T. R . P a r k i n , O n f i r s t a p p e a r -a n c e of p r i m e d i f f e r e n c e s , M a t h .C o m p . 2 1 ( 1 9 6 7 ) 4 8 3 - 4 8 8 , R . P .B r e n t , T h e f i r s t o c c u r r e n c e o f l a r g eg a p s b e t w e e n s u c c e s s i v e p r i m e s ,M a t h . C o m p . 2 7 ( 1 97 3 ) 9 5 9 - 9 6 3 .

    ( 12 ) T h e d a t a f o r t h i s g r a p h a r e t a k e nf r o m L e h m e r ' s t a b l e o f p r i m e n u m b e r s( D .N . L e h m e r , L i s t o f P r i m e N u m b e r sf r o m I t o 1 0 , O O 6 , 7 2 1 , H a f n e r P u b -l i s h i n g C o . , N e w Y o r k , 1 9 5 6 ) .( 13 ) T h i s a n d t h e f o l l o w i n g g r a p h w e r em a d e u s i n g t h e v a l u e s o f ~( x) f o u n di n D . C. M a p e s , F a s t m e t h o d f o r c o m -p u t i n g t h e n u m b e r o f p r i m e s l e s st h a n a g i v e n l i m i t , M a t h . C o m p . 1 7( 1 9 63 ) 1 7 9 - 1 85 . I n c o n t r a s t t o

    L e h m e r ' s d a t a u s e d i n t he p r e v i o u sg r a p h , t h e s e v a l u e s w e r e c a l c u l a t e df r o m a f o r m u l a f o r ~ (x ) a n d n o t b yc o u n t i n g t h e p r i m e s u p t o x .( 14 ) S. S k e w e s , O n t h e d i f f e r e n c e ~ ( x) -l i ( x ) ( I ), J . L o n d o n M a t h . S o c , 8( 1 9 33 ) 2 7 7 - 2 8 3 . S k e w e s ' p r o o f o ft h i s b o u n d a s s u m e s t h e v a l i d i t y o ft h e R i e m a n n h y p o t h e s i s w h i c h w e d i s -c u s s la t e r. T w e n t y - t w o y e a r s l a t e r( O n t h e d i f f e r e n c e ~ ( x ) - l i ( x ) ( I I ) ,P r o c . L o n d . M a t h . S o c . ( 3) 5 ( 1 9 5 5)4 8 - 7 0) h e p r o v e d w i t h o u t u s i n g t h eR i e m a n n h y p o t h e s i s t h a t t h e r e e x i s t sa n x l e s s t h a n t h e ( y e t m u c h l a r g e r )b o u n d 1 0 1 0 1 0 9 6 4

    f o r w h i c h ~ ( x) > L i ( x ) . T h i s b o u n dh a s b e e n l o w e r e d t o 1 0 1 0 5 2 9 . 7b y C o h e n a n d M a y h e w a n d t o 1 . 6 5 x i O 1165b y L e h m a n ( On th e d i f f e r e n c e~ ( x) - l i ( x ) , A c t a A r i t h m . 1 1 ( 1 9 66 )3 9 7 - 4 1 0 ). L e h m a n e v e n s h o w e d t h a tt h e r e i s a n i n t e r v a l o f a t l e a s t1 05 00 n u m b e r s b e t w e e n 1 . 5 3 a n d 1 . 6 5 x i O I 16 5 w h e r e ~ ( x ) i s l a r g e rt h a n L i ( x ) . A s a c o n s e q u e n c e o f h i si n v e s t i g a t i o n , i t a p p e a r s l i k e l yt h a t t h e r e i s a n u m b e r n e a r 6 . 6 6 3x 1 0 3 70 w i t h ~ ( x ) > L i ( x ) a n d t h a tt h e r e i s n o n u m b e r l e s s t h a n 1 02 0w i t h t h i s p r o p e r t y .

    ( 15 ) N a m e l y ( as c o n j e c t u r e d b y G a u s s i n1 7 9 6 a n d p r o v e d b y M e r t e n s i n 1 8 74 )

    7 1 = l o g l o g x + C + rp < x p

    w h e r e ~ (x ) 0 a s x t e n d s t o i n f i n -i t y a n d C ~ 0 . 2 6 1 4 9 7 i s a c o n s t a n t .T h i s e x p r e s s i o n i s s m a l l e r t h a n 3 .3w h e n x = 1 0 9 , a n d e v e n w h e n x = 1 0 18i t s t i l l l i e s b e l o w 4 .

    (16) P . L . C h e b y s h e v , R e c h e r c h e s n o u v e l l e ss u r le s n o m b r e s p r e m i e r s , P a r i s ,1 8 5 1 , C R P a r i s 2 9 ( 1 8 4 9 ) 3 9 7 - 4 0 1 ,7 3 8 - 7 3 9. F o r a m o d e r n p r e s e n t a t i o n( i n G e r m a n ) o f C h e b y s h e v ' s p r o o f ,s e e W . S c h w ar z , E i n f U h r u n g i n M e t h o -d e n u n d E r g e b n i s se d e r P r i m z a h l t h e o -r i e B I - H o c h s c h u l t a s c h e n b u c h 2 7 8 / 2 7 8 a ,M a n n h e i m 1 9 6 9 , C h a p t . I I . 4 , P. 42 -4 8 .

    ( 17 ) T h e l a r g e s t p o w e r o f p d i v i d i n g p !i s p [ n / P ] + [ n T p 2 ] + . . . , w h e r e [x ] i st h e l a r g e s t i n t e g e r S x . T h u s i n t h en o t a t i o n o f t h e l e m m a w e h a v e

    V p = r~17. I V ] - [ ~ - [ i )E v e r y s u m m a n d i n t h i s s u m i s e i t h e rO o r I a n d i s c e r t a i n l y O f o rr > ( l o g n / l o g p ) ( s i n c e t h e n [ n / p r ]= O ) . T h e r e f o r e 9 p S ( l o g n / l o g p ) ,f r o m w h i c h t h e c l ~ i m f o l l o w s .

    ( 18 ) T h e d e f i n i t i o n o f ~ ( s) a s I + I / 2 s+ I / 3 s + . .. g i v e n a b o v e m a k e s s e n s eo n l y w h e n s is a c o m p l e x n u m b e rw h o s e r e a l p a r t i s l a r g e r t h a n I( s i n c e t h e s e ri e s c o n v e r g e s f o rt h e s e v a l u e s o f s o n l y ) a n d i n t h i sd o m a i n ~ (s ) h a s n o z e r o e s . B u t t h ef u n c t i o n ~ (s ) c a n b e e x t e n d e d t o af u n c t i o n f o r a l l c o m p l e x n u m b e r s s ,s o t h a t i t m a k e s s e n s e t o s p e a k o fi t s r o o t s i n t h e w h o l e c o m p l e xp l a n e . T h e s i m p l e s t w a y t o e x t e n dt h e d o m a i n o f d e f i n i t i o n o f ~ (s ) a tl e a s t t o t h e h a l f , p l a n e R e ( s) > Oi s t o u s e t h e i d e n t i t y

    ( 1 - 2 1 - s ) ~ ( s ) = I + 1 + I _~ + . . .2 s 3 s

    - 2 + 4- 6 s .. .= ( _ I ) n - 1

    n=l n s 'w h i c h i s v a l i d f o r R e ( s ) > I , a n dt o o b s e r v e t h a t t h e s e r i e s o n t h er i g h t c o n v e r g e s f o r a l l s w i t h p o s -i t i v e r e a l p a r t . W i t h t h i s , t h e " i n -t e r e s t i n g " r o o t s o f t he z e t a fu n c -t i o n , i . e . t h e r o o t s p = B + i y w i t hO < 8 < I, c a n b e c h a r a c t e r i z e d i n

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    an elementary way by the two equa-tions

    n~ I (-I)n-In 8n~ 1 (-I)n-In 8

    cos (y log n) = O,

    sin (yl og n) = O.

    The sum over the roots p in Rie-mann's formula is not absolute lyconverge nt and therefore must besummed in the proper order (i.e.,according to increasing ab solutevalue of Im(p)).Finally, I should mention that,although Riema nn stated the exactformul a for ~(x) al read y in 1859,it wasn 't prov ed until 1895 (by vonMangoldt).

    (19) These roots were ca lcul ated al read yin 1903 by Gra m (J.-P. Gram, Surles zeros de la fonct ion ~(s) deRiema nn, Ac ta Math. 27 (1903) 289 -304). For a very nice presenti on ofthe theory of Riemann' s zeta func-tion, see H.M. Edward s, R iema nn'sZeta Function, Acade mic Press, NewYork, 1974.

    (20) Namel y the Rie man n hypoth esis im-plies (and in fact is equi vale nt tothe statement) that the error inGauss's ap proxi mation Li(x) to ~(x)is at most a constan t timesxl/2.1og x. At the present it iseven unknown whethe r this error issmaller than xC for any constantc < I.

    (21) This and the following graphs aretaken from H. Ries el and G. G6hl,Some calculations related to Rie-mann's prime number formula, Math.Comp. 24 (1970) 969 - 983.

    %/

    The 1977 Mathematics Calendar has found aworthy successor:

    19

    PsychologyCalendar '78containing a selection of twelve themeswhich portray something of the excitingspirit which pervades psyc hological in-quiry:

    DreamingCognitive ContoursSocial Dominance in MonkeysMcCollough EffectDynamic Effects of Repetitive PatternsHarlow and PiagetShadow and LightMemory Span TestWilhelm Wundt Father of PsychologyRecognit ion of FacesSplit BrainJerome Bruner's Theory of CognitiveLearning

    P.S. No cause for concern. The next MathematicsCalendar will appear in 1979.