A P P E N D I X : On C o m p o s i t e N u m b e r s sy
J° L. N i c o l a s
H i g h l y c o m p o s i t e n u m b e r s n are p o s i t i v e i n t e g e r s s a t i s f y i n g
d(n) ~ d(m) for all m < n, (I)
w h e r e d is t h e d i v i s o r f u n c t i o n . S r i n i v a s a R a m a n u j a n s t u d i e d h i g h l y c o m p o s i t e n u m b e r s in g r e a t d e t a i l a n d his long p a p e r [3] is q u i t e famous. But t h e r e was m u c h w o r k on h i g h l y c o m p o s i t e n u m b e r s a n d r e l a t e d t o p i c s t h a t R a m a n u j a n did not p u b l i s h . D u r i n g his c e n t e n n i a l in D e c e m b e r 1987, t h e f i r s t p u b l i s h e d c o p y [2] of h i s L o s t N o t e b o o k and o t h e r u n p u b l i s h e d p a p e r s w a s r e l e a s e d and in this i m p r e s s i v e v o l u m e a m a n u s c r i p t of R a m a n u j a n on h i g h l y c o m p o s i t e n u m b e r s
( p r e v i o u s l y u n p u b l i s h e d ) is i n c l u d e d (pages 2 8 0 - 3 0 8 ) . It is to b e noted, h o w e v e r , t h a t at the t o p of p a g e 295 of [2] the w o r d s - " M i d d l e of a n o t h e r p a p e r " is not h a n d w r i t t e n by R a m a n u j a n . A short a n a l y s i s of t h i s m a n u s c r i p t on h i g h l y c o m p o s i t e n u m b e r s is g i v e n in [i] po 238- 239.
The t a b l e on p a g e 280 of [2] is not a list of h i g h l y c o m p o s i t e n u m b e r s . This t a b l e a l m o s t c o i n c i d e s w i t h the list of l a r g e l y c o m p o s i t e n u m b e r s n w h i c h s a t i s f y the w e a k e r i n e q u a l i t y
d(n) > d(m) for all m ~ n. (2)
Note t h e s l i g h t d i f f e r e n c e b e t w e e n (I) a n d (2). T h e r e are o n l y four l a r g e l y c o m p o s i t e n u m b e r s w h i c h w e r e o m i t t e d by R a m a n u j a n in this table, namely, 4200, 151200, 415800, 4 9 1 4 0 0 . Also, as J. P. M a s s i a s h a s p o i n t e d out, the n u m b e r 1 5 0 8 0 in t h i s t a b l e is n o t l a r g e l y c o m p o s i t e .
19
In t h i s u n p u b l i s h e d m a n u s c r i p t R a m a n u j a n a l s o h a s s o m e v e r y i n t e r e s t i n g r e s u l t s o n g(n), t h e s u m o f t h e d i v i s o r s of n. In t h i s c o n t e x t w e p o i n t o u t a r e s u l t d u e to R o b i n [4] t h a t ~(n) g e Y n l o g l o g n for n ) 5041. H e r e y is E u l e r ' s c o n s t a n t . M o r e p r e c i s e l y h e s h o w e d t h a t
w h e r e
~(N) 1
N l o g l o g N 4 e Y e x p { 2 ( i - / ~ ) + c + 0 ( 2 )}'
~ log x / x log x
c = y + 2 - log 4 ~.
(3)
l o g N = [ log p + 0(/~) = x + 0 (/~ l o g 2 x ) (4)
p 4 x p = p r i m e
u n d e r t h e a s s u m p t i o n of t h e R i e m a n n H y p o t h e s i s . U s i n g (4) w e m a y r e w r i t e (3) as
~(~) 1
N i o g l o g N 4 e Y { l + 2 ( i - / ~ ) + c + 0 ( ... 2)}. (5)
~ N l o g l o g N ~i-o-~ ( l o g l o g N )
R a m a n u j a n w r o t e d o w n a s i m i l a r f o r m u l a a b o u t s e v e n t y y e a r s e a r l i e r
~(N) (see [2], w i t h t h e n o t a t i o n Z _ I ( N ) f o r t h e m a x i m a l o r d e r of
p. 303):
r i m (Z_I(N) - e Y l o g l o g N) ~i--o~ ( e Y ( 2 / 2 + y - log 4~). (6)
U n f o r t u n a t e l y (5) a n d (6) d o n o t a g r e e ; it s e e m s t h a t in f o r m u l a (382) o f R a m a n u j a n ([2], p. 303) t h e s i g n of t h e t e r m 2 ( / ~ - l ) / / l o g N is w r o n g a n d so t h e r i g h t h a n d s i d e of (6) s h o u l d r e a d
e Y ( y - l o g 4~ + 2 ( 2 - ~ ) ) .
T h e w r o n g s i g n s e e m s to c o m e f r o m R a m a n u j a n ' s a n a l y s i s of h i s f o r m u l a (377) of [2]. As R a m a n u j a n e x p l a i n s at t h e b e g i n n i n g of §71, p. 302 of [2], t h e t e r m (logN) 1/2- S / l o g l o g N a r i s e s f r o m f o u r t e r m s of f o r m u l a (377) a n d in f o r m u l a (379) t h e c o e f f i c i e n t of t h i s t e r m h a s t h e w r o n g s i g n !
n we h a v e
In (3), N is a c o l l o s s a l y a b u n d a n t n u m b e r o f p a r a m e t e r x a n d for s u c h
20
In t h e s a m e m a n u s c r i p t R a m a n u j a n h a s a v e r y n i c e e s t i m a t i o n o f t h e m a x i m a l o r d e r o f ~ ( n ) / n s f o r a l l s, w h i c h is n o t a t a l l e a s y t o o b t a i n . T h i s r e s u l t o f R a m a n u j a n o n t h e m a x i m a l o r d e r o f
~ ( n ) / n s f o r s ¢ 1 u n d e r t h e a s s u m p t i o n o f t h e R i e m a n n H y p o t h e s i s is n e w ( a n d h a s n o t y e t b e e n r e d i s c o v e r e d ! ) a n d it w i l l d e f i n i t e l y b e w o r t h w h i l e t o l o o k i n t o t h i s f u r t h e r . I h o p e t o d o t h i s o n a l a t e r o c c a s i o n .
R E F E R E N C E S
i)
2)
3)
4)
J. L. N i c o l a s , O n h i g h l y c o m p o s i t e n u m b e r s , in " R a m a n u j a n R e v i s i t e d " , P r o c . C e n t e n a r y C o n f e r e n c e , U n i v . I l l i n o i s , U r b a n a , A c a d e m i c P r e s s , N Y ( 1 9 8 7 ) , p. 2 1 5 - 2 4 4 .
S. R a m a n u j a n , " T h e L o s t N o t e b o o k a n d o t h e r u n p u b l i s h e d p a p e r s " , N a r o s a P u b l i s h i n g H o u s e , N e w D e l h i ( 1 9 8 7 ) , p. 2 8 0 - 3 0 8 .
S. R a m a n u j a n , H i g h l y c o m p o s i t e n u m b e r s , P r o c . L o n d o n M a t h . S o c . , 2, 14 ( 1 9 1 5 ) , p. 3 4 7 - 4 0 9 .
G. R o b i n , G r a n d e s v a l e u r s d e la f o n c t i o n s o m m e d e s d i v i s e u r s et h y p o t h e s e d e R i e m a n n , J. M a t h . p u r e s et a p p l . , 63 ( 1 9 8 4 ) ,
p. 1 8 7 - 2 1 3 .
D e p a r t m e n t o f M a t h e m a t i c s U n i v e r s i t e C l a u d e - B e r n a r d L y o n i, 6 9 6 2 2 V i l l e u r b a n n e C e d e x
F r a n c e
