APPENDIX:
On C o m p o s i t e
Numbers
sy
J° L. Nicolas
Highly
composite
d(n)
where
~ d(m)
numbers
famous.
numbers
topics
in D e c e m b e r
and other
volume
in great
a manuscript
of another
papers
long paper
on h i g h l y
composite
did not publish.
was
copy
r e l e a s ed
of R a m a n u j a n
unpublished)
is included
is not h a n d w r i t t e n
of this m a n u s c r i p t
on h i g h l y
[3] is q u i t e
numbers
During his
and in this
composite
(pages
295 of
centennial
Notebook
numbers
It is to be
[2] the words
numbers
and
impressive
280-308).
by Ramanujan.
composite
studied h i g h l y
[2] of his Lost
on h i g h l y
that at the top of page
paper"
Ramanujan
and his
the first p u b l i s h e d
unpublished
however,
satisfying
(I)
Srinivasa
detail
that Ramanujan
1987,
(previously
noted,
function.
But there was much work
related
integers
for all m < n,
d is the divisor
composite
n are p o s i t i v e
A short
is given
- "Middle
analysis
in [i] po
238-
239.
The table
on page
numbers.
This
composite
numbers
d(n)
Note the
largely
table,
table
has p o i n t e d
composite.
out,
coincides
with
the
the w e a k e r
composite
list of largely
inequality
for all m ~ n.
numbers
4200,
[2] is not a list of h i g h l y
satisfy
difference
composite
namely,
almost
n which
> d(m)
slight
280 of
between
which
151200,
the number
(2)
(I) and
(2).
were
omitted
415800,
491400.
15080 in this
There are only
by R a m a n u j a n
Also,
table
four
in this
as J. P. Massias
is not largely
19
In this
interesting
context
unpublished
results
we p o i n t
for n ) 5041.
on
out
Here
manuscript
g(n),
the
a result
a l s o has
sum of the d i v i s o r s
due
y is E u l e r ' s
Ramanujan
to R o b i n
[4]
constant.
that
More
some
very
of n.
~(n)
In this
g eY nloglogn
precisely
he
showed
that
~(N)
NloglogN
4 eYexp
{2(i-/~) + c + 0
~
log x
where
c = y + 2 - log
In
(3),
1
(
/x log
(3)
2 )}'
x
4 ~.
N is a c o l l o s s a l y
abundant
number
of p a r a m e t e r
x and
for
such
n we h a v e
log N =
under
[
log p + 0(/~) = x + 0 (/~ log2x)
p4x
p=prime
t h e a s s u m p t i o n of the R i e m a n n H y p o t h e s i s .
Using
rewrite
(3)
~(~)
with
p.
down
notation
2(i-/~) + c
~ N
loglogN
a similar
Z_I(N)
(Z_I(N)
Unfortunately
+ 0
formula
for the
so the
([2],
right
eY(y-log
The w r o n g
(377)
of
[2],
formula
sign
term
(377)
the w r o n g
As
and
sign!
N)
(6) do not
p.
303)
the
hand
side
of
1
(
.............. 2)}.
~i-o-~ (loglogN)
about
maximal
seventy
years
of ~(N)
order
(5)
earlier
(see
[2],
~i--o~ ( e Y ( 2 / 2
agree;
sign
(6)
+ y - log
it seems
of the
should
term
that
4~).
(6)
in f o r m u l a
2(/~-l)//logN
(382)
is w r o n g
read
4~ + 2 ( 2 - ~ ) ) .
seems
[2].
the
- eYloglog
(5) a n d
of R a m a n u j a n
of
+
303):
rim
and
4 eY{l
wrote
the
(4) we m a y
as
NioglogN
Ramanujan
(4)
to come
Ramanujan
from
Ramanujan's
explains
at the b e g i n n i n g
(logN) 1/2- S / l o g l o g N
in f o r m u l a
(379)
analysis
the
arises
from
coefficient
four
of his
of §71,
terms
of this
formula
p.
302
of
term has
20
In t h e
same m a n u s c r i p t
the m a x i m a l
obtain.
order
This
of
~ ( n ) / n s for all
result
of R a m a n u j a n
~ ( n ) / n s for s ¢ 1 u n d e r
new
(and h a s
worthwhile
the
not y e t b e e n
to
look
Ramanujan
into
has
s, w h i c h
of the
rediscovered!)
further.
and
I hope
nice
estimation
is not at all
on t h e m a x i m a l
assumption
this
a very
easy
to
Hypothesis
is
order
Riemann
it w i l l
of
of
definitely
to do this
be
on a later
occasion.
REFERENCES
i)
J.
L.
Nicolas,
Revisited",
Academic
2)
3)
4)
Proc.
Press,
S. R a m a n u j a n ,
Narosa
On h i g h l y
Centenary
NY
(1987),
"The L o s t
Publishing
Highly
2,
p.
(1915),
New
in
Univ.
"Ramanujan
Illinois,
Urbana,
215-244.
and o t h e r
Delhi
composite
unpublished
(1987),
numbers,
p.
Proc.
papers",
280-308.
London
Math.
Soc.,
347-409.
G. Robin,
Grandes
hypothese
de R i e m a n n ,
p.
numbers,
Conference,
p.
Notebook
House,
S. R a m a n u j a n ,
14
composite
valeurs
J.
de
la f o n c t i o n
Math.
pures
somme
et appl.,
187-213.
Department
Universite
Lyon
i,
69622
of M a t h e m a t i c s
Claude-Bernard
Villeurbanne
France
Cedex
des
63
diviseurs
(1984),
et