7
sin
The First 50 Million
Prime Numbers*
Don Zagier
I -
for
~F(s)
s
sin -s
;
the a l g e b r a i s t
it is
"the c h a r a c t e r i s t i c
of a f i n i t e
field"
or
"a p o i n t
in Spec
~"
or
To my parents
"a n o n - a r c h i m e d e a n
valuation";
the 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 b e r s i n d u c t i v e l y b y the r e c u r s i o n (I)
Pn+1
=
1
[I - log2( ~ +
n
r~1
(-1)r
ISil<'~'<irSn
2 pi1"''pir
([x] = b i g g e s t
)]
- I
integer
S x);
and, f i n a l l y , the l o g i c i a n s h a v e r e c e n t iv b e e n d e f i n i n q the p r i m e s as t h e p o s i t i v e v a l u e s o f the p o l y n o m i a l (2)
F(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,
r,s,t,u,v,w,x,y,z)
=
I w o u l d l i k e to t e l l y o u t o d a y a b o u t a
subject which, although I have not worked
in it m y s e l f , has a l w a y s e x t r a o r d i n a r i l y
c a p t i v a t e d me, a n d w h i c h has f a s c i n a t e d
mathematicians from the earliest times
until the present - namely, the question
o f t h e d i s t r i b u t i o n of p r i m e n u m b e r s .
You certainly all know what a prime
n u m b e r is: it is a n a t u r a l n u m b e r b i g g e r
t h a n 1 w h i c h is d i v i s i b l e b y no o t h e r
n a t u r a l n u m b e r e x c e p t for I. T h a t a t
l e a s t is the n u m b e r t h e o r i s t ' s d e f i n i tion; o t h e r m a t h e m a t i c i a n s
sometimes have
other definitions. For the function-theo r i s t , for i n s t a n c e , a p r i m e n u m b e r is
an 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 tion
::The following article is a revised version of
the author's inaugural lecture ~Antrittsvorlesung) held on May 5, 1975 at Bonn University.
Additional remarks and references to the literature may be found at the end.
Translated from the German by R. Perlis..The
original German version will also be published
in Beihefte zu Elemente der Mathematik No. 15,
Birkh~user Verlag, Basel.
[k + 2] [1 - (wz+h+j-q) 2 (2n+p+q+z-e) 2 - (a2y2-y2+1-x2)2 ({e4+2e3}{a+1}2+1-o2) 2 (16{k+1}3{k+2}{n+1}2+1-f2)2
({(a+u~-u2a) 2-1}{n+4dy}2+1-{x+cu}2) 2
- (ai+k+1-l-i) 2 ({gk+2g+k+1}{h+j}+h-z) 2 (16r2y~{a2-1}+1-u2) 2 (p-m+l{a-n-1}+b{2an+2a-n2-2n-2}) 2 (z-pm+pla-p21+t{2ap-p2-1}) 2 (qTx+y{a-p-1}+s{2ap+2a-p2-2~-2})2
( a ~ 1 2 - 1 2 + 1 - m 2 ) 2 - (n+l+v-y) ].
But I hope that you are satisfied with
the f i r s t d e f i n i t i o n t h a t I gave.
T h e r e a r e two f a c t s a b o u t 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 of w h i c h I h o p e
to c o n v i n c e y o u so o v e r w h e l m i n g l y t h a t
t h e y w i l l b e p e r m a n e n t l y e n g r a v e d in y o u r
h e a r t s . T h e f i r s t is that, d e s p i t e t h e i r
s i m p l e d e f i n i t i o n a n d r o l e as the b u i l d ing b l o c k s of t h e n a t u r a l n u m b e r s , the
p r i m e n u m b e r s b e l o n g to t h e m o s t a r b i trary and ornery objects studied by mathe m a t i c i a n s : t h e y g r o w like w e e d s a m o n g
the n a t u r a l n u m b e r s , s e e m i n g to o b e y no
o t h e r l a w t h a n t h a t o f c h a n c e , a n d nob o d y c a n p r e d i c t w h e r e the n e x t o n e w i l l
sprout. T h e s e c o n d f a c t is e v e n m o r e
a s t o n i s h i n g , for it s t a t e s j u s t t h e o p posite: that the prime numbers exhibit
s t u n n i n g r e g u l a r i t y , t h a t t h e r e a r e laws
governing their behaviour, and that they
obey these laws with almost military precision.
8
To support the first of these claims,
let me begin by showing you a list of the
prime and c o m p o s i t e numbers up to 100
(where apart from 2 I have listed only
the odd numbers).
prime
2
3
5
7
11
13
17
19
23
29
31
37
41
compo s i te
43
47
53
59
61
67
71
73
79
83
89
97
9
15
21
25
27
33
35
39
45
49
51
55
57
63
65
69
75
77
81
85
87
91
93
95
99
2127 - 1 =
170141183460469231731687303715884105727.
Not u n t i l 1951, with the appearance of
electronic computers, were larger prime
numbers discovered. In the a c c o m p a n y i n g
table, you can see the data on the successive title-holders (3). At t h e m o m e n t ,
the lucky fellow is the 6 0 0 2 - d i g i t number 219937 - I (which I w o u l d n o t care to
write down); if you d o n ' t believe me, you
can look it up in the G u i n n e s s book of
records.
The l a r g e s t known prime number
and lists of the primes among the 1OO
numbers i m m e d i a t e l y preceding and following 10 million:
The prime numbers between 9,999,900 and
10,000,000
9,999,901
9,999,907
9,999,929
9,999,931
9,999,937
9,999,943
9,999,971
9,999,973
9,999,991
The prime numbers between 10,0OO,OOO and
10,0OO,100
10,0OO,019
I0,OOO,079
I hope you will agree that there is
no apparent r e a s o n why one number is
prime and another not. To the contrary,
upon looking at these numbers one has
the feeling of being in the presence of
one of the inexplicable secrets of creation. That even m a t h e m a t i c i a n s have not
penetrated this secret is perhaps m o s t
c o n v i n c i n g l y shown by the ardour w i t h
w h i c h they search for bigger and bigger
primes - with n u m b e r s which grow regularly, like squares or powers of two, nobody would ever bother writing d o w n examples larger than the p r e v i o u s l y known
ones, but for prime numbers people have
gone to a great deal of trouble to do
just that. For example, in 1876 Lucas
proved that the number 2 127 - I is prime,
and for 75 years this remained u n s u r p a s sed - which is perhaps not surprising
when one sees this number:
2127-
I
~ ( 2 1 4 8 + I)
114(2127180(2127-
I) + I
I)2+ I
~
~ ,
0
0
~-,~
~'~
~o
~O
~,~
o
39
1876
Lucas
44
1951
Ferrier
41 ~
79 ~
1951
Miller.+
Wheeler +
EDSAC I
1952
Lehmer +
Robinson
+ SWAC
2521 - I
157
2607 - I
21279 - I
22203 - I
183 I
386
664
22281 - I
687
23217 - 1
969
24253 - I
24423 - I
1281 ~
1332 #
29689 - I
2917
29941 - I
2993 ~
211213 - I
3376
219937 - 1
6002
1957
1961
Riesel +
BESK
Hurwitz +
Selfridge
+ IBM 7090
1963
Gillies +
ILIAC 2
1971
Tuckerman
+ IBM 360
More interesting, however, is the
question about the laws g o v e r n i n g prime
numbers. I have already shown you a list
of the prime numbers up to 1OO. Here is
the same information p r e s e n t e d g r a p h i c a l ly. The function d e s i g n a t e d ~(x) (about
which I will be c o n t i n u a l l y speaking from
now on) is the number of prime numbers
not e x c e e d i n g x; thus ~(x) begins with O
and jumps by I at every prime number 2,
3, 5 etc. A l r e a d y in this picture we can
see that, despite small oscillations,
~(x) by and large grows quite regularly.
9
no e x c e p t i o n to this rule. It is not difficult to flnd an empirical formula w h i c h
gives a good d e s c r i p t i o n of the growth
of the prime numbers. B e l o w 1OO there are
25 primes, that is, o n e - f o u r t h of the
numbers; b e l o w 10OO there are 168, or
about one-sixth; up to 10,OO0 there are
1229 prime numbers, i.e. a b o u t one-eighth.
If we extend this list, computing the
p r o p o r t i o n of prime numbers to natural
n u m b e r s up to one hundred thousand, one
million, etc., then we find the following table (in which the v a l u e s of ~(x),
listed so n o n c h a l a n t l y here, r e p r e s e n t
thousands of hours of dreary calculation).
~(x)
0
X
I
,,,.
100
50
But when I extend the domain of x - v a l u e s
from a hundred to fifty thousand, then
this r e g u l a r i t y b e c o m e s b r e a t h - t a k i n g l y
clear, for the graph now looks like this:
6000
5000
4000
3000
2000
1000
/
0
J
I
t
l
i
10000
20000
30000
40000
50000
x
For me, the smoothness with which this
curve climbs is one of the m o s t a s t o n i s h ing facts in m a t h e m a t i c s .
Now, w h e r e v e r nature reveals a pattern, there are sure to crop up scientists looking for the explanation. The
r e g u l a r i t y observed in the primes forms
x
~(x)
x/~(x)
10
1OO
1OOO
10,OO0
1OO,OOO
1,OOO,OOO
10,0OO,0OO
100,0OO,OOO
1,O00,0OO,OOO
10,OOO,OOO,OOO
4
25
168
1,229
9,592
78,498
664,579
5,761,455
50,847,534
455,052,512
2.5
4.0
6.0
8.1
10.4
12.7
15.O
17.4
19.7
22.0
Here we see that the ratio of x to
~(x) always jumps by a p p r o x i m a t e l y 2.3
when we go from a power of 10 to the
next. M a t h e m a t i c i a n s immediately recognize 2.3 as the logarithm of 10 (to the
base e, of course). Thus we are led to
c o n j e c t u r e that
x
~(x) ~ log x
where the sign ~ m e a n s that the ratio
~(x)/(x/log x) tends to I as x goes to
infinity. This r e l a t i o n s h i p (which was
not proved until 1896) is k n o w n as the
prime number ~eorem. Gauss, the g r e a t e s t
m a t h e m a t i c i a n of them all, d i s c o v e r e d it
at the age of fifteen by studying prime
number tables contained in a book of logarithms that had been given to him as a
present the previous year. T h r o u g h o u t his
life Gauss w a s keenly interested in the
d i s t r i b u t i o n of the prime n u m b e r s and he
m a d e e x t e n s i v e calculations. In a letter
to Enke (4) he d e s c r i b e s how he "very often used an idle quarter of an hour to
count through another chiliad [i.e., an
interval of 1,OO0 numbers] here and there"
~ntil finally he had listed all the prime
numbers up to 3 m i l l i o n (!) and compared
their d i s t r i b u t i o n w i t h the formula w h i c h
he had conjectured.
The prime number theorem s~ates that
~(x) is a s y m p t o t i c a l l y - i.e., with a relative error of 0% - equal to x/log x.
But if we compare the g r a p h of the function x/log x with that of ~(x), then we
10
see that, a l t h o u g h the function x / l o g x
q u a l i t a t i v e l y m i r r o r s the b e h a v i o u r of
~(x), it c e r t a i n l y does not agree with
~(x) s u f f i c i e n t l y well to explain the
smoothness of the latter:
6000
5000
4000
3000
6000
2000
5000
1000
&O00
~
~t~(x
I
10 000
log x
3000
X
20 000
30 000
40 000
501000 .
2000
1000
I
i
I
I
I
10 000
20 000
30 000
40 000
50 000
X
v
Therefore it is natural to ask for better
approximations. If we take another look
at our table of the ratios of x to ~(x),
we find that this ratio is almost exactly log x - I. W i t h a more careful calcul a t i o n and with m o r e detailed data on
~(x), Legendre (5) found in 1808 that a
p a r t i c u l a r l y good a p p r o x i m a t i o n is obtained if in place of I we subtract
1.08366 from log x, i.e.
x
~(x) % log x - 1.08366
Another good a p p r o x i m a t i o n to ~(x),
w h i c h was first given by Gauss, is obtained by taking as starting point the
empirical fact that the frequency of
prime numbers near a very large number x
is almost exactly I/log x. From this,
the number of prime numbers up to x
should be a p p r o x i m a t e l y given by the
logari~mic sum
_
LS(X)
I
I
log 2 + log---~
+
+
"'"
I
log-----~
or, what is e s s e n t i a l l y the same (6), by
the logari~mic integral
x
Li(x)
I
= 2/ lo--~-~t.
If we now compare the graph of Li(x) with
that of ~(x), then we see that w i t h i n the
accuracy of our picture the two c o i n c i d e
iexactly.
T h e r e is no point in showing you the picture of L e g e n d r e ' s a p p r o x i m a t i o n as well,
for in the range of the graph it is an
even better a p p r o x i m a t i o n to ~(x).
There is one more a p p r o x i m a t i o n which
I would like to mention. Riemann's research on prime numbers suggests that
the p r o b a b i l i t y for a large number x to
be prime should be even closer to 1/log x
if one c o u n t e d not only the prime numbers
but also the powers of primes, counting
the square of a prime as half a prime,
the cube of a prime as a third, etc. This
leads to the a p p r o x i m a t i o n
(x) + 89 ~(/~)
+ ~ ~(3~s
+ ... = Li(x)
or, equivalently,
(x) = Li(x)
- 89 ~ i ( ~ )
- 89 ~i(3/~)
_
. . .
(7)
The f u n c t i o n on the right side of this
formula is denoted by R(x), in honour of
Riemann. It represents an a m a z i n g l y good
a p p r o x i m a t i o n to ~(x), as the following
values show:
X
~ (X)
R(X)
I00,0OO,O00
200,000,000
300,000,000
400,000,000
500,000,000
600,000,000
700,000,000
800,000,000
900,000,000
1,000,OO0,000
5,761,455
11,078,937
16,252,325
21,336,326
26,355,867
31,324,703
36,252,931
41,146,179
46,009,215
50,847,534
5,761,552
11,O79,090
16,252,355
21,336,185
26,355,517
31,324,622
36,252,719
41,146,248
46,009,949
50,847,455
For those in
tle function
that R(x) is
given by the
ries
R(x)
= I +
where ~(n+1)
tion (8).
the audience who know a littheory, perhaps I m i g h t add
an entire function of log x,
rapidly c o n v e r g i n g power se-
~
I
n=1 n~ (n+1)
(log x) n
n!
'
is the R i e m a n n
zeta func-
11
At this point I should emphasize that
Gauss's and Legendre's approximations to
~(x) were obtained only empirically, and
that even Riemann, although he was led
to his function R(x) by theoretical considerations, never proved the prime number theorem. That was first accomplished
in 1896 by Hadamard and (independently)
de la Vall~e Poussin; their proofs were
based on Riemann's work.
While still on the theme of the pred i c t a b i l i t y of the prime numbers, I would
like to give a few m o r e numerical examples. As already mentioned, the probability for a number of the order of magnitude x to be prime is roughly equal to
I/log x; that is, the number of primes
in an interval of length a about x should
be approximately a/log x, at least if
the interval is long enough to make statistics meaningful, but small in comparison to x. For example, we would expect
to find around 8142 primes in the interval between 100 m i l l i o n and 100 m i l l i o n
plus 150,000 because
Interval
Prime
numbers
I
O
1OO,OO0,OOO1OO,150,OOO
1,OOO,O00,OOO1,OOO,150,OOO
10,OOO,O00,OOO10,OOO,150,OOO
100,OOO,O00,OOO100,000,150,000
1,000,000,000,0001,000,000,150,000
10,000,000,000,00010,000,000,150,000
100,000,000,000,000100,000,000,150,000
1,000,000,000,000,0001,000,000,000,150,000
log
150,OOO
(1OO,O00,OO0)
150,OOO
= 18.427
150~000
8142
I
~
8154
~
584 601
7238 7242 461 466
6514 6511
5922
374 389
5974 309 276
5429 5433 259 276
5011
5065 221 208
4653 4643 191
4343 4251
186
166 161
. % 8142
Correspondingly, the probability that
two random numbers near x are both prime
is approximately I/(log x) 2. Thus if one
asks how many prime twins (i.e. pairs of
primes differing by 2, like 11 and 13 or
59 and 61) there are in the interval from
x to x+a then we m i g h t expect approxim a t e l y a/(log x) 2. Actually, we should
expect a bit more, since the fact that n
is already prime slightly changes the
chance that n+2 is prime (for example
n+2 is then certainly odd). An easy heuristic argument (9) gives C.a/(log x) 2
as the expected number of twin primes in
the interval [x, x+a] where C is a .constant with value about 1.3 (more exactly:
C = 1.3203236316...). Thus between 1OO
m i l l i o n and 1OO m i l l i o n plus 150 thousand
there should be about
(1.32''')(18.427)2=
Prime
twins
584
As you can see, the agreement with the
theory is extremely good. This is especially surprising in the case of the
prime pairs, since it has not yet even
been proved that there are infinitely
many such pairs, let alone that they are
distributed according to the conjectured
law.
I want to give one last illustration
of the predictability of primes, namely
the problem of the gaps between primes.
If one looks at tables of primes, one
sometimes finds unusually large intervals, e.g. between 113 and 127, which
don't contain any primes at all. Let g(x)
be the length of the largest prime-free
interval or "gap" up to x. For example,
the largest gap below 200 is the interval from 113 to 127 just mentioned, so
g(2OO) = 14. Naturally, the number g(x)
grows very erratically, but a heuristic
argument suggests the asymptotic formula (II)
g(x)
pairs of prime twins. Here are data computed by Jones, Lal and Blundon (io) giv-ing the exact number of primes and prime
twins in this interval, as well as in
several equally long intervals around
larger powers of 10:
~
(log x) 2
In the following picture, you can see
how well even the wildly irregular function g(x) holds to the expected behaviour.
12
700
This picture, I think, shows w h a t the
person who decides to study number theory has let himself in for. As you can
see, for small x (up to a p p r o x i m a t e l y I
million) Legendre's a p p r o x i m a t i o n
x/(log x - 1.O8366) is c o n s i d e r a b l y better than Gauss's Li(x), but after 5 million Li(x) is better, and it can be shown
that Li(x) stays better as x grows.
600
500
&O0
300
200
100
0
10
102
10 3
10 4
10 5
106
107
108
109
1010
1011
1012
But u p to 10 m i l l i o n there are only
some 600 thousand prime numbers; to s h o w
you the promised 50 m i l l i o n primes, I
have to go not to 10 m i l l i o n but all the
way out to a b i l l i o n (American style:
10 ). In this range, the g r a p h of
R(x) - ~(x) looks like this(13) :
600~
R(x)-X(x)
/~
.A,.,, ,A
-300|
Up to now, I have s u b s t a n t i a t e d m y
claim about the o r d e r l i n e s s of the primes
m u c h m o r e t h o r o u g h l y than m y c l a i m about
their disorderliness. Also, I have not
yet fulfilled the promise of m y title to
s h o w you the first 50 m i l l i o n primes, but
have only shown you a few thousand. So
here is a graph of ~(x) compared w i t h the
a p p r o x i m a t i o n s of Legendre, Gauss, and
Riemann up to 10 m i l l i o n (12) . Since these
four functions lie so close together that
their graphs are i n d i s t i n g u i s h a b l e to the
naked eye - as we already saw in the picture up to 50,000 - I have plotted only
the differences b e t w e e n them:
AA
, fL
--
.;o
O~< x ~< 1,000,000,000
The o s c i l l a t i o n of the function K ( x ) - ~ ( x )
become larger and larger,;but even for
these a l m o s t i n c o n c e i v a b l y large values
of x they never go beyond a few hundred.
In c o n n e c t i o n with these data I can
m e n t i o n yet another fact about the number of prime numbers ~(x). In the picture up to 10 million, G a u s s ' s approximation was always bigger than ~(x). That
remains so until a billion, as you can
see in the picture on the following page
(in w h i c h the above data are plotted logarithmically).
X(X)+3OO-
z(x}+200i
~(x)+l1~176
3~'(x]1
~(x)-1ool
Legendre
V~
i,fV Vm
"
R,emann
,
x (m rmlilons)
M
-- Rlemann " V ~ / ~
/
Riemann
~ '
" -
~N
13
verges, i.e. e v e n t u a l l y exceeds any prev i o u s l y given number. His proof, which
is also very simple, used the function
200C
100C
~(s)
=
I
+
i_
+
2s
Lt(x)-R(x)
"~///
500
/
t
200
10C
10
I
20
I
30
i
50
I
70
I
1
100
200
x (in millions)
J
500
I
1000
S u r e l y this graph gives us the impression that with i n c r e a s i n g x the difference Li(x) - ~(x) grows steadily to infinity, that is, that the logarithmic integral Li(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 s
the number of primes up to x (this w o u l d
agree with the o b s e r v a t i o n that R(x) is
a better a p p r o x i m a t i o n than Li(x), since
R(x) is always smaller than Li(x)). But
this is false: it can be shown that there
are points where the o s c i l l a t i o n s of
R(x) - ~(x) are so big that ~(x) a c t u a l ly becomes larger than Li(x). Up to now
no such numbers have been found and perhaps none ever will be found, but Littlewood proved that they exist and Skewes(14)
proved that there is one that is smaller
than
I__ +
...
,
3s
whose importance for the study of ~(x)
was fully r e c o g n i z e d only later, with
the work of Riemann. It is amusing to rem a r k that, a l t h o u g h the sum of the reciprocals of all primes is divergent, this
sum over all the known primes (let's say
the first 50 million) is smaller than
four(15).
The first major result in the direction of the prime number t h e o r e m was
proved by C h e b y s h e v in 1850(16). He
showed that for s u f f i c i e n t l y large x
x x < ~(x)
0.89 log
< 1.11
x
log x
i.e., the prime number t h e o r e m is correct with a relative error of at most
11%. His proof uses binomial c o e f f i c i e n t s
and is so p r e t t y that I c a n n o t r e s i s t at
least sketching a simplified v e r s i o n of
it (with somewhat worse constants).
In the one direction, we will prove
x
(x) < 1.7 ~
.
This i n e q u a l i t y is valid for x < 12OO.
Assume i n d u c t i v e l y that it has been
proved for x < n and consider the m i d d l e
binomial c o e f f i c i e n t
10101034
(a number of which H a r d y once said that
it was surely the b i g g e s t that had ever
served any definite purpose in m a t h e m a t ics). In any case, this example shows
h o w u n w i s e it can be to base c o n c l u s i o n s
about primes solely on numerical data.
In the last part of my lecture I would
like to talk about some theoretical results about ~(x) so that you don't go
away w i t h the feeling of having seen only
e x p e r i m e n t a l math. A n o n - i n i t i a t e would
c e r t a i n l y think that the property of
being prime is m u c h too random for us to
be able to prove a n y t h i n g about it. This
was refuted already 2,200 years ago by
Euclid, who proved the existence of infinitely m a n y primes. His argument can
be formulated in one sentence: If there
were only finitely m a n y primes, then by
m u l t i p l y i n g them together and adding I,
one would get a number which is not div i s i b l e by any prime at all, and that is
impossible. In the 18th century, Euler
proved more, namely that the sum of the
r e c i p r o c a l s of the prime numbers di-
Since
22n=
n
(I+1)2n= (o2n)+(~n)+'''e(~n)+'''+(2n2n)
this c o e f f i c i e n t
other hand
is at m o s t 22n. On the
2n
(2n) !
(2n) x(2n-1) x...x2xl
(n ) = ~
= (nx(n-1)X.
x2xl)
"
Every prime p smaller than 2n appears in
the numerator, but c e r t a i n l y no p bigger
than n can appear in the denominator.
is d i v i s i b l e by every prime b e t w e e n n and
2n:
n<p<2n
P
2n
(n) 9
But the p r o d u c t has ~(2n) - ~(n)
each bigger than n, so we get
factors,
14
(2n) -~ (n)
n
or,
<
taking
~(2n)
2n
(n)
~
p<
n<p<2n
ram,
ways
ered
mula
22n
<
logarithms
- ~(n)
< 2n log 2 < 1 39
n
log n
"
~
B y induction, the t h e o r e m is v a l i d
so ~(n) < I .7 (n/log n), and a d d i n g
relations gives
for n,
these
(n > 1200).
H e n c e the t h e o r e m
Since
~(2n+I)
is v a l i d also for
< 1.7 l o g
2n.
n
+ I < 3.09 ~ o g n + I
< ~(2n)
2n+I
(2n+i")
(n > 1200)
it is a l s o v a l i d for 2n+I, c o m p l e t i n g the
induction.
For t h e b o u n d in the o t h e r d i r e c t i o n ,
w e need a s i m p l e l e m m a w h i c h c a n b e
p r o v e d e a s i l y u s i n g the w e l l - k n o w n form u l a for the p o w e r of p w h i c h d i v i d e s n!
(17) :
Lemma: Let p be a prime. If pVp is the largest power of p
then
pVP
Corollary:
(~),
dividing
+ 89
"
n
2n
< 3.09 ~
< 1.7
log n
log (2n)
~(2n)
he d i d s o m e t h i n g w h i c h is in m a n y
m u c h m o r e a s t o n i s h i n g - he d i s c o v a n exact f o r m u l a for ~(x). This forhas the f o r m
=
Li(x)
-
+ 89
Z
P
Li
+
(x p)
w h e r e the sum r u n s o v e r the r o o t s of the
zeta f u n c t i o n ~(s)(18) . T h e s e r o o t s
(apart f r o m the s o - c a l l e d "trivial" r o o t s
0 = -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 to the formula) are
c 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 lie between 0 a n d 1. The f i r s t t e n of them are
as follows(19):
I
Pl = ~ + 1 4 . 1 3 4 7 2 5
i,
I
P2 = ~ + 2 1 . 0 2 2 0 4 0
i,
I
P3 = ~ + 2 5 . 0 1 0 8 5 6
i,
I
P4 = ~ + 3 0 . 4 2 4 8 7 8
i,
I
P5 = ~ + 3 2 . 9 3 5 0 5 7
i,
Pl ~ 89 - 14.134725
i,
I
P2 = ~ - 2 1 . 0 2 2 0 4 0
i,
1
P3 = ~ - 2 5 . 0 1 0 8 5 6
i,
54 = 89 - 3 0 . 4 2 4 8 7 8
i,
55 = 89 - 3 2 . 9 3 5 0 5 7
i.
< n.
coefficient"(~)
Evergbinemial
satisfies
(~) =
~
p~n
pVp _< n~(n)
If w e add t h e i n e q u a l i t y
for all b i n o m i a l
of the c o r o l l a r y
coefficients
(~)
w i t h g i v e n n, t h e n w e find
2n =
(I+I) n =
and t a k i n g
n
Z
k=O
(~) S
logarithms
(n+l).n ~(n)
It is e a s y to s h o w t h a t w i t h e a c h r o o t
its c o m p l e x c o n j u g a t e a l s o appears. B u t
that the real p a r t of e v e r y r o o t is exa c t l y I/2 is still u n p r o v e d : this is the
famous Riemann hypothesis, which would
have f a r - r e a d h i n g c o n s e q u e n c e s for number theory(20). It has b e e n v e r i f i e d for
7 m i l l i o n roots.
W i t h t h e h e l p of the R i e m a n n f u n c t i o n
R(X) i n t r o d u c e d a b o v e w e c a n w r i t e R i e m a n n ' s f o r m u l a in the f o r m
gives
~(x)
~(n)
> n log 2
-
'log
n
> 2
n
3 log n
log
-
log
=
R(x)
-
Z
R ( x p)
P
(n+1)
n
(n > 200).
In closing, I w o u l d like to say a few
w o r d s a b o u t R i e m a n n ' s work. 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 the p r i m e n u m b e r t h e o -
The k th a p p r o x i m a t i o n to ~ (x) w h i c h
f o r m u l a y i e l d s is the f u n c t i o n
Rk(X)
= R(x)
+ TI(X)
+ T2(x)
this
+ . . . + Tk(X) ,
15
where Tn(X) = -R(xPn) - R(x pn) is the
c o n t r i b u t i o n of the nth pair of roots of
the zeta function. For each n the function Tn(x) is a smooth, oscillating function of x. The first few look like this
(21):
~
0
~
AA
A
A
0.3
0.2 i
o~
~
0
N
= T1 (x)
-0.1
It follows that Rk(x) is also a smooth
function for each k. As k grows, these
functions approach ~(x). Here, for example, are the graphs of the 10th and 29th
approximations,
-0.2
-0.3
0.2
i/
~
N
= T2 (X)
0.1
~
15--
O.
N
T3
(x)
-0"II
-0.2
0.2
0.1
0
-0.1
-0.2
~/
N
~~o
Tz. (x)
X
I
50
I
100
..-
16
Xtx)
.............. R10(x)
R29(x)
R29(x)
I
I
50
t00
a n d if w e c o m p a r e t h e s e c u r v e s w i t h the
g r a p h of ~(x) u p to 100 (p. 9) we g e t
the following picture:
Remarks
(I)
J.M. G a n d h i , F o r m u l a e for the n t h
prime, Proc. W a s h i n g t o n S t a t e Univ.
Conf. o n N u m b e r Theory, W a s h i n g t o n
S t a t e Univ., P u l l m a n , Wash., 1971
96 - 106
(2) J.P. Jones, 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 of the s e t o f p r i m e n u m b e r s ,
N o t i c e s of t h e A M S 22 (1975) A 326.
I
I
50
100
X
I h o p e t h a t w i t h this a n d the o t h e r
p i c t u r e s I h a v e shown, I h a v e c o m m u n i c a t ed a c e r t a i n i m p r e s s i o n of the i m m e n s e
ibeauty o f the p r i m e n u m b e r s and of the
iendless s u r p r i s e s w h i c h t h e y h a v e in
istore for us.
(3) T h e r e is a g o o d r e a s o n w h y so m a n y
of t h e n u m b e r s in t h i s l i s t h a v e
the f o r m M k = 2k - 1: A t h e o r e m of
L u c a s s t a t e s that M k (k>2) is p r i m e
if a n d o n l y if M k d i v i d e s Lk-1,
w h e r e the n u m b e r s L n a r e d e f i n e d ind u c t i v e l y b y L I = 4 a n d L n + I = L~ - 2
(so L 2 = 14, L 3 = 194, L 4 = 37634,
...) a n d h e n c e it is m u c h e a s i e r to
t e s t w h e t h e r M k is p r i m e t h a n it is
to t e s t a n o t h e r n u m b e r of the same
order of magnitude.
T h e p r i m e n u m b e r s o f the f o r m
2K - I (for w h i c h k i t s e l f m u s t n e c -
17
e s s a r i l y be prime) are called Mersenne primes (after the French m a t h ematlcian M e r s e n n e who in 1644 gave
9
a list
of such p r i m e s up to 10 79 ,
c o r r e c t up to 1018 ) and play a role
in connection w i t h a c o m p l e t e l y different p r o b l e m of number theory.
Euclid d i s c o v e r e d that when 2P - I
is prime then the number 2P -I (2P-I)
is "perfect", i.e. it equals the
sum of its proper d i v i s o r s (e.g.
6 = I + 2 + 3 , 2 8 = I + 2 + 4 + 7 +
%4,
496
=
I +
2
+
4
§
8
+
16
+
31
(lOg X) "t dt
R(x) = 7 t r(t+i)~(t+l)
o
(~(s) = the Riemann zeta function
and F(s) = the gamma function) and
R ( e 2 ~ X ) ' = ~2 (~
2 x
+
62 + 124 + 248) and Euler showed
that every even p e r f e c t number has
this form. It is u n k n o w n whether
there are any odd perfect numbers
at all; if they exist, they must be
at least 10 IQ0 . There are exactly
24 values of p < 20,000 for w h i c h
2P - 1 is prime.
(4)
(8) R a m a n u j a n has given the following
a l t e r n a t i v e forms for this function:
Gauss, Werke II (1892) 444 For a d i s c u s s i o n of the history of the various a p p r o x i m a t i o n s to
~(x), in which an E n g l i s h translation of this letter also appears,
see L.J. Goldstein: A history of
the prime number theorem, Amer.
Math. Monthly, 80 (1973) 599 - 615.
C.F.
447.
(5) A.M. Legendre, E s s a i sur la theorie
de Nombres, 2nd edition, Paris,
1808, p. 394.
_
~2
( 12x
4
3
+ 3--~4x
5
+ ~ x6+ . .
2525
+ 40x 3 + --~--x +...
(9) Namely: The p r o b a b i l i t y that for a
r a n d o m l y chosen pair (m, n) of numbers both m and n are ~ 0 (mod p)
is o b v i o u s l y [ (p-1)/p] 2 , while for
a r a n d o m l y chosen number n, the.
p r o b a b i l i t y that n and n+2 are both
O (mod p) is I/2 for p = 2 and
(p-2)/p for p ~ 2. Thus the probability for n and n+2 m o d u l o p to be
prime twins differs by a factor
- 1.5 < Li(x)
< Ls(x),
i.e. the d i f f e r e n c e b e t w e e n Li(x)
and Ls(x) is bounded. We should also
m e n t i o n that the l o g a r i t h m i c integral is often d e f i n e d as the C a u c h y
principal value
for p r 2 and by 2 for p = 2 from
the c o r r e s p o n d i n g p r o b a b i l i t y for
two i n d e p e n d e n t numbers m and n. Altogether, we have therefore improved
our c h a n c e s by a factor
p2 _ 2p
C = 2
Xdt
+
s oTG~q
I-5
2
p
(6) M o r e precisely
Li(x) = o§ log t
)
(Bk = k th Bernoulli number; the symbol ~ m e a n s that the d i f f e r e n c e of
the two sides tends to O as x grows
to infinity). See G.H. Hardy, Ramanujan: Twelve Lectures on Subjects
S u g g e s t e d by His Life and Work, Cambridge U n i v e r s i t y Press, 1940, Chapter 2.
p-2
Ls(x)
1
=
lim /i-~
dt
)
but this d e f i n i t i o n d i f f e r s from
that given in the text only by a
constant.
(7) The c o e f f i c i e n t s are formed as follows: the c o e f f i c i e n t of Li(n/x) is
+ 1/n if n is the p r o d u c t of an e v e n
number of d i s t i n c t primes, -I/n if
n is the p r o d u c t of an odd number
of d i s t i n c t primes, and 0 if n contains m u l t i p l e prime factors.
p>2
p prime
p2 _ 2p + I
= 1.32032. For a somewhat more
careful p r e s e n t a t i o n of this argument, see Hardy and Wright, An Int r o d u c t i o n to the Theory of Numbers,
C l a r e n d o n Press, Oxford, 1960,
w 22.20 (p. 371 - 373)
(10) M.F. Jones, M. Lal, and W.J. Blundon,
S t a t i s t i c s on certain large primes,
Math. Comp. 21 (1967) 103 - 107.
(11) D. Shanks, On maximal gaps b e t w e e n
successive primes, Math. Comp. 18
(1964) 646 - 651. The g r a p h of g(x)
was m a d e from the tables found in
18
w h e r e ~(x) § 0 as x t e n d s to infinity and C ~ 0 . 2 6 1 4 9 7 is a constant.
T h i s e x p r e s s i o n is s m a l l e r than 3.3
w h e n x = 109 , and e v e n w h e n x = 1018
it s t i l l lies b e l o w 4.
the f o l l o w i n g papers: L.J. L a n d e r
and T.R. P a r k i n , On f i r s t a p p e a r ance of p r i m e d i f f e r e n c e s , M a t h .
Comp. 21 (1967) 483 - 488, R.P.
Brent, The f i r s t o c c u r r e n c e o f large
gaps b e t w e e n s u c c e s s i v e p r i m e s ,
Math. Comp. 27 (1973) 959 - 963.
(16)
(12) The d a t a for this g r a p h are t a k e n
f r o m L e h m e r ' s table of p r i m e n u m b e r s
(D.N. L e h m e r , L i s t of P r i m e N u m b e r s
f r o m I to 1 0 , O O 6 , 7 2 1 , H a f n e r P u b l i s h i n g Co., N e w York, 1956).
(13) T h i s and the f o l l o w i n g g r a p h w e r e
m a d e u s i n g t h e v a l u e s of ~(x) found
in D.C. M a p e s , F a s t m e t h o d for comp u t i n g the n u m b e r of p r i m e s l e s s
than a g i v e n limit, Math. Comp. 17
(1963) 179 - 185. In c o n t r a s t to
L e h m e r ' s d a t a u s e d in the p r e v i o u s
graph, t h e s e v a l u e s w e r e c a l c u l a t e d
from a f o r m u l a for ~(x) and n o t by
c o u n t i n g the p r i m e s u p to x.
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 s
sur les n o m b r e s p r e m i e r s , Paris,
1851, CR P a r i s 29 (1849) 397 - 401,
738 - 739. For a m o d e r n p r e s e n t a t i o n
(in German) of C h e b y s h e v ' s proof,
see W. Schwarz, E i n f U h r u n g in M e t h o den und Ergebnisse der Primzahltheorie B I - H o c h s c h u l t a s c h e n b u c h
278/278a,
M a n n h e i m 1969, Chapt. II.4, P. 42 48.
(17) T h e l a r g e s t p o w e r of p d i v i d i n g p!
is p [ n / P ] + [ n T p 2 ] + . . . , w h e r e [x] is
the l a r g e s t i n t e g e r S x. Thus in the
n o t a t i o n of the l e m m a w e h a v e
Vp = r~17.( I V ]
(14) S. Skewes, O n the d i f f e r e n c e ~(x) li(x) (I), J. L o n d o n Math. Soc, 8
(1933) 277 - 283. Skewes' p r o o f of
this b o u n d a s s u m e s the v a l i d i t y of
the R i e m a n n h y p o t h e s i s w h i c h w e discuss later. T w e n t y - t w o y e a r s l a t e r
(On the d i f f e r e n c e ~(x) - li(x) (II),
Proc. Lond. Math. Soc. (3) 5 (1955)
48 - 70) he p r o v e d w i t h o u t u s i n g the
R i e m a n n h y p o t h e s i s that t h e r e e x i s t s
an x less t h a n the (yet m u c h larger)
bound
101010964
for w h i c h ~(x) > Li(x).
has b e e n l o w e r e d to
This b o u n d
1010529.7
by C o h e n a n d M a y h e w and to 1 . 6 5 x i O 1165
by L e h m a n (On the d i f f e r e n c e
~(x) - li(x), A c t a A r i t h m . 11 (1966)
397 - 410). L e h m a n e v e n s h o w e d t h a t
t h e r e is a n i n t e r v a l of at l e a s t
10500 n u m b e r s b e t w e e n 1.53•
and 1 . 6 5 x i O I165 w h e r e ~(x) is larger
than Li(x). A s a c o n s e q u e n c e o f his
i n v e s t i g a t i o n , it a p p e a r s l i k e l y
t h a t t h e r e is a n u m b e r n e a r 6 . 6 6 3
x10370 w i t h ~(x) > Li(x) a n d t h a t
there is no n u m b e r less t h a n 1020
w i t h this 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 by G a u s s in
1796 and p r o v e d by M e r t e n s in 1874)
7 1 = log log x + C + r
p<x p
- [~
-
[i)
E v e r y s u m m a n d in this sum is e i t h e r
O o r I and is c e r t a i n l y O for
r > (log n / l o g p) (since then [n/p r]
= O). T h e r e f o r e 9p S (log n / l o g p),
f r o m w h i c h the 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 of ~(s) as 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 e
o n l y w h e n s is a c o m p l e x n u m b e r
w h o s e r e a l p a r t is l a r g e r than I
(since the series c o n v e r g e s for
t h e s e v a l u e s of s only) and in this
d o m a i n ~(s) has no zeroes. B u t the
f u n c t i o n ~(s) can b e e x t e n d e d to a
f u n c t i o n for all c o m p l e x n u m b e r s s,
so t h a t it m a k e s s e n s e to s p e a k of
its r o o t s in the w h o l e c o m p l e x
plane. The s i m p l e s t w a y to e x t e n d
the d o m a i n of d e f i n i t i o n of ~(s) at
l e a s t to the h a l f , p l a n e Re(s) > O
is to u s e the i d e n t i t y
(1-21-s)~(s)
-
2
+ 4-~
= I + 1
+ I_~ + ...
2s
3s
6s
...
=
(_I)n-1
n=l
ns
'
w h i c h is v a l i d for Re(s) > I, and
to o b s e r v e that the s e r i e s o n the
r i g h t c o n v e r g e s for a l l s w i t h p o s i t i v e r e a l part. W i t h this, the "int e r e s t i n g " r o o t s o f the zeta function, i.e. the r o o t s p = B + iy w i t h
O < 8 < I, can be c h a r a c t e r i z e d in
19
an elementary way by the two equations
n~ I (-I)n-I
n8
c o s ( y l o g n) = O,
n~ 1 (-I)n-I
n8
sin(ylog n) = O.
The sum over the roots p in Riemann's formula is not a b s o l u t e l y
c o n v e r g e n t and t h e r e f o r e m u s t be
summed in the proper order (i.e.,
according to i n c r e a s i n g absolute
value of Im(p)).
Finally, I should m e n t i o n that,
a l t h o u g h Riemann stated the exact
formula for ~(x) already in 1859,
it w a s n ' t proved until 1895 (by von
Mangoldt).
The 1977 Mathematics Calendar has found a
worthy successor:
Psychology
Calendar '78
containing a selection of twelve themes
which p o r t r a y something of the exciting
spirit w h i c h pervades p s y c h o l o g i c a l inquiry:
Dreaming
Cognitive Contours
(19) These roots were calculated a l r e a d y
in 1903 by Gram (J.-P. Gram, Sur
les zeros de la f o n c t i o n ~(s) de
Riemann, Acta Math. 27 (1903) 289 304). For a v e r y nice p r e s e n t i o n of
the theory of R i e m a n n ' s zeta function, see H.M. Edwards, Riemann's
Zeta Function, A c a d e m i c Press, N e w
York, 1974.
Social D o m i n a n c e
in Monkeys
M c C o l l o u g h Effect
Dynamic Effects of Repetitive Patterns
Harlow and Piaget
Shadow and Light
Memory Span Test
W i l h e l m W u n d t Father of P s y c h o l o g y
(20) Namely the R i e m a n n hypothesis implies (and in fact is e q u i v a l e n t to
the statement) that the error in
Gauss's a p p r o x i m a t i o n Li(x) to ~(x)
is at most a c o n s t a n t times
x l / 2 . 1 o g x. At the present it is
even unknown w h e t h e r this error is
smaller than xC for any c o n s t a n t
c < I.
R e c o g n i t i o n of Faces
Split Brain
Jerome Bruner's Theory of Cognitive
Learning
(21) This and the f o l l o w i n g graphs are
taken from H. Riesel and G. G6hl,
Some c a l c u l a t i o n s r e l a t e d to Riem a n n ' s prime number formula, Math.
Comp. 24 (1970) 969 - 983.
%/
P.S. No cause for concern. The next Mathematics
Calendar will appear in 1979.