Full resolution (TIFF) - On this page / på denna sida - Sidor ...
<< prev. page << föreg. sida << >> nästa sida >> next page >>
Below is the raw OCR text
from the above scanned image.
Do you see an error? Proofread the page now!
Här nedan syns maskintolkade texten från faksimilbilden ovan.
Ser du något fel? Korrekturläs sidan nu!
This page has never been proofread. / Denna sida har aldrig korrekturlästs.
o G. H. HARDY:
that some theorems, as we say, ’lie deep’, and others nearer
to the surface. If anyone produces an elementary proof of
the prime number theorem, he will show that these views are
wrong, that the subject does not hang together in the way
we have supposed, and that it is time for the books to be
cast aside and for the theory to be rewritten.
You are probably familiar with the general idea of the
’sieve’ or ’crible’ of Eratosthenes. We write down all the
integers
i , 2, 3, 4, 5, 6, 7, 8, 9, 10, n, 12, 13, 14, 15,
= = == (8)
up to x. We erase (or underline), first i ; then every even
number after 2; then multiples of 3, except 3 itself; and so
on, repeating the process for every p. The process comes to
an end when/ exceeds ^x, for then only primes are left.
Suppose now that /1? /2, ...,/, are the first r primes,
that x is large and r fixed, and that we use the sieve for
these primes only. The number of numbers left is
approximately
/ i W i \ / i \
x\i – r – . I . . . i – ;
I PJ\ A/ I A/’
it is easy to see that the error in this enumeration is at most
2r -(- r. If jcr (x) is the number of numbers, not exceeding x,
and prime or not divisible by any of these r primes, then
(9)
We have supposed so far that r is fixed. The result would
be much more interesting if we could suppose that r is a
function of x. Let us assume provisionally that this is
legitimate, and take r to be the largest prime not exceeding}^.
We obtain
by Mertens’ formula (5). This formula is false (so that our
assumption was illegitimate), and it is significant that it is
<< prev. page << föreg. sida << >> nästa sida >> next page >>
