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G, H. HARDY:
the product extending now over all odd primes. We thus
obtain the formule
p|n
where / is an odd prime divisor of n, and this we may
reasonably call ’Sylvester’s formula’.
Sylvester’s formula is certainly wrong. It is correct,
apparently, in its most obviously interesting parts, in its crude
order of magnitude, and in the irregularly oscillating factor
which depends upon the arithmetic structure of n. It is the
constant factor ^Ae~c tliat cannot be correct. It is interesting
to pause for a moment to consider how such a negative result
can be established.
There is no mystery at all about the average value of
\(n). It is quite easy to prove x) that
w
n 2(log?z)2
If now we assume any asymptotic formula for v(V), we can
test it by its compatibility with (8). We may assume a
formula of Sylvester’s type, but with an unspecified constant
factor £; and we find, as the test of compatibility, that B-2A.
Thus the only possible formula of this type is
/ x 2An l V lp - i\ , x
VW=(To^lH7="2J’ (9)
p|n
Sylvester’s formula must be wrong, if not in principle, then
by a constant multiplier 2e~c = 1-123 ...*). The formula (9),
on the other hand, seems almost certainly correct.
Sylvester’s contribution to the problem passed unnoticed
for nearly 50 years. I turn now to the writers, Merlin, Brun,
and Stäckel, who have attacked it recently, and particularly
to Brun. The work of Mr Littlewood and myself belongs to
a different circle ot ideas, and I shall speak of it later.
l) The actual theorem is due to Landau (1900). *
a) There is considerably more difficulty in testing the discrepancy by
comparison with the facts. Shah and Wilson have given a very clear and
interesting discussion of this point.
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