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GOLDBACH S THEOREM. 3
We have however to exclude all cases in which either m
or m’ is a multiple of 3. If 3 rø1), the two sets of cases thus
excluded are the same, while if 3 -f n the effect of the double
exclusion is cumulative. W should therefore expect divisibility
by 3 to increase v(»); and a similar argument applies to any
other prime, so that v (n) should be largest when n is
composed of a large number of different small prime factors. All
these rough expectations prove to be in complete accordance
with the facts.
Sylvester (1871) was the first mathematician to suggest an
asymptotic formula for v (n). Sylvester’s rule is stated in
words, and when translated into symbols is as follows :
(3)
where TI (n) is the number of primes not exceeding n, and the
product extends over all primes p for which 3 ^ p 5^ n and
/ \ n. We know, though Sylvester did not, that
n (n) ~ - – (4)
v ’ log;/ w
This is the famous ’Primzahlsatz’, and we can use it to simplify
(3). We also require a theorem of Mertens (1874), to the
effect that
n/ I \ ? - ^
l1 -jr 1°^’ (5)
where C is Euler’s constant. This theorem, which lies much
less deep than (4), shows that
nip - 2\ T~T / T \TT/ r\ 2^~c
^ U=i) = 1111 -(7^Tp)pl 111 -7) ~ icg^’
where
~»––-." / T \
(6)
J) Following Landau, I write ’3 n for *« is divisible by 3’, and ’3 -f n to
express the contrary.
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