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GOLDBACH’S THEOREM. 15
nr-1
M(»)~(f-?)\S> (39>
where
5= yfeH-^)= > (B^Kw. (4o>
h, k
(40
and h, in (41), is less than and prime to k.
The series S, the ’singular series’, is the series on which
the solution of all these problems depends. It can, as I said
before, be summed in finite terms1). If r and n are of
opposite parity, then 6* - ö. If they are of the same parity, then
s = 2 c, jj p^yzp-^F–) - (42)
P l n
where
m-r ! (__ T V \
(43)
If r = 3, we are led to (27), and if r - 2 to (9), the powers
of log n which appear in these formulae being introduced in
the passage from N to v; and there are similar formulae for
every value of r.
There is unfortunately a vital difference between the case
r = 2, which corresponds to Goldbach’s theorem, and all the
rest. We have to fill in the skeleton which I have presented
to you, and to transform it into an accurate proof; and in
doing this we find ourselves compelled to suppose that r>2.
It only remains that I should explain to you shortly the reason
for this regrettable limitation2).
Our work depends upon a system of approximate formulae
for /(#), each valid near a particular point of the unit circle. It
will be sufficient for my purpose to consider the point x - u
If x = e~~y, and y is positive, our formula for f(x] is of the form
’) The summation is a simple but entertaining exercise in elementary
algebra, which I must be content to take for granted.
z) The explanation which follows must be taken merely as a first
approximation to the truth.
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