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IO G. H. HARDY:
so that vr(X) is the number of representations of n as a sum
of r primes1). We have
(19)
V ^
by Cauchy’s theorem, the path of integration C being the
circle x = /?, where ö < 7? < I. We take
/?=!–. (20)
We do not base our analysis, however, on the actual formulae
that I have written. It is more convenient to use the formulae
f(x) = x? log 2 + x* log 3 + ^5 log 5 + . . . - £^P log/, (21)
(22)
Here
^.(w) = Z log/x log/2 - . . log/r, (24)
where the summation applies to all sets pl9 /2, . . ., /r whose
sum is n. Goldbach’s theorem asserts that va(#) (or, what is
the same thing, that N2 (n}) is positive for n = 4, 6, . . . . Our
object is the more comprehensive one of finding an asymptotic
formula for \T(ri)\ and it is easy to show that, if we can find
such a formula for Nr(n\ we can deduce one for \T(n) by
division by (log ri)r 2).
Our fundamental idea is the same as that which has guided
us in our work on Waring’s problem. The unit circle is a
barrier of singularities for /(#)., we may say, roughly, that/Car)
becomes large when x approaches the circle. There are
however certain special points of the circle in whose
neighbourhood f(pc] is largest, and whose contributions to the integral
(25) are of dominating importance. These points are the
’rational points h, k\ for which
li vr(») has no connection with the vr(») of (12).
2) If ö <C & < i, we have (i - ö) log n <C log/ <C log n for nearly all the
primes in question.
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