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16 G. H. HARDY: GOLDBACH’S THEOREM.
where 8 may be any positive number. Thus (f(x)}r consist of
two parts, a term jj/~~r, which is exactly known, and an error
term whose order is, so far as our analysis shows, not less
than 0(j~~ir); and each of these terms gives rise to a
corresponding term iii the final formula. The contribution of the
dominant term can be found precisely, and is of order nr~l.
The contribution of the error term can only be estimated in a
cruder manner, and the best that we can say about it is that
it is of order 0(n^r}. This error must, if our approximation is
to succeed, be smaller than the dominant term; and this requires
that r- i ^> \r or r^>2. In fact I have done rather more
than justice to our method. It would appear, from what I have
stated, that we stand on the very margin of success. But
there are further complications in the analysis, and we fail by
a power «t.
I implied, when I was discussing the methods of Merlin
and Brun, that it was hardly reasonable to suppose that they
could possibly succeed. You may naturally ask me what I
think of the prospects of our own, and the question is one to
which I find it rather difficult to reply. You must make me,
for the moment, a present of hypothesis R. I presume that
the hypothesis of Riemann will some day be proved.
Hypothesis R will, I am sure, be proved within a week from then,
and the proofs will be substantially the same. There is nothing
whatever to suggest that, in these respects, one Z-function
behaves unlike another.
Apart from this I would reply that, the hypothesis once
proved or granted, I see no particular reason why our method
should not succeed. It seems to me adequate for the problem;
the ideas which underlie it are not too easy and lie sufficiently
deep. It fails in detail, and not in principle, even as it is; the
failure is not a failure of the method, but of the analytical
powers of Mr. Littlewood and myself. The method seems to
embody the essential features of the problem, and leads,
naturally and inevitably, to what is plainly the real result. I
believe that, when the problem is solved, it will be solved in
some such way as this.
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