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GOLDE ACH’S THEOREM. 5
The writings of these three mathematicians have a striking
common characteristic: they use elementary methods only.
You may ask me what an ’elementary’ method is, and I must
explain precisely what I understand by this expression. I do
not mean an easy or a trivial method; an elementary method
may be quite desperately ingenious and subtle. I am using
the word in a definite and technical sense, and in this I am
only following the common usage of arithmeticians. I mean,
by an elementary method, a method which makes no use of
the notion of an analytic function. And the question that I
wish to put to you is this: is it reasonable, in the present
state of mathematical knowledge, to hope to obtain an
elementary proof of Goldbach’s theorem?
If I reply to this question in the negative, as I must and
shall, if I say that I am compelled to regard all such efforts as
foredoomed to failure, I trust that you will not misunderstand
me. I cannot believe that the methods of Merlin and Brun are
sufficiently powerful or sufficiently profound to lead to a solution
of the problem. But I am very far from meaning that I regard
their work as devoid of interest and value. There is much in
Brun’s work in particular that seems to me very beautiful,
and some of his theorems ought, I think, to find their way
into every book on the theory of numbers.
We have however to take account both of the history and
the logical structure of our subject. Let us turn back then
for a moment to its central theorem, the ’Primzahlsatz’ or
’prime number theorem’ expressed by the equation (4). It
seems plain that this must be at any rate an easier theorem
than Goldbach’s theorem. No elementary proof is known, and
one may ask whether it is reasonable to expect one. Now
we know that the theorem is roughly equivalent to a theorem
about an analytic function, the theorem that Riemann’s
Zeta-functionl) has no zeros on a certain line ?) A proof of such
a theorem, not fundamentally dependent upon the ideas of
the theory of functions, seems to me extraordinarily unlikely.
It is rash to assert that a mathematical theorem cannot be
proved in a particular way; but one thing seems quite clear.
We have certain views about the logic of the theory; we think
J) £(j; = £(c + it).
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