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GOLDBACH’S THEOREM. 13
/(*) = 2 log/ e-P» =~y-s r (s) - ds (30)
which is substantially (though not exactly)
The integrals are taken along a line parallel to the imaginary
axis and passing to the right of the point s = i.
The subject of integration has poles when s = i, when s
is zero or a negative integer, and at all the complex zeros of
£(j), which we denote generally by p. If we assume Riemann’s
hypothesis, every p has the real part i. It is natural to
suppose that, if we move the path of integration further and
further to the left, and apply Cauchy’s theorem, we shall
express f (of) in the form of an infinite series, in which the most
important terms will be the terms corresponding to the residues
for i and p. If we denote these terms by
then the term corresponding to p is of order y. i when y is
small, and we may hope that the order of the series will be
much the same. In this case we shall prove that f(pf)^y~~ *,
which is the simplest case of (25), and that the order of
magnitude of the error is not notably greater than that of y-\.
It is evident that our estimate of the error will depend
essentially on our assumption.
Passing now to the general case, I denote by X(#) the
arithmetical function of n which is equal to log n when n is
prime and otherwise to zero. We have now
(33)
\R J
We write
n ~ mk + /(/=: 1,2, ...,£; m = o, i , 2, . . . ),
and we obtain
(34)
Z,m
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