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14 G. H. HARDY:
Thus f (x) is expressed as the sum of a finite number of
functions, each of which plainly depends on the distribution
of primes in an arithmetical progression mk -j- /. We
are thus led to make an assumption, analogous to the
hypothesis of Riemann, about all the functions L(s) on which this
distribution depends; in other words, we are led to hypothesis
R. Unless we make some such assumption our difficulties,
already considerable, will be terribly increased.
We therefore assume hypothesis R. We now write X- e~~y
and express f(x\ by means of the formulae (33) and (34), in
the form of an integral
r
(35)
where Z(s) is a linear combination of the logarithmic derivatives
L’(s)\L(s) of the Z-functions associated with modulus k. The
dominant term of f(oc\ in the neighbourhood of a^,*, is the
residue of the integrand for s~= I, and a simple calculation
shows that this is the function which appears on the right
hand side of (25).
We now return to the integral (23), and consider the
separate contributions of the arcs £ft>A. If we substitute for f(x),
on %htk, the approximation given by (25), and transform the
integral by the substitutions
we obtain
Y~fenYdY.
The path of integration is now a segment of a straight line,
parallel to the imaginary axis in the plane of F, and passing
to the right of the origin.
We can substitute for this segment the complete straight
line of which it is part, the error involved in this
approximation proving to be unimportant. The integral can then be
evaluated in finite terms, and (37) assumes the form
.T> <*>
We have finally to sum with respect to k and k, and we
obtain the formula
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