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G. H. HARDY: A NEW PROOF FOR THE ZETA-FUNCTION. Ji
A new proof of the functional equation for the
Zeta-function.
By G. H. Hardy.
I. It is well known that
oc
X /
2,
!)*.*. X ’si n l
4 ,"="
if kiK^x<^(k-\-\]n. If we multiply (i) by xs~l, where Ö < .y < i,
and integrate over the interval (ö, oo), assuming for the moment
that we way invert the order of integration and summation,
we obtain
(2)
4 -^ J ^-/ 2;w+i ,1
/;=() to m=0 o
1 V, w C
JT > - IH
4 -A J
\T7
F (j) sin -J- sn y
or
say. This equation, proved in the first instance when o<><O,
will hold, when considered as a relation between analytic
functions, over any region throughout which these functions
exist.
The series for Ö (A viz.
/f=0
is convergent if only d - H (s) < i. When ö << - i, it is equal to
^l+S jjl+*
’____ ( js___ 2s . j ^s__. . . . \ - -._ (j__21^’s) 2! fj")"
and it effects the analytic continuation of this function for
o<C I. The series for W (j), on the other hand, is convergent
when ö > ö, and equal to
6*
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