| dbp:proof
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- With some simple algebra performed on finite sums, we can write for any complex s
Now if and , the factor multiplying is zero, and
where denotes a special Riemann sum approximating the integral of over .
For i.e., , we get
Otherwise, if , then , which yields (en)
- If is real and strictly positive, the series converges since the regrouped terms alternate in sign and decrease in absolute value to zero. According to a theorem on uniform convergence of Dirichlet series first proven by Cahen in 1894, the function is then analytic for , a region which includes the line
. Now we can define correctly, where the denominators are not zero,
or
Since is irrational, the denominators in the two definitions are not zero at the same time except for , and the function is thus well defined and analytic for except at . We finally get indirectly that when : (en)
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