The order of $\zeta'/\zeta$
Assume the Riemann Hypothesis. Define for to be the greatest lower bound of the numbers for which
holds as for all . It is a theorem that is a
convex function of which is continuous and decreasing for with for all It can be shown that
. There is an analogous function which can be defined for
and it can be shown that this analogous function is,
in fact, equal to . See Titchmarsh for all of these facts.
Which bound is corrct?
If the smaller bound is the correct one, then near the half-line we see that
for . On the other hand,
Thus, if the smaller bound holds, then there is a jump at 1/2. It could be that the
smaller bound holds to the right of the critical-line but that there is
radically different behavior on the critical line.
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