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

for . 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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