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Finding Maximum without a plot


I should like to find the maximum of 

Abs[Zeta[1/2 + I*t]]/t^(1/4)

for t large, say, t< 10^10.

The trouble with NMaximize is that (sometimes) only local maxima are picked up. 

When plotting, say

Plot[Abs[Zeta[1/2 + I*t]/t^(1/4)], {t, 3, 10^5}, PlotRange -> Full]

I can see that, around t = 20,000, there is a value above 1.5. This value is not picked up when plotting the same function in the range {t, 3, 10^6}.

Does anyone have any solutions other than plotting blocks of 10^k, 10^(k+1)?



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