Finding Maximum without a plot

*To*: mathgroup at smc.vnet.net*Subject*: [mg129798] Finding Maximum without a plot*From*: Tim Trudgian <tim.trudgian at gmail.com>*Date*: Thu, 14 Feb 2013 02:11:19 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

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)?