Inverse Fourier transform of real and imaginary part of Riemann zeta function on critical line

Mathematica 8

The spikes are at x-values equal to logarithms of 1,2,3,…

Clear[a, c, p]
Monitor[a = Table[(Zeta[1/2 + I*t]), {t, 0, 1000, 0.01}];, t]
dct = FourierDCT[Re[a], 3];
dst = FourierDST[Im[a], 3];
ListLinePlot[dct[[1 ;; 800]], ImageSize -> Large, PlotRange -> {-20, 200}]
ListLinePlot[dst[[1 ;; 800]], ImageSize -> Large, PlotRange -> {20, -200}]

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