Fourier transform of real and imaginary part of Riemann zeta function equal to one half

The frequencies appear to be logarithms of the natural numbers: 1,2,3,…

The first plot is the Fourier Cosine Transform of the real part of the Riemann zeta function along the line in the complex plane with real part equal to one half.

The second plot is the Fourier Sine Transform of the imaginary part of the Riemann zeta function along the line in the complex plane with real part equal to one half.

Mathematica 8

Clear[a, dct, dst]
Monitor[a = Table[(Zeta[1/2 + I*t]), {t, 0, 1000, 0.01}];, t]
dct = FourierDCT[Re[a]];
dst = FourierDST[Im[a]];
ListLinePlot[dct[[1 ;; 800]], ImageSize -> Large, PlotRange -> {-20, 100}]
ListLinePlot[dst[[1 ;; 800]], ImageSize -> Large, PlotRange -> {20, -100}]

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