How plot the Riemann zeta zero spectrum with the Fourier transform? [closed]

Question

This question is closed as off-topic and duplicate of How plot the Riemann zeta zero spectrum with the Fourier transform in Mathematica?

In the paper "The Riemann Hypothesis" by J. Brian Conrey in figure 6 there is a plot of the Fourier transform of the error term in the prime number theorem. See the plot to the left in the image below:

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In a blog post called Primes out of Thin Air written by Chris King there is a Matlab program that plots the spectrum. See the plot to the right at the beginning of the post. A translation into Mathematica is possible:

Mathematica:

scale = 10^6;
 start = 1;
 fin = 50;
 its = 490;
 xres = 600;
 y = N[Accumulate[Table[MangoldtLambda[i], {i, 1, scale}]], 10];
 x = scale;
 a = 1;
 myspan = 800;
 xres = 4000;
 xx = N[Range[a, myspan, (myspan - a)/(xres - 1)]];
 stpval = 10^4;
 F = Range[1, xres]*0;

For[t = 1, t <= xres, t++,
 For[yy = 0, yy <= Log[x], 1/stpval++,
 F[[t]] =
 F[[t]] +
 Sin[t*myspan/xres*yy]*(y[[Floor[Exp[yy]]]] - Exp[yy])/Exp[yy/2];
 ]
 ]
 F = F/Log[x];
 ListLinePlot[F]

However, this is as I understand it the matrix formulation of the Fourier sine transform and it is therefore very costly to compute. I do NOT recommend running it because it already crashed my computer once.

Is there a way in Mathematica utilising the Fast Fourier Transform, to plot the spectrum with spikes at x-values equal to imaginary part of Riemann zeta zeros?

I have tried the commands "FourierDST" and "Fourier" without success. The problem seems to be that the variable "yy" in the code is included in both "Sin[t*myspan/xres*yy]" and "(y[[Floor[Exp[yy]]]] - Exp[yy])/Exp[yy/2]".