How to produce Riemann zeta zero spectrum with the Fourier transform in Mathematica?

Question

All:

I post a question generating Riemann Zeta zero spectrum using Mathematica on board of mathematica.stackexchange.com:

https://mathematica.stackexchange.com/questions/95294/can-anyone-re-produce-this-result-related-to-the-spectrum-of-riemann-zeta-using

Can anyone help ?

It is related to a plot of paper "The Riemann Hypothesis" by J. Brian Conrey: the Fourier transform of the error term in the prime number theorem.

https://i.stack.imgur.com/hrz15.jpg , spectrum of zeta zeros

Beside Mathematica problems, I also have questions regarding the math behind it:

A: Do we do Fourier analysis on Chebyshev function itself or the error term, (Chebyshev - x) ?

B: Why do we need to calculate 10^6 points of Chebyshev function for this purpose ? Can we do with some less points, such as: 10^4 ?

C: How is the number of data points for Chebyshev function related to the resolution of frequency domain of zero spectrum ?

D: Can we use built Fast Fourier Transform to do above analysis or do we have to implement discrete sine transform ourselves as in the code.