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I haven't achieved my ultimate goal yet (which I described in an earlier reply in this discussion), but I think derivation of Fourier series for prime number counting functions is a significant achievement in of itself, and so I've decided to generate and post a notebook illustrating the derivation of Fourier series for prime number counting functions, This is going to take a bit of time, so please be patient and stay tuned to my discussions.

I included a PDF file versus a notebook for a couple of reasons. The first reason is I figured not everyone has access to Mathematica, but nearly everyone has access to a PDF reader. The second reason is you can't run evaluations on the notebook because the notebook doesn't include the definitions of the underlying functions which calculate the Fourier series. This document was intended to illustrate the high-level results (which I consider somewhat beautiful in of themselves) to a wide audience versus to bury the reader in the details and complexity of the mathematics which would require a significant amount of additional explanation.