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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.

I began a fairly intense investigation of prime number counting functions and the Riemann hypothesis nearly a year ago (i.e. at the beginning of last October). I hadn’t originally planned on sharing any results until my investigation was complete, but I found these particular results to be so beautiful that I couldn’t resist sharing them.

I believe these Fourier series have the potential to provide considerable insight into formulas for prime number counting functions. My ultimate goal is to illustrate how the components of Riemann’s formula for his prime counting function and the components of von Mangoldt’s formula for Chebyshev’s prime counting function evolve from these Fourier series. I have not yet achieved this goal, so my investigation is still a work in progress.

Determination of how the individual zeta zero components of Riemann’s formula and von Mangoldt’s formula evolve from these Fourier series is of particular interest as this might lead to a proof of the Riemann hypothesis. Proof of the Riemann hypothesis is one of Clay Mathematical Institute’s seven millennium math problems, and there’s a substantial award ($1 million) associated with a proof of the Riemann hypothesis, so please forgive me for not being anxious to share the details of nearly a year’s worth of intense effort.

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.