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.