If you're interested in prime number theory and the Riemann hypothesis, I'm guessing you'll probably be interested in the attached document in which I illustrate Fourier series for prime number counting functions including Riemann's prime number counting function and Chebyshev's prime number counting function. I also included a couple of plots from the document below which I believe best illustrate the essence of the information contained within the document.
Fourier Series for Riemann's Prime Counting Function:
Fourier Series for Chebyshev's Prime Counting Function:
I added the Raspberry Pi group to this post because I created the attached document and performed all of the evaluations contained within the document using the Mathematica version which comes bundled with the Raspbian operating system. A complete evaluation of the attached document takes approximately one hour with Mathematica running on a Raspberry Pi 3 Model B.
@Steven Clark could you please also attach the notebook? We cannot run code in PDF.
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.
@Steven Clark I understand that, and appreciate the beauty. But this is why it is important to play with it. It is the main Wolfram programming community, - most of the people here will have Mathematica. You could keep PDF as an attachment and add notebook as a 2nd attachment so people have a choice. You could add functions' definitions in additional section and make it InitializationCell if you'd like to. I think those functions are interesting too.
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.
Even if it is preliminary work, the object of the forum is to show work in the Wolfram Language, and a notebook or cdf file is the way to do this. Not a pdf file. It is understandable that for preliminarty work one might not wish to go into all details, and that's fine. But if you only want to show pictures, sans code used to create them, then this forum is really not the right venue.
As for using Fourier transforms on functions from number theory, might want to have a look at Prime Numbers and the Riemann Hypothesis if you've not seen it already.
Daniel: Thanks for the reference. I reviewed an online copy of this reference at the beginning of the year. Could you please point me to a reference with respect to the objectives of the forum? I can't seem to find them anywhere.
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.