The primary purpose of this website is the illustration of the genuine natural Fourier series for prime counting functions. In early September of 2016 I posted a few illustrations to the Wolfram Community. There are now a significant number of additional illustrations posted on this website (see PAGES to the right or below depending on the device you’re using to access the website and the orientation of your screen). Please see page 1 for further clarification on what I mean by genuine natural Fourier series.
Pages 16 and above are intended for those with a more passionate interest in prime number theory and the details of the underlying mathematics. You must be a registered user and logged onto this website to access the content of these pages.
The plot in the header above illustrates the Fourier series for the first-order derivative of the first Chebyshev function which takes a step of $\log p$ at each prime the form $x=p$. The plot in the header was generated using the minimum frequency evaluation limit $f=1$, where $f$ is assumed to be a positive integer. The reference function shown in orange in the plot is $2\ f\log x$. The plot in the header illustrates an example of a more general result, which is that for positive integer values of $x$ the Fourier series for the first-order derivative of a prime counting function evaluates to $2\ f$ times the step size of the associated prime counting function at the positive integer $x$.
Up to mid February of 2017 I was primarily using blog posts to convey updates to my website, and pages to convey technical content. I was also conveying updates to my website on this page which seemed a bit redundant. Since the outline of the pages and the list of updates on this page had both grown quite long, I decided to experiment with using blog posts to convey technical content as well as updates to my website. I’ve adopted the general guideline that all blog posts related to technical content will be marked with the Category “Fourier Series for Prime Counting Functions”, and all blog posts related to website updates will be marked with the category “Website Updates”. I’m also planning to mark blog posts containing technical content with additional categories to aid others in finding information of interest.