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).
This website currently illustrates the high-level results of my investigation versus the details of the underlying mathematics for a couple of reasons. First, the details of the underlying mathematics are still under investigation for a possible proof of the Riemann Hypothesis. Second, I believe the mathematical beauty of the high-level results will likely be appreciated by a wider audience than the details of the underlying mathematics.
I recommend first taking a look at the illustrations of Fourier series for the first-order derivatives of the prime counting functions on page 5 (and it’s children pages) because I believe these are more informative and inherently more beautiful and easier to relate to than the illustrations of Fourier series for the prime counting functions on page 4 and their second-order derivatives on page 6. If you’re familiar with the Riemann zeta function $\zeta(s)$ and the concept of zeta zeros, I also recommend you take a look at pages 9 and 10 and their associated children pages.
Fourier Series Terminology:
I’d like to clarify a point with respect to what I’m calling a Fourier series on this website. Fourier series are normally used to represent periodic functions. The Fourier series which I’ve derived for prime counting functions are really infinite sets of Fourier series, each with an infinite set of harmonics. Since the range of x values illustrated on this website is small, I only needed to evaluate a small number of the infinite sets of Fourier series to obtain a reasonable approximation. As the range of x values being illustrated increases, the number of Fourier series being evaluated needs to increase as well to obtain a reasonable approximation.
Genuine Natural Fourier Series:
When I use the term genuine natural Fourier series, I mean the Fourier series illustrated on this web site exhibit fundamental relationships with their associated prime counting functions. For example, the Fourier series for the first-order derivatives of the prime counting functions evaluate to exactly $2\ f$ times the step-size of the associated prime counting function at positive integer values of $x$.
Methods for Derivation of Fourier Series:
I’ve defined three methods for derivation of Fourier series for prime counting functions. This website initially focused on illustrating the Fourier series for prime counting functions derived via the initial method 1, but I’m now in the process of adding illustrations for Fourier series derived via methods 2 and 3. The three methods provide slightly different results which can most easily be seen in plots of the first-order derivatives of the Fourier series for prime counting functions. At positive integer values of $x$, the first-order derivatives derived from all three methods evaluate exactly to $2\ f$ times the step-size of the associated prime counting function. The difference between the three methods is noticable in the way the first-order derivatives derived from each method oscillate between positive integer values of $x$.
In his seminal 10 page paper “On the Number of Primes Less Than a Given Magnitude” published in 1859, Riemann applied a Fourier inversion to derive a formula for a prime-power counting function from the non-trivial zeros of a zeta function, and for this reason I believe the discovery of the genuine natural Fourier series for prime counting functions may ultimately prove to be the most significant advance in prime number theory since Riemann’s seminal paper of 1859.
In this paper, Riemann hypothesized that the real portion of every non-trivial zeta zero is one half, which has become known as the Riemann Hypothesis. Many consider Mathematics to be the Queen of the Sciences, Number Theory to be the Queen of Mathematics, and proof of the Riemann Hypothesis to be the Holy Grail of Number Theory. In fact proof of the Riemann Hypothesis is one of the Clay Mathematics Institute’s seven millennium math problems (see the link below), and there’s a substantial award (i.e. $1 million) associated with a proof of the Riemann Hypothesis.
Steven Foster Clark