The Laplace and Fourier transforms of the Fourier series representations of $U(x)$ and it’s first and second-order derivatives are now discussed and illustrated on page 12 and its children.
The hierarchy of functions derived from Fourier series representations of prime counting functions is now illustrated on page 14 and it’s children. You must be a registered user and logged onto this website to access the content of these pages.
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. Pages 4 and 5 summarize the number of formulas I’ve derived for the base prime counting functions and their first-order derivatives respectively. Page 4.10 will illustrate three formulas for Riemann’s prime-power counting function $J[x]$, and page 4.11 will illustrate six formulas for the second Chebyshev function $\psi[x]$. Page 5.10 will illustrate six formulas for the first-order derivative $J'[x]$ of Riemann’s prime-power counting function, and page 5.11 will illustrate six formulas for the first-order derivative $\psi'[x]$ of the second Chebyshev function.
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.
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 $2f\log x$. The plot in the header illustrates an example of a more general result, which is that for an integer value of $x$ the Fourier series for the first-order derivative of a prime counting function evaluates to $2f$ times the step size of the associated prime counting function at the integer $x$.