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This website illustrates Fourier series for a step function, two staircase functions, and six prime counting functions all of which are defined below.

**$U[x]$ – Step function which takes unit steps at $|x|=1$.**

**$S[x]$ – Staircase function which takes a step of $1$ at each positive integer.**

**$T[x]$ – Staircase function which takes a step of $\log[n]$ at each positive integer $x=n$.**

**$\pi[x]$ – Base Prime Counting Function that takes a step of 1 at each prime integer $x=p$.**

**$J[x]$ – Riemann’s Prime-Power Counting Function that takes a step of $1/n$ at prime-powers of the form $x=p^n$.**

**$K[x]$ – Simple Prime-Power Counting Function that takes a step of $1$ at prime-powers of the form $x=p^n$.**

**$\vartheta[x]$ – First Chebyshev Function that takes a step of $\log[p]$ at each prime integer $x=p$.**

**$\psi[x]$ – Second Chebyshev Function that takes a step of $\log[p]$ at prime-powers of the form $x=p^n$.**

**$M[x]$ – Third “Chebyshev-like” Function that takes a step of $n\log[p]$ at prime-powers of the form $x=p^n$.**

The relationships between the rational-step functions $S[x]$, $\pi[x]$, $J[x]$, and $K[x]$ and the log-step functions $T[x]$, $\vartheta[x]$, $\psi[x]$, and $M[x]$ are expressed in terms of their first order derivatives below.

$\quad T'[x]=\log[x]\ S'[x]\\$

$\quad\vartheta'[x]=\log[x]\ \pi'[x]\\$

$\quad\psi'[x]=\log[x]\ J'[x]\\$

$\quad M'[x]=\log[x]\ K'[x]$

The primary purpose of the pages below is to provide background information for those who might not be as familiar with prime number theory. If you’re already familiar with the base prime counting function $\pi[x]$, Riemann’s prime-power counting function $J[x]$ and his associated formula, the first Chebyshev function $\vartheta[x]$, and the second Chebyshev function $\psi[x]$ and von Mangoldt’s associated formula, there’s really nothing new in pages 3.4, 3.5, 3.7, and 3.8 below so you can probably skip these pages without much loss of information.

If you’re already familiar with these four prime counting functions, I recommend you at least take a look at page 3.6 for the simple prime-power counting function $K[x]$ and page 3.9 for the third “Chebyshev-like” function $M[x]$. I’m not sure if anyone has investigated these functions before, but I think the simple prime-power counting function $K[x]$ is more fundamental in a couple of ways than the base prime counting function $\pi[x]$ and Riemann’s prime-power counting function $J[x]$ which I attempt to illustrate on page 8. Since I don’t know if these two functions $K[x]$ and $M[x]$ have been investigated before and I’m not aware of any official names for them, I made up a couple of names for these two functions.

I also highly recommend you take a look at page 3.1 for the unit-step function $U[x]$. I didn’t originally fully appreciate the value of the Fourier series representation of $U[x]$ and it’s derivatives, but I’ve come to realize it may by far be the most important Fourier representation illustrated on this website for reasons explained on page 3.1.

In the pages below, I provide mathematical definitions and illustrations for the counting functions. In some cases, the mathematical definitions reference mathematical functions defined in the Wolfram Language. If you’re not familiar with these the functions, you can find definitions for them on the following web site: http://reference.wolfram.com/language/?source=nav. The easiest way to find information on a function is to type it’s name in the search box.