# 5.10 Illustration of Six Formulas for $J_b'[x]$

I’ve defined three methods for derivation of Fourier series for prime number counting functions, and this page illustrates the formulas derived for $J_b'[x]$ via each of these three methods.

This page also illustrates three additional formulas derived for $J_b'[x]$ from the following relationship.

$$J_b'[x]=\frac{\psi_b'[x]}{\log[x]}$$

The following six plots illustrate the formulas for $J_b'[x]$ (blue) using the evaluation frequency $f=1$. The horizontal grid-lines in the plots below are at $2\ f$, $\frac{2\ f}{2}$, $\frac{2\ f}{3}$, and $\frac{2\ f}{4}$. All of the formulas illustrated here exhibit the same exact convergence to $\frac{2\ f}{n}$ at prime-powers of the form $x=p^n$ and to $0$ at the remaining positive integer values of $x$.

The following two plots illustrate the formulas derived for $J_b'[x]$ via method 1. Note that for the first plot below the magnitude of the oscillation beneath the $x$-axis seems relatively constant as $x$ increases, whereas for the second plot below the magnitude of the oscillation beneath the $x$-axis seems to decrease as $x$ increases.

The following two plots illustrate the formulas derived for $J_b'[x]$ via method 2.

The following two plots illustrate the formulas derived for $J_b'[x]$ via method 3.