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

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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.


First-Order Derivative of Riemann's Prime-Power Counting Function (i.e. j'[x]) Evaluated at f=1.
Method 1: $J_b'[x]$ evaluated at $f$=$1$.

$J'(x)=\frac{\psi'(x)}{\log(x)}$ evaluated at $f$=$1$.
Method 1: $J_b'[x]=\frac{\psi_b'[x]}{\log[x]}$ evaluated at $f$=$1$.

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


Method 2: $J'(x)$ evaluated at $f$=$1$.
Method 2: $J_b'[x]$ evaluated at $f$=$1$.

Additional plot coming soon.


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


Method 3: $J'(x)$ evaluated at $f$=$1$.
Method 3: $J_b'[x]$ evaluated at $f$=$1$.
Additional plot coming soon.