5.5 $J_b'[x]$: First-Order Derivative of Riemann’s Prime-Power Counting Function

Navigation Links: – Next Page – Parent Page


The Fourier series for the first-order derivative $J_b'[x]$ of Riemann’s prime counting function evaluates to $\frac{2\ f}{n}$ at prime powers of the form $x=p^n$ where $f$ is the evaluation frequency limit. The horizontal grid-lines in the plots below are at $2\ f$ {$1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$} since $20$ is the maximum value of $x$ for these plots and $Floor[\log[2,\ 20]]=4$ (except the first plot only includes the first three grid-lines and the last plot does not include any grid-lines).


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

First-Order Derivative of Riemann's Prime-Power Counting Function (i.e. j'[x]) Evaluated at f=1.
$J_b'[x]$ evaluated at $f=1$.
First-Order Derivative of Riemann's Prime-Power Counting Function (i.e. j'[x]) Evaluated at f=2.
$J_b'[x]$ evaluated at $f=2$.
First-Order Derivative of Riemann's Prime-Power Counting Function (i.e. j'[x]) Evaluated at f=3.
$J_b'[x]$ evaluated at $f=3$.


First-Order Derivative of Riemann's Prime-Power Counting Function (i.e. j'[x]) around x=19 Evaluated at f={2, 4, 8}.
$J_b'[x]$ around $x=19$ evaluated at $f={2, 4, 8}$.