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

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).