5.8 $\psi_b'[x]$: First-Order Derivative of Second Chebyshev Function

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The Fourier series for the first-order derivative $\psi_b'[x]$ of the second Chebyshev function evaluates to $2\ f\ \log[p]$ at prime powers of the form $x=p^n$ where $f$ is the evaluation frequency limit. The reference function in the following plots is $2\ f\ \log[Abs[x]]$. The horizontal grid-lines in the plots below are at $2\ f\ \log[2]$ and $2\ f\ \log[3]$ since $20$ is the maximum value of $x$ for these plots and $Floor[\log[20]]=3$ (except the last plot does not include any grid-lines).


A point worth mentioning here is the negative value of the evaluation at $x=0$. Evaluation parameters can be selected to make this evaluation either positive or negative, so evaluation parameters were selected to to minimize the absolute value of the deviation of the evaluation from zero at $x=0$ in order to bring the evaluation at $x=0$ closer to the order of magnitude of the rest of the evaluations.


First-Order Derivative of the Second Chebyshev Function (i.e. l'[x]) Evaluated at f=1.
$\psi_b'[x]$ evaluated at $f=1$.

First-Order Derivative of the Second Chebyshev Function (i.e. l'[x]) Evaluated at f=1.
$\psi_b'[x]$ evaluated at $f=1$.
First-Order Derivative of the Second Chebyshev Function (i.e. l'[x]) Evaluated at f=2.
$\psi_b'[x]$ evaluated at $f=2$.
First-Order Derivative of the Second Chebyshev Function (i.e. l'[x]) Evaluated at f=3.
$\psi_b'[x]$ evaluated at $f=3$.


First-Order Derivative of the Second Chebyshev Function (i.e. l'[x]) around x=19 Evaluated at f={2, 4, 8}.
$\psi_b'[x]$ around x=19 evaluated at $f={2, 4, 8}$.