5.9 $M_b'[x]$: First-Order Derivative of Third “Chebyshev-like” Function

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The Fourier series for the first-order derivative $M_b'[x]$ of the third “Chebyshev-like” function evaluates to $2\ f\ n\ \log[p]$ at prime-powers of the form $x=p^n$ where $f$ is the evaluation frequency limit. The reference function in the following plots is $2\ f\ \log[Abs[x]]$.


First-Order Derivative of the Third "Chebyshev-like" Function (i.e. m'[x]) Evaluated at f=1.
$M_b'[x]$ evaluated at $f=1$.

First-Order Derivative of the Third "Chebyshev-like" Function (i.e. m'[x]) Evaluated at f=1.
$M_b'[x]$ evaluated at $f=1$.
First-Order Derivative of the Third "Chebyshev-like" Function (i.e. m'[x]) Evaluated at f=2.
$M_b'[x]$ evaluated at $f=2$.
First-Order Derivative of the Third "Chebyshev-like" Function (i.e. m'[x]) Evaluated at f=3.
$M_b'[x]$ evaluated at $f=3$.


First-Order Derivative of the Third "Chebyshev-like" Function (i.e. m'[x]) around x=19 Evaluated at f={2, 4, 8}.
$M_b'[x]$ around x=19 evaluated at $f={2, 4, 8}$.