5.7 $\vartheta_b'[x]$: First-Order Derivative of First Chebyshev Function

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The Fourier series for the first-order derivative $\vartheta_b'[x]$ of the first Chebyshev function evaluates to $2\ f\ \log[p]$ at primes of the form $x=p$ 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 First Chebyshev Function (i.e. c'[x]) Evaluated at f=1.
$\vartheta_b'[x]$ evaluated at $f=1$.

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


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