6.5 $J”[x]$: Second-Order Derivative of Riemann’s Prime-Power Counting Function

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The first plot below illustrates $J”[0]=0$ and $J”[x]$ is an odd function of $x$. The next three plots illustrate $J”[x]$ in the right=half plane at $f={1, 2, 3}$. The last plot below focuses around $x=19$ with evaluation frequencies $f={2, 4, 8}$.


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

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


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