Evolution of Zeta Zeros from First-Order Derivative $\psi'[x]$ of Second Chebyshev Function

The inspiration for the approach illustrated in this post is the book “Prime Numbers and the Riemann Hypothesis” by Barry Mazur and William Stein.


The first two plots below illustrate $\psi'[e^{Abs[t]}]-1$  (where $\psi'[x]$ is the Fourier series for the first-order derivative of the second Chebyshev function) at a frequency evaluation limit of $f=1$. The orange reference function in these two plots is $2\ f\log\ e^{Abs[t]}$.

l'[E^Abs[t]] with Evaluation Limits {t, -Log[20], Log[20]} and f=1.
$\psi'[e^{Abs[t]}]-1$ with Evaluation Limits $-\log[20]\le t\le\log[20]$ and $f=1$.
l'[E^t] with Evaluation Limits {t, 0, Log[20]} and f=1.
$\psi'[e^t]-1$ with Evaluation Limits $0\le t\le\log[20]$ and $f=1$.


The next two plots below illustrate $e^{-Abs[t/2]}\left(\psi'[e^{Abs[t]}]-1\right)$ (where $(\psi'[x]$ is the Fourier series for the first-order derivative of the second Chebyshev function) at a frequency evaluation limit of $f=1$. The orange reference function in these two plots is $2\ f\ e^{-Abs[t/2]}\log\ e^{Abs[t]}$.

E^(-t/2) l'[E^t] with Evaluation Limits {t, -Log[20], Log[20]} and f=1.
$e^{-Abs[t/2]}\left(\psi'[e^{Abs[t]}]-1\right)$ with Evaluation Limits $-\log[20]\le t\le\log[20]$ and $f=1$.
E^(-t/2) l'[E^t] with Evaluation Limits {t, 0, Log[20]} and f=1.
$e^{-t/2}\left(\psi'[e^t]-1\right)$ with Evaluation Limits $0\le t\le\log[20]$ and $f=1$.


The last plot below illustrates the evolution of the zeta zeros from the real portion of the Fourier transform of $e^{-t/2}\left(\psi'[e^t]-1\right)$ (where $\psi'[x]$ is the Fourier series for the first-order derivative of the second Chebyshev function) at a frequency evaluation limit of $f=1$. The Fourier transform is evaluated along the critical line (i.e. $Re[s]=\frac{1}{2}$).

$Re[FourierTransform[e^{-t/2}\left(\psi'[e^t]-1\right),\ t,\ s]]$

Real Part of Fourier Transform of E^(-t/2) l'[E^t] with Frequency Evaluation Limit f=1.
$Re[FourierTransform[e^{-t/2}\left(\psi'[e^t]-1\right),\ t,\ s]]$ Evaluated along Critical Line (i.e. $Re[s]=\frac{1}{2}$) with Frequency Evaluation Limit $f=1$.

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