4.11 Illustration of Six Formulas for $\psi[x]$

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I’ve defined three methods for derivation of Fourier series for prime number counting functions, and this page illustrates the formulas derived for $\psi[x]$ via each of these three methods.


This page also illustrates three additional formulas derived for $\psi[x]$ from the following relationship. The value $t=1$ is used as the lower integration bound because the formula derived for the indefinite integral exhibits infinities at $t=0$.

$\psi[x]=\int_{1}^{x}\log[t]\ J'[t]\ dt$


The following six plots illustrate the formulas for $\psi[x]$ (blue) using the evaluation frequency $f=4$. The $\psi[x]$ function (orange) and linear function x (green) are shown as references. Note the plots here all look fairly similar.


The following two plots illustrate the formulas derived for $\psi[x]$ via method 1.


Additional plot coming soon.


 

$\psi(x)=\int_{1}^{x}J(x)dx$ evaluated at f=4.
Method 1: $\psi[x]=\int_{1}^{x}\log[t]\ J'[t]\ dt$ evaluated at $f=4$.

The following two plots illustrate the formulas derived for $\psi[x]$ via method 2.


Additional plots coming soon.


The following two plots illustrate the formulas derived for $\psi[x]$ via method 3.


Additional plots coming soon.