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

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.

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

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