5.11 Illustration of Six Formulas for $\psi_b'[x]$

Navigation Links: – First Page – Parent Page


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


This page also illustrates three additional formulas derived for $\psi_b'[x]$ from the following relationship.

$\psi_b'[x]=\log[x]\ J_b'[x]$


The following six plots illustrate the formulas for $\psi_b'[x]$ (blue) using the evaluation frequency $f=1$. The function $2\ f\ \log[x]$ (orange) is shown as a reference. The horizontal grid-lines in the plots below are at $2\ f\ \log[2]$ and $2\ f\ \log[3]$. All of the formulas illustrated here exhibit the same exact convergence to $2\ f\ \log[p]$ at prime-powers of the form $x=p^n$ and to $0$ at the remaining positive integer values of $x$.


The following two plots illustrate the formulas derived for $\psi_b'[x]$ via method 1. Note that for the first plot below the magnitude of the oscillation beneath the $x$-axis seems relatively constant as $x$ increases, whereas for the second plot below the magnitude of the oscillation beneath the $x$-axis seems to grow as $x$ increases.


First-Order Derivative of the Second Chebyshev Function (i.e. l'[x]) Evaluated at f=1.
Method 1: $\psi_b'[x]$ evaluated at $f$=$1$.

$\psi'(x)=\log(x)\ J'(x)$ evaluated at $f$=$1$.
Method 1: $\psi_b'[x]=\log[x]\ J'[x]$ evaluated at $f$=$1$.

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


Method 2: $\psi'(x)$ evaluated at $f$=$1$.
Method 2: $\psi_b'[x]$ evaluated at $f$=$1$.

Additional plot coming soon.


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


Method 3: $\psi'(x)$ evaluated at $f$=$1$.
Method 3: $\psi_b'[x]$ evaluated at $f$=$1$.

Additional plot coming soon.