10.2.2 Illustration of Derived Formula for $\zeta'[s]$ along Critical Line $s=\frac{1}{2}+i t$

Navigation Links: – First Page – Parent Page


The following plots illustrate the real and imaginary parts of the method 1 formulas derived for $\zeta'[s]$, $\zeta_a'[s]$, and $\zeta_b'[s]$ evaluated along the critical line $s=\frac{1}{2}+i\ t$ (orange) . The Zeta'[s] function defined in the Wolfram Language is shown as a reference (blue). The red dots in the plot correspond to the evaluation of the method 1 formulas for $\zeta'[s]$, $\zeta_a'[s]$, and $\zeta_b'[s]$ for $t$ equal to the imaginary portion of the zeta zeros. These plots illustrate the method 1 formula derived for $\zeta'[s]$ does not seem to converge to $\zeta'[s]$ for $s=\frac{1}{2}+i\ t$ due to the oscillation in the $\zeta_a'[s]$ component, but the $\zeta_b'[s]$ component seems to converge to $\zeta'[s]$ fairly quickly as the magnitude of $t$ increases as $\zeta_b'[s]$ seems to be well converged to $\zeta'[s]$ by the first zeta zero. Note that whereas $\zeta_a'[s]$ exhibited a low frequency oscillation as well as a high frequency oscillation in the plots for $s=1+i\ t$ on page 10.2.1, $\zeta_a'[s]$ does not seem to exhibit the low frequency oscillation in the plots below for $s=\frac{1}{2}+i\ t$.


Real Part

Method 1 formula for $Re[\zeta'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Re[\zeta'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Re[\zeta_a'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Re[\zeta_a'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Re[\zeta_a'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Re[\zeta_b'[\frac{1}{2}+i t]]$ (orange).


Imaginary Part

Method 1 formula for $Im[\zeta'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Im[\zeta'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Im[\zeta_a'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Im[\zeta_a'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Im[\zeta_b'[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Im[\zeta_b'[\frac{1}{2}+i t]]$ (orange).