11.3.1 Illustration of Derived Formula for $\zeta[s]$ along Convergence Boundary Line $s=1+i t$

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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 convergence boundary line $s=1+i\ t$ (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 $s=1+i\ t$ 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 $\zeta_a[s]$ seems to exhibit a low frequency oscillation as well as a high frequency oscillation when evaluated at $s=1+i\ t$.


Real Part

Method 1 formula for $Re[\zeta[1+i t]]$ (orange)..
Method 1 formula for $Re[\zeta[1+i t]]$ (orange).
Method 1 formula for $Re[\zeta_a[1+i t]]$ (orange).
Method 1 formula for $Re[\zeta_a[1+i t]]$ (orange).
Method 1 formula for $Re[\zeta_b[1+i t]]$ (orange).
Method 1 formula for $Re[\zeta_b[1+i t]]$ (orange).


Imaginary Part

Method 1 formula for $Im[\zeta[1+i t]]$ (orange).
Method 1 formula for $Im[\zeta[1+i t]]$ (orange).
Method 1 formula for $Im[\zeta_a[1+i t]]$ (orange).
Method 1 formula for $Im[\zeta_a[1+i t]]$ (orange).
Method 1 formula for $Im[\zeta_b[1+i t]]$ (orange).
Method 1 formula for $Im[\zeta_b[1+i t]]$ (orange).