11.3.2 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 11.3.1, $\zeta_a[s]$ 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 $Re[\zeta_b[\frac{1}{2}+i t]]$ (orange).
Method 1 formula for $Re[\zeta_b[\frac{1}{2}+i t]]$ (orange).


Imaginary Part

Method 1 formula for $Im[\zeta_a[\frac{1}{2}+i t]]$ (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).


The plot below illustrates a the relationship between the first harmonic of the method 1 formula for $\zeta(s)$ and zero Crossings of $\zeta\left(\frac{1}{2}+i\ t\right)$. Note there is a one-to-one correspondence between the peaks of the real part of the first harmonic for $\zeta(s)$ (orange) and the absolute value of $\zeta\left(\frac{1}{2}+i\ t\right)$ (blue). The formula for $\zeta(s)$ only converges for $Re(s)>1$, but a subset of the formula (which includes the first harmonic) seems to converge fairly rapidly along the critical line and seems to be well converged by the first zeta zero (which was illustrated in the previous plots). This seems to suggest the relationship illustrated below will hold up as $t\to\infty$ as it only seems logical the fundamental harmonic of $\zeta(s)$ will track the basic oscillation of $\zeta(s)$.

Relationship Between First Harmonic of $\zeta(s)$ and Zero Crossings of $\zeta\left(\frac{1}{2}+i\ t\right)$