A useful approximate functional relationship between the fourier analysis sidebands of the Riemann Siegel Z function and its application for robust spectral filtering of partial Euler Products when Re(s) -> 1 or 0.

First quiescent region $N_1=\sqrt{\frac{t}{2\pi}}$ spectrally filtered truncated partial Euler Product based approximations of the Riemann Siegel Z function away from the critical line can be improved (in accuracy, numerical stability and runtime) by co-opting a heuristic zeroth order functional relationship between the true Riemann Siegel Z function fourier analysis sidebands. For example, with fourier analysis of the true Riemann Siegel Z function spectrum for fixed Re(s) along the interval t=t0+(-819.1,819.2) via Pari-GP FFT functions using a 32768 point fourier grid with spacing $\Delta t=0.05$ when Re(s) > 0.5 the weaker fourier sideband $F(-w) \approx \overline{F(w)}\cdot \left|\mychi(\sigma+I\cdot 2\pi\cdot e^{\left(2\cdot w\right)})\right|$ and for Re(s) < 0.5 the weaker sideband $F(w) \approx \overline{F(-w)}\cdot \left|\mychi(1-(\sigma+I\cdot 2\pi\cdot e^{\left(2\cdot w\right)}))\right|$ where $|\mychi|$ is the (absolute magnitude of the) multiplicative factor in the Riemann Zeta functional equation but its use occurs in the frame of reference of the Riemann-Siegel Z function fourier transform using $w_i=\frac{i\cdot 4\cdot\pi}{32768\cdot 0.05}\equiv\frac{i\cdot 4\cdot\pi}{n\cdot \Delta t}, i \in [-n/2+1,n/2]$. This functional relationship is particularly useful for grid search calculations of Riemann Zeta extreme peaks on the line s=1+It via partial Euler Product based calculations.