A useful approximation of the Riemann Zeta function, away from the real axis, using spectral filtering of the partial Euler product.

By performing precise spectral filtering of the fourier transform of the Riemann Zeta function partial Euler Product over a finite interval on the critical line, the subsequent inverse fourier transform provides a useful approximation of the Riemann Zeta function. Two different algorithms are used for spectral filtering depending on whether the Euler Product calculation uses truncation at the first or second quiescent region of the corresponding Riemann Zeta Dirichlet series. The spectral filtering algorithm for $N_1=\sqrt{\frac{t}{2\pi}}$ truncation of the primes in the Euler product calculation involves (i) independent fourier transform calculations of the Euler Product analogues of the two Riemann-Siegel zeroth order components and (ii) a linear combination of the two spectral filtered transforms whereby the fourier transform of the $\prod \frac{1}{(1-\frac{1}{p^s})}$ term supplies the low frequency components and the fourier transform of the $\chi(s) \cdot \prod \frac{1}{(1-\frac{1}{p^{(1-s)}})}$ term supplies the high frequency components for a combined spectral function upon which the inverse fourier transform is extracted.