Numerical Behavior of the Riemann Zeta Function Using Real-to-Complex Conversion

The Riemann Zeta function ζ(s) lies at the heart of analytic number theory, encoding the distribution of primes through its non-trivial zeros. This paper introduces a direct computational framework for evaluating Z(t) = eiθ(t)ζ( 1/2 + it) at large imaginary parts t, employing a real-to-complex number conversion and a novel valley scanner algorithm. The method efficiently identifies zeros by tracking minima of |Z(t)|, achieving stability and precision up to t ≈ 10^20 with moderate computational cost, using AWS EC2 computation. Results are compared against known Andrew Odlyzko zero datasets, validating the method’s accuracy while simplifying the high-t evaluation pipeline.